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+{"text":"\\section{Introduction}\n\nIt is rarely possible to model realistic physical systems with exact\nsolutions to the equations of some general underlying theory.\nDespite this, many interesting problems deviate only slightly from a\nmodel problem that can be understood exactly. Such solutions are\nusually tractable only because of symmetry assumptions. Once they're\nunderstood, perturbation theory may be used to understand many\nsystems that ``almost'' satisfy the appropriate symmetry principle.\nWhile this statement has a clear intuitive meaning, quantifying it\ncan be difficult. It is also not necessarily obvious how -- or if it\nis meaningful -- to uniquely propagate a symmetry from solutions\nwhere it is exact into the perturbations that break it. These issues\nare particularly important in the context of conservation laws. As\nan example, one might want to know how to construct approximately\nconserved quantities that are unique generalizations of some exact\ncounterpart in a similar system. There would hopefully be a sense in\nwhich such quantities varied slowly for some class of small\nperturbations.\n\nSome steps towards understanding problems like these are explored\nhere in the context of affine collineations (of which Killing\nvectors are special cases) associated with curved spacetimes. While\nthese kinds of symmetries rarely exist, there are various senses in\nwhich approximate replacements can usually be introduced. One method\nfor finding vector fields that are ``almost Killing'' is to write\ndown an action whose value provides some sense for how nearly a\nparticular flow preserves the metric \\cite{Matzner}. There are\nimportant caveats to this interpretation, although the final result\nis that any vector field extremizing such an action satisfies a\nfairly simple generalization of Killing's equation. Various reasons\nhave been given for suggesting other extensions as well\n\\cite{AlmostStat,YanoBochner,Komar1,Komar2,YorkSym}. While any\ngenuine Killing vectors that might exist are solutions to all of\nthese equations, it is not usually clear how the remaining fields\nshould be interpreted. This problem arises even in flat spacetime.\n\nWhat form an approximate symmetry should take is highly dependent on\nits intended use. One application is in the estimation of a black\nhole's angular momentum. This requires finding rotational Killing\nfields on certain 2-spheres foliating a horizon. Various methods\nhave therefore been developed for defining such objects using only\nthe intrinsic geometry of these surfaces\n\\cite{S2Conformal,S2KT,S2Killing}. The concept of approximate\nKilling fields has also been adapted for use on initial data sets\nused in $3+1$ splits of Einstein's equation \\cite{DainInitData}.\n\nThe approach taken here is to define a set of vector fields in a\nfour dimensional volume that can all be viewed as analogs of known\nsymmetries in Minkowski spacetime. The physical interpretation is\nthat these fields may be viewed as generators of approximate\nsymmetries by a specified observer. Any sufficiently small region\nnear a particular point can be made to look nearly flat. Some\nstructures from Minkowski spacetime may be therefore be introduced\nvery near this point. Approximate symmetries that take advantage of\nthis fact are proposed in Sect. \\ref{Sect:Symmetries}. It is then\nshown in Sect. \\ref{Sect:GAC} that analogous objects can also be\nintroduced near an observer's worldline. These sorts of vector\nfields can actually be extended in a non-perturbative way to finite\nregions around the point or worldline from which they were\nconstructed. A well-defined subset provides a precise analog of the\nPoincar\\'{e} group. Translations, rotations, and boosts very near an\nobserver extend in a useful way to cover large portions of the\nspacetime. Any exact symmetries that may exist are included as\nspecial cases. Some connections to conservation laws are discussed\nin Sect. \\ref{Sect:ConsLaws}, and a simple example involving\ngravitational plane wave is finally presented in Sect.\n\\ref{Sect:Example}.\n\n\n\\subsection*{Exact symmetries}\n\nGiven some spacetime $(\\mathcal{M},g_{ab})$, there are several types\nof exact symmetries that may be discussed. The most common of these\ntake the form of vector fields whose associated diffeomorphisms\npreserve some geometric structure. The most ubiquitous examples are\nthe Killing fields. Their flows preserve the metric. Vector fields\n$Y^a_{\\mathrm{K}}$ with this property satisfy\n\\begin{equation}\n  \\Lie_{Y_{\\mathrm{K}}} g_{ab} = 0 . \\label{KillingDef}\n\\end{equation}\nAny solutions that may exist can be used to find conserved\nquantities associated with geodesics or matter distributions\n\\cite{Wald}, identify mass centers \\cite{SchattStreub1}, simplify\nEinstein's equation \\cite{SimplifyEinstWSyms, ExactSolns}, classify\nits solutions \\cite{ExactSolns, Hall}, and so on.\n\nThis utility has (among other reasons) motivated various\ngeneralizations of \\eqref{KillingDef}. Perhaps the simplest of these\narises from considering flows that preserve the metric only up to\nsome constant factor:\n\\begin{equation}\n  \\Lie_{Y_{\\mathrm{H}} } g_{ab} = 2 c g_{ab} .\n  \\label{HomotheticDef}\n\\end{equation}\nAny $Y^a_{\\mathrm{H}}$ satisfying this equation with constant $c$ is\nknown as a homothetic vector field or homothety. Allowing the\ndilation factor $c$ to vary would define a conformal Killing vector.\nThese objects preserve the metric up to an arbitrary multiplicative\nfactor. The standard Killing vectors are special cases of either\nclass. Like them, conformal and homothetic vector fields usually do\nnot exist. Their presence can be very useful, however. The existence\nof a proper (non-Killing) homothetic vector is often used to define\na notion of geometric self-similarity, for example. Such objects\ntherefore appear in certain models of gravitational collapse and\ncosmology. They are also related to the appearance of critical\nphenomena in general relativity \\cite{SelfSimilar}.\n\nA simple generalization of the homotheties can be found by\nconsidering vector fields that satisfy\n\\begin{equation}\n  \\nabla_a \\Lie_{Y_{\\mathrm{A}}} g_{bc} = 0 .\n  \\label{AffineDef}\n\\end{equation}\nSolutions to this equation are known as affine collineations. They\nare the generators of infinitesimal affine transformations. Killing\nand homothetic vector fields are special cases. All of the affine\ncollineations may be interpreted geometrically as preserving the\nLevi-Civita connection. This means that $\\Lie_{Y_{\\mathrm{A}}}$ and\n$\\nabla_a$ commute when acting on arbitrary tensor fields. Geodesics\nand their affine parameters are also preserved under the action of\nany $Y_{\\mathrm{A}}^a$. Although this might seem to be a significant\ngeneralization of the Killing vector concept, solutions rarely\nexist. The only non-flat vacuum spacetimes that admit non-homothetic\naffine collineations are the \\textit{pp}-waves \\cite{RareAffine}.\nSimilarly, it has been shown that proper homotheties cannot exist in\nany asymptotically flat vacuum spacetime with positive Bondi mass\n\\cite{RareHomothety}. Despite these results, interesting affine\ncollineations can occasionally be identified in geometries that are\nnot Ricci-flat. Doing so provides a number of simplifications for\nvarious problems. Some of these derive from the fact that\n\\begin{equation}\n  K_{ab} = \\Lie_{Y_{\\mathrm{A}}} g_{ab} \\label{AffineKT}\n\\end{equation}\nis a second-rank Killing tensor; i.e.\n\\begin{equation}\n  \\nabla_{(a} K_{bc)} = 0. \\label{KTDef}\n\\end{equation}\nIt should be noted that not all symmetric tensors satisfying this\nequation can be derived from affine collineations. One\ncounterexample is the Killing tensor associated with Carter\nconstants in Kerr.\n\nTransformations generated by affine collineations can be viewed as\nmapping geodesics into geodesics. They preserve the affine\nparameters of each curve. Dropping this latter requirement recovers\nthe so-called projective collineations. A precise definition may be\nfound in \\cite{Hall}, although it will not be needed here. One of\ntheir interesting consequences is that they leave invariant the\nprojective curvature tensor:\n\\begin{equation}\n  \\Lie_{Y_{\\mathrm{P}}} \\big( R^{a}{}_{bcd} - \\frac{2}{3} \\delta^{a}_{[c} R_{d]b} \\big) = 0.\n\\end{equation}\nVector fields satisfying this equation are not always projective,\nhowever.\n\nThe list of definitions here could keep growing as new fields are\nadded that preserve more and more geometric structures.\nInterestingly, the quantities introduced so far all share a very\nuseful characteristic that does not easily generalize: the space of\nvector fields in each of the mentioned classes has finite dimension.\nFurthermore, any single element is uniquely determined by its value\nand the values of its first one or two derivatives at a single\npoint. These properties are well-known for Killing fields. Four\ndimensional spacetimes (which is all that will be considered here)\nadmit a maximum of 10 linearly independent Killing vectors. At most\none homothety can exist that is not itself Killing. The maximum\nnumber of (not necessarily proper) conformal Killing vectors is 15,\nand the affine collineations total no more than 20. Finally, the\nspace of projective collineations has a maximum of 24 dimensions\n\\cite{Hall}. Properties like these do not hold for vector fields\nwhose flows leave invariant the Riemann, Ricci, or Einstein\ncurvature tensors of a given spacetime. Despite this, the class of\napproximate symmetries introduced below is constructed so as to have\nfinite dimension. Any given member is fixed by its value together\nwith the values of its first derivatives at a point. Unlike the\nexact symmetries, these objects always exist at least in some finite\nregion. After fixing a reference frame, the space of approximate\nsymmetries has exactly 20 dimensions. Ten of these will be\nidentifiable as generalized Killing vectors, while the remaining ten\nwill be related to more general affine collineations.\n\nIt has already been remarked that the presence of Killing fields\nimplies the existence of various conserved quantities. The same can\nalso be said for more general collineations. Extensive discussions\nof exact symmetries and associated integrals of the geodesic\nequation may be found in \\cite{GeodesicConsts}. Many of these\nsymmetries are non-Noetherian in the sense that they preserve the\nequations of motion, but not the action. Despite this, their\npresence allows constants of motion to also be assigned to arbitrary\nstress-energy distributions satisfying Einstein's equation\n\\cite{KatzinLevine, Collinson}. As will be discussed in Sect.\n\\ref{Sect:ConsLaws}, generalizations of these quantities can be\nassociated with any approximate affine collineations that are\nidentified.\n\n\\section{Symmetries near a point}\\label{Sect:Symmetries}\n\nGeneric symmetries in general relativity are usually discussed in\nthe context of asymptotically flat spacetimes. There then exist\napproximate notions of isometry that improve as one approaches\ninfinity \\cite{Wald}. Generalizations of these ideas also exist for\ngeometries with somewhat more complicated (but still highly\nsymmetric) asymptotic behavior like that of anti-de Sitter\n\\cite{AsymptAdS}. The assumption of a simple limiting form for the\ngeometry makes it convenient to invariantly describe certain\nproperties of a spacetime in terms of ``measurements at infinity.''\nQuantities that may be identified as a spacetime's total energy or\nangular momentum appear naturally, for example.\n\nWhile useful in many contexts, these ideas do not always translate\ninto observations made by physical observers. Measurements like\nthose expected from gravitational wave detectors do come very close\nto fitting into this formalism. Others can require a more local\ndescription. In particular, it is sometimes important to understand\nwhat given observers would experience inside strongly curved regions\nof spacetime. Abstracting the concept of an observer to a timelike\nworldline $\\Gamma$, vector fields may be introduced in (say) some\nconvex neighborhood $W$ of $\\Gamma$ that act like approximations to\nKilling fields or more general collineations. This is always\npossible, and the symmetries these vectors generalize become exact\non $\\Gamma$ itself. Limiting collineations can evidently be useful\non scales that are either very large or very small. It is much less\nclear how to easily describe systems at intermediate distances.\n\n\\subsection{Motivation}\n\nThe idea of a local symmetry just outlined is best introduced by\nfirst considering vector fields $\\psi^a(x,\\gamma)$ that generalize\nthe affine collineations in some reasonable way near a fixed\nreference point $\\gamma$. Let these objects be defined inside a\nnormal neighborhood $N$ of this point. It is intuitively obvious\nthat vector fields may always be chosen such that $\\nabla_a \\LieS\ng_{bc}$ vanishes at $\\gamma$. While this condition is reasonable to\nrequire, it is not very interesting by itself. Much more can be said\nif each vector field in this class is uniquely fixed in $N$ by\nknowledge of\n\\begin{equation}\n  \\psi^\\sfa(\\gamma,\\gamma) , \\qquad \\nabla_\n  \\sfa \\psi^{\\mathsf{b}}(\\gamma,\\gamma) . \\label{InitDataFirst}\n\\end{equation}\nIt will be assumed that each $\\psi^a$ depends linearly on this\ninitial data with no degeneracy. This implies that there are always\n$4+16=20$ linearly independent vector fields defined about any given\npoint in a four dimensional spacetime. Note that indices in\n\\eqref{InitDataFirst} have been written in a sans-serif font to\nemphasize that they are associated with the preferred point\n$\\gamma$.\n\nApproximate affine collineations with the appropriate properties may\nbe constructed by projecting symmetries of the tangent space\n$T_\\gamma N$ into $N$ using the exponential map. Consider the linear\ntransformations\n\\begin{equation}\n  X^\\sfa \\rightarrow X^\\sfa + \\epsilon B_{{\\mathsf{b}}}{}^{a} X^{\\mathsf{b}}\n  \\label{XFormVect}\n\\end{equation}\nof vectors $X^\\sfa$ in this space parameterized by an arbitrary\ntensor $B_{{\\mathsf{b}}}{}^{\\sfa}$. Being a vector space, $T_\\gamma N$ has a\npreferred origin. Adding another term to \\eqref{XFormVect} to shift\nthat origin would be awkward. The translational symmetries that such\na procedure might produce are certainly important, although\ngenerating them requires a more subtle treatment described below.\nFor now, consider only the vector fields\n\\begin{equation}\n  \\Psi^\\sfa = X^{\\mathsf{b}} B_{{\\mathsf{b}}}{}^{\\sfa}\n  \\label{AffineTangent}\n\\end{equation}\nassociated with homogeneous transformations of the given form. These\nclearly satisfy\n\\begin{equation}\n  \\frac{\\partial}{\\partial X^\\sfa} \\Lie_\\Psi g_{{\\mathsf{b}} {\\mathsf{c}}} = 2 \\frac{\\partial}{\\partial\n  X^\\sfa} \\left( g_{{\\mathsf{d}}({\\mathsf{c}}}(\\gamma) \\frac{\\partial}{\\partial X^{{\\mathsf{b}})}} \\Psi^{{\\mathsf{d}}} \\right)\n  =0,\n\\end{equation}\nso they are affine within $T_\\gamma N$ in the sense of\n\\eqref{AffineDef}. Such transformations can be made to induce shifts\n$x \\rightarrow x + \\epsilon \\psi$ in spacetime points associated\nwith vectors $X^\\sfa$ via\n\\begin{equation}\n  x = \\exp_\\gamma X .\n  \\label{ExpMap}\n\\end{equation}\n\nA simple relation between $\\psi^a$ and $\\Psi^\\sfa$ is found by\nintroducing Synge's world function $\\sigma(x,y) = \\sigma(y,x)$. This\ntwo-point scalar returns one-half of the squared geodesic distance\nbetween its arguments. The assumption that $N$ be a normal\nneighborhood of $\\gamma$ ensures that $\\sigma(x,\\gamma)$ is uniquely\ndefined for all points $x$ in this region. Many of its properties\nare reviewed in \\cite{Synge, PoissonRev}. Most importantly for the\nproblem at hand, the first derivative of the world function\neffectively inverts the exponential map. Any set $\\{ \\gamma, x, X^a\n\\}$ satisfying \\eqref{ExpMap} is related via\n\\begin{equation}\n  X_{\\sfa} = - \\sigma_{\\sfa}(x,\\gamma),\n  \\label{XDef}\n\\end{equation}\nwhere the common shorthand $\\sigma_\\sfa = \\nabla_\\sfa \\sigma =\n\\partial \\sigma\/ \\partial \\gamma^\\sfa$ has been used. The\nright-hand side of (\\ref{XDef}) generalizes the concept of a\nseparation vector between two points. It is useful in that a\nstraightforward expansion shows that linear transformations of the\nform (\\ref{XFormVect}) effectively shift spacetime points by an\namount parameterized with a vector $\\psi^{a}(x,\\gamma)$ satisfying\n\\begin{equation}\n\\Psi^{\\sfa} = - \\sigma^{\\sfa}{}_{a} \\psi^{a} . \\label{PsiTopsi}\n\\end{equation}\nIf the various components of $X^\\sfa$ as defined in (\\ref{XDef}) are\nused as coordinates, the bitensor $-\\sigma^{\\sfa}{}_{a} = - g^{\\sfa\n{\\mathsf{b}}}\n\\partial^2 \\sigma\/ \\partial x^a \\partial \\gamma^{{\\mathsf{b}}}$\nreduces to the identity. Components of $\\psi^a$ and $\\Psi^{\\sfa}$\nare therefore identical in normal coordinate systems of this type.\nIn general, it is useful to introduce\n\\begin{equation}\n  H^{a}{}_{\\sfa} = [ - \\sigma^{\\sfa}{}_{a} ]^{-1} \\label{HDef}\n\\end{equation}\nas the matrix inverse of the operator appearing in \\eqref{PsiTopsi}.\nThis always exists in the regions considered here. Using\n\\eqref{AffineTangent} now shows that\n\\begin{equation}\n \\psi^{a}(x,\\gamma) = - H^{a}{}_{\\sfa} \\sigma_{{\\mathsf{b}}} B^{{\\mathsf{b}} \\sfa} .\n \\label{PsiPart}\n\\end{equation}\nHolding $\\gamma$ fixed, this equation defines a 16-parameter family\nof vector fields generated by $B_{\\sfa {\\mathsf{b}}}=\\nabla_\\sfa \\psi_{\\mathsf{b}}\n(\\gamma,\\gamma)$. Every such $\\psi^a$ vanishes at $\\gamma$. It also\nsatisfies \\eqref{AffineDef} at this point. In flat spacetime, these\nvector fields coincide everywhere with exact affine collineations.\n\nNot all such symmetries are included in (\\ref{PsiPart}), however.\nThe four translational Killing fields are missing. These can be\nobtained by considering transformations that directly shift the base\npoint $\\gamma$. Perturbations of this form cannot leave $X^\\sfa$\nfixed, as the initial and final vectors must be elements of\ndifferent spaces. Introducing some $A^\\sfa$, we therefore demand\nthat $X^\\sfa$ be parallel-transported along the curve that $\\gamma$\nfollows under the one-parameter family of transformations\n\\begin{equation}\n  \\gamma \\rightarrow \\gamma + \\epsilon A .\n\\end{equation}\nUsing this together with \\eqref{XDef} and the homogeneous\ntransformation \\eqref{XFormVect} generates the full 20-parameter\nfamily of approximate affine collineations\n\\begin{equation}\n  \\psi^{a} = H^{a}{}_{\\sfa} ( \\sigma^{\\sfa}{}_{{\\mathsf{b}}} A^{\\mathsf{b}} - \\sigma_{\\mathsf{b}} B^{{\\mathsf{b}} \\sfa}\n  ) . \\label{JacobiFirstDer}\n\\end{equation}\nGiven any $A^{\\sfa}$ and $B^{\\sfa {\\mathsf{b}}}$, these objects all satisfy\n\\begin{equation}\n  \\nabla_\\sfa \\LieS g_{{\\mathsf{b}} {\\mathsf{c}}} (\\gamma) = 0 . \\label{JacobiAffine}\n\\end{equation}\nThe initial data\n\\begin{equation}\n  A^\\sfa = \\psi^\\sfa(\\gamma,\\gamma) , \\qquad B^{\\sfa {\\mathsf{b}}} =\n  \\nabla^\\sfa\n  \\psi^{\\mathsf{b}}(\\gamma,\\gamma) \\label{InitialData}\n\\end{equation}\ndetermine $\\psi^a(x,\\gamma)$ throughout $N$. In Minkowski spacetime,\none finds that\n\\begin{equation}\n    \\psi^\\alpha = A^\\alpha + (x-\\gamma)^\\beta B_{\\beta}{}^{\\alpha}\n\\end{equation}\nin the usual coordinates. These coincide exactly with all of the\naffine collineations in this geometry.\n\nIn general, vector fields satisfying (\\ref{AffineDef}) in a curved\nspacetime also have the form (\\ref{JacobiFirstDer}) for some\n$A^\\sfa$ and $B^{\\sfa {\\mathsf{b}}}$. This is most easily seen by noting\nthat vector fields with the given form have been obtained before as\ngeneral solutions to the equation of geodesic deviation (also known\nas the Jacobi equation) \\cite{Dix70a}\n\\begin{equation}\n  \\sigma^{b} \\sigma^{c} ( \\nabla_{b} \\nabla_{c} \\psi_{a} -\n  R_{abc}{}^{d} \\psi_{d} ) = 0. \\label{Jacobi}\n\\end{equation}\nFor any fixed $x$, this is an ordinary differential equation along\nthe geodesic connecting that point to $\\gamma$. Solving it\nrepeatedly for all geodesics in $N$ passing through this origin\nreproduces the vector fields (\\ref{JacobiFirstDer}). It is clear\nthat such solutions always exist as long as the geometry is\nreasonably smooth. These are the spacetime's Jacobi fields about\n$\\gamma$. The bitensors $H^{a}{}_{\\sfa} \\sigma^{\\sfa}{}_{{\\mathsf{b}}}$ and\n$H^{a}{}_{\\sfa} \\sigma_{\\mathsf{b}}$ are known as Jacobi propagators.\n\nSolutions to the geodesic deviation equation effectively map one\ngeodesic into another while preserving the affine parameters of both\ncurves. It was noted above that this is the defining characteristic\nof affine collineations. The difference is that such vector fields\nmust map \\textit{every} geodesic into another geodesic. This\nintuitive argument makes it clear that all affine collineations --\nor Killing fields as special cases -- must be solutions of\n(\\ref{Jacobi}). The proof follows from noting that second\nderivatives of any exact affine collineation $Y_{\\mathrm{A}}^a(x)$\nmust satisfy\n\\begin{equation}\n  \\nabla_{b} \\nabla_{c} Y^a_{\\mathrm{A}} = - R_{bdc}{}^{a} Y^d_{\\mathrm{A}} .\n    \\label{Del2Affine}\n\\end{equation}\nThis result is clear from \\eqref{Del2General}, and is actually\nequivalent to \\eqref{AffineDef}. Substituting it into (\\ref{Jacobi})\nshows that all affine collineations are indeed special cases of\nJacobi fields. As expected, $Y^a_{\\mathrm{A}}$ satisfies the\ngeodesic deviation equation along all geodesics; even those that do\nnot pass through $\\gamma$. This point illustrates precisely how\n\\eqref{Jacobi} generalizes the equation defining an affine\ncollineation. It is simply \\eqref{Del2Affine} contracted into\n$\\sigma^b \\sigma^c$. Alternatively, the Jacobi equation is\nequivalent to \\eqref{LieSig7}.\n\nTo summarize, the following is now evident:\n\\begin{theorem}\\label{Thm:JacobiBasic}\n  Let $N$ be a normal neighborhood of some point $\\gamma$. Define a Jacobi\n  field $\\psi^a(x,\\gamma)$ to be a solution of \\eqref{Jacobi} throughout this region. It is explicitly given by\n  \\eqref{JacobiFirstDer} for some initial data with the form\n  \\eqref{InitialData}. The set of all Jacobi fields about a fixed $\\gamma$ forms a 20-dimensional group in\n  four spacetime dimensions. Each element satisfies $\\mathcal{L}_\\psi \\nabla =0$ at\n  $\\gamma$. Furthermore, all affine collineations are members of\n  this group.\n\\end{theorem}\nThese properties motivate our identification of the Jacobi fields as\ngeneralizations of affine collineations near $\\gamma$.\n\nFurther results that strengthen this decision are derived in the\nappendix. Even though few Jacobi fields are genuine affine\ncollineations, all can be interpreted as exact symmetries of certain\nquantities connected with the spacetime's geometric structure:\n\\begin{theorem}\\label{Thm:JacobiExactSyms}\n  Given a Jacobi field defined as in theorem \\ref{Thm:JacobiBasic},\n  it is always true that\n  \\begin{equation}\n    \\LieS \\sigma^\\sfa = \\LieS \\sigma^a = \\LieS \\sigma^{\\sfa}{}_{a} =\n    \\LieS H^{a}{}_\\sfa =0 ,\n  \\end{equation}\n  where one of the arguments in each of these equations is taken to be the\n  origin $\\gamma$.\n\\end{theorem}\nLie derivatives on two-point tensor fields are defined to act\nindependently on each argument. See \\eqref{LieSig1}, for example.\nQuantities appearing in this theorem are all important in Riemann\nnormal coordinate systems parameterizing arbitrary points $x$ by the\ncomponents of $X^\\sfa = -\\sigma^\\sfa (x,\\gamma)$. In terms of a more\ndirect interpretation of the Jacobi fields as approximately\nsatisfying \\eqref{AffineDef}, Lie derivatives of the metric with\nrespect to an arbitrary $\\psi^a$ are strongly constrained by the\nidentities \\eqref{LieSig6}-\\eqref{LieSig9}.\n\nStatements of this sort do not exhaust the connections between the\nJacobi equation and a spacetime's symmetries. There is a sense in\nwhich higher-rank Killing tensors that may exist are also solutions\nto the geodesic deviation equation \\cite{CavigliaBasic}.\nFurthermore, projective collineations can be shown solve an\ninhomogeneous form of (\\ref{Jacobi}) proportional to $\\sigma_a$\n\\cite{Caviglia}. These observations will not be discussed any\nfurther here, although it is possible that they could be used to\ngeneralize the present framework.\n\n\\subsection{Special cases} \\label{Sect:SpecialJacobi}\n\nIt is often useful to single out a subset of the Jacobi fields\ndistinguished by antisymmetric $B_{\\sfa {\\mathsf{b}}} = \\nabla_\\sfa\n\\psi_{\\mathsf{b}}$. These may be said to generalize only the Killing fields\nof a given spacetime. Distinguishing them with a subscript ``K,''\nthey clearly satisfy $\\Lie_{\\psi_{\\mathrm{K}}} g_{\\sfa {\\mathsf{b}}}(\\gamma)\n= 0$ as well as (\\ref{JacobiAffine}). Such objects form a\n10-dimensional group that may be thought of as a generalization of\nthe Poincar\\'{e} group. They have been suggested before as useful\ngenerators for the linear and angular momenta of extended matter\ndistributions \\cite{Dix70a, Dix74, Dix79}. Fixing a hypersurface\n$\\Sigma$ that passes through $\\gamma$ and the worldtube of some\nwell-behaved spatially-compact stress-energy distribution $T^{ab}$,\nlet\n\\begin{equation}\n  p_\\sfa(\\gamma,\\Sigma) A^\\sfa + \\frac{1}{2} S_{\\sfa{\\mathsf{b}}}(\\gamma,\\Sigma) B^{[\\sfa{\\mathsf{b}}]} = \\int_\\Sigma T^{a}{}_{b}\n  \\psi^b_{\\mathrm{K}} \\rmd S_a .\n  \\label{DixMomenta}\n\\end{equation}\nVarying the 10 free parameters here determines the four linear\nmomenta $p^\\sfa$ and six angular momenta $S^{\\sfa{\\mathsf{b}}} =\nS^{[\\sfa{\\mathsf{b}}]}$. Explicit formulae are easily found using\n(\\ref{JacobiFirstDer}). They coincide with standard definitions in\nflat spacetime (where all $\\psi^a_{\\mathrm{K}}$ are Killing).\n\nTheorem \\ref{Thm:JacobiExactSyms} is easily expanded for these\nvector fields:\n\\begin{corollary}\\label{Thm:JacobiExactSymsKilling}\n    Given any Jacobi field $\\psi^a_{\\mathrm{K}}$ satisfying\n     $\\mathcal{L}_{\\psi_{\\mathrm{K}}} g_{\\sfa {\\mathsf{b}}} =0$,\n    \\begin{equation}\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma_\\sfa =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma_a =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} \\sigma_{\\sfa a} =\n      \\mathcal{L}_{\\psi_{\\mathrm{K}}} H^{a \\sfa} = 0.\n    \\end{equation}\n    Again, one argument in each of these equations is assumed to be $\\gamma$.\n\\end{corollary}\nThis follows from the well-known identity \\cite{Synge, PoissonRev}\n\\begin{equation}\n  \\sigma^\\sfa \\sigma_\\sfa = \\sigma^{a} \\sigma_{a} = 2 \\sigma ,\n  \\label{SigIdent2}\n\\end{equation}\nand its first derivative\n\\begin{equation}\n  \\sigma^{\\sfa}{}_{a} \\sigma^{a} = \\sigma^\\sfa .\n  \\label{SigIdent1}\n\\end{equation}\nBy definition, $\\sigma = g_{\\sfa {\\mathsf{b}}}(\\gamma) X^\\sfa X^{\\mathsf{b}}\/2$ is\none-half of the squared geodesic distance between $\\gamma$ and $x$.\nKilling-type Jacobi fields based at $\\gamma$ therefore drag both\narguments of $\\sigma(x,\\gamma)$ in such a way that distances are\npreserved.\n\nIt is also possible to identify Jacobi fields that act like\nhomotheties near $\\gamma$. These are distinguished by letting\n\\begin{equation}\n  B_{(\\sfa {\\mathsf{b}})} = \\frac{1}{2} \\Lie_{\\psi_{\\mathrm{H}}} g_{\\sfa {\\mathsf{b}}}(\\gamma) = c g_{\\sfa\n  {\\mathsf{b}}} (\\gamma). \\label{HomB}\n\\end{equation}\nAs in \\eqref{HomotheticDef}, $c$ is an arbitrary constant. For\nsimplicity, the purely Killing components of some prospective\n$\\psi^a_{\\mathrm{H}}$ can be removed by setting $A_\\sfa = B_{[\\sfa\n{\\mathsf{b}}]} = 0$ and $c \\neq 0$. Substitution into (\\ref{JacobiFirstDer})\nthen shows that\n\\begin{equation}\n  \\psi^{a}_{\\mathrm{H}} = - c H^{a}{}_{\\sfa} \\sigma^\\sfa  = c\n  \\sigma^{a}. \\label{JacobiHom}\n\\end{equation}\nThis second equality follows from contracting $\\delta^{a}_{b} =\n-H^{a}{}_{\\sfa} \\sigma^{\\sfa}{}_{b}$ with $\\sigma^{b}$ and using\n\\eqref{SigIdent1}.\n\nThe simplicity of (\\ref{JacobiHom}) is interesting, although perhaps\nnot surprising. It is consistent with the interpretation of\n$-\\sigma^\\sfa$ as a ``separation vector'' between $x$ and $\\gamma$.\nAs has been noted before, $\\sigma^{\\sfa}{}_{b} \\rightarrow -\n\\delta^{\\alpha}_{\\beta}$ in a normal coordinate system. The\ncomponents of $\\sigma^{a}$ would therefore be equal to $X^\\alpha$.\nThe normal coordinate functions themselves act as components of an\napproximately homothetic vector field. This is unique up to a\nconstant factor and the addition of Killing-type Jacobi fields. It\ngeneralizes the dilations of flat spacetime.\n\nGeneralized Killing tensors of various types can also be generated\nfrom Jacobi fields. In analogy to \\eqref{AffineKT}, let\n\\begin{equation}\n  \\mathcal{K}_{ab} = \\LieS g_{ab} .\n  \\label{KTApprox1}\n\\end{equation}\nThese objects exactly satisfy \\eqref{KTDef} at $\\gamma$, and\npresumably approximate it near this point. It is straightforward to\nwrite down other objects which also have this property. For example,\ntwo (possibly identical) Killing-type Jacobi fields $\\psi^a$ and\n$\\bar{\\psi}^a$ can be used to define\n\\begin{equation}\n  \\mathcal{K}'_{ab} = \\psi_{(a}\n  \\bar{\\psi}_{b)} .\n  \\label{KTApprox2}\n\\end{equation}\nThis expression clearly generalizes to approximate Killing tensors\nof any rank. Exact second-rank Killing tensors probably exist that\ncannot be written in either of these forms, so it is unclear how\nuseful they are.\n\nVery near $\\gamma$, it is possible to approximate the Jacobi fields\nexplicitly. This will be especially useful in Sect.\n\\ref{Sect:ConsLaws} below, where a notion of gravitational current\nis introduced with respect to a given vector field. Consider a\nTaylor expansion of $\\LieS g_{ab}$ in powers of $X^\\sfa$. The first\ntwo terms in this series are trivially obtained from\n\\eqref{JacobiAffine} and \\eqref{InitialData}. Better approximations\ninvolve third and higher derivatives of $\\LieS g_{ab}$ in the limit\n$x \\rightarrow \\gamma$. The lowest order interesting terms can be\nfound from \\eqref{Del2Lieg} and \\eqref{Del3Lieg}. Making use of\n\\eqref{Sig4Coinc}, the final results are that\n\\begin{equation}\n  \\fl \\quad \\LieS g_{ab} \\simeq \\sigma^{\\sfa}{}_{a} \\sigma^{{\\mathsf{b}}}{}_{b} \\big[ \\LieS g_{\\sfa {\\mathsf{b}}} - \\frac{1}{3} X^{\\mathsf{c}} X^{\\mathsf{d}} ( \\LieS R_{\\sfa {\\mathsf{c}} {\\mathsf{b}} {\\mathsf{d}}}\n  + \\frac{1}{2} X^{\\mathsf{f}} \\LieS \\nabla_{\\mathsf{f}} R_{\\sfa {\\mathsf{c}} {\\mathsf{b}} {\\mathsf{d}}}) \\big] + \\Or(X^4) , \\label{LieGExpandJacobi}\n\\end{equation}\nand\n\\begin{eqnarray}\n  \\fl \\quad \\nabla_c \\LieS g_{ab} \\simeq - \\frac{2}{3} \\sigma^{\\sfa}{}_{a} \\sigma^{{\\mathsf{b}}}{}_{b}\n  \\sigma^{{\\mathsf{c}}}{}_{c} X^{\\mathsf{d}} \\big[ R_{{\\mathsf{d}} {\\mathsf{c}}\n  (\\sfa}{}^{{\\mathsf{f}}} \\LieS g_{{\\mathsf{b}}) {\\mathsf{f}}} + g_{{\\mathsf{f}}(\\sfa} \\LieS R_{{\\mathsf{b}}) {\\mathsf{d}} {\\mathsf{c}}}{}^{\\mathsf{f}}  + \\frac{3}{4}\n  X^{\\mathsf{f}}  \\nonumber\n  \\\\\n  \\fl \\qquad ~ \\times \\big( \\frac{1}{3} g_{\\mathsf{h} (\\sfa} \\LieS\n  \\nabla^{\\mathsf{h}} R_{{\\mathsf{b}}) {\\mathsf{d}} {\\mathsf{f}} {\\mathsf{c}}} - g_{\\mathsf{h} \\sfa } \\LieS \\nabla_{\\mathsf{d}} R_{{\\mathsf{f}} ({\\mathsf{b}}\n  {\\mathsf{c}})}{}^{\\mathsf{h}} - g_{\\mathsf{h} {\\mathsf{b}}} \\LieS \\nabla_{\\mathsf{d}}\n  R_{{\\mathsf{f}} (\\sfa {\\mathsf{c}})}{}^{\\mathsf{h}} \\big) \\big] + \\Or(X^3).\n  \\label{LieGGradJacobi}\n\\end{eqnarray}\nLie derivatives here are evaluated at $\\gamma$, so they only involve\n$A_\\sfa$, $B_{\\sfa {\\mathsf{b}}}$, $g_{\\sfa {\\mathsf{b}}}$, $R_{\\sfa {\\mathsf{b}} {\\mathsf{c}}\n{\\mathsf{d}}}$, and its first two derivatives. The factors of\n$\\sigma^{\\sfa}{}_{a}$ in front of these equations are used as a\nconvenient means for converting tensors at $\\gamma$ into tensors at\n$x$. It is perhaps more typical to use parallel propagators\n$g^{\\sfa}{}_{a}$ for this purpose \\cite{PoissonRev}, although the\naforementioned simplicity of $\\sigma^{\\sfa}{}_{a}$ in normal\ncoordinates makes it an attractive alternative. There is very little\ndifference at low orders regardless. $\\sigma^{\\sfa}{}_{a}$ can be\nfreely interchanged with $-g^{\\sfa}{}_{a}$ in\n\\eqref{LieGGradJacobi}. This is also possible in\n\\eqref{LieGExpandJacobi} when $B_{(\\sfa {\\mathsf{b}})} = 0$.\n\nApproximations like these are not useful over regions where the\ncurvature changes significantly, or on length scales approaching the\ncurvature radius. An alternative approach is to expand the various\nbitensors built from $\\sigma$ using its definition as an integral\nalong a geodesic. Simplifications can often be introduced by\nignoring all terms nonlinear in the Riemann tensor. A general method\nfor this type of weak-field procedure may be found in \\cite{Synge,\ndeFelice}. Specific details involved with expanding the Jacobi\nfields in this way will not be given here.\n\n\\section{Symmetries near a worldline}\\label{Sect:GAC}\n\nThe Jacobi fields just discussed generalize the idea of a Killing\nfield or more general affine collineation in a normal neighborhood\nof a given point. This is useful for some purposes, although it does\nnot have a very clear physical interpretation. The choice of origin\nshould presumably correspond to a preferred point, although there\nare few of these that might arise in practice. It is often more\nuseful to base the idea of an approximate symmetry off of a given\ntimelike worldline rather than a single point. This could correspond\nto the path of some observer. In some cases, the physical system\npicks out preferred reference frames. A binary star system\nexperiencing no mass transfer can admit three center-of-mass frames\n(rigorously defined in \\cite{Dix70a, EhlRud,CM}), for example. Two\nof these are associated with the individual stars, while the third\ndescribes the system as a whole. There are also preferred observers\nin most cosmological models. Expressing a system's dynamics in terms\nof quantities associated with these frames has an obvious physical\ninterpretation. The distinction between approximate symmetries\ndefined with respect to a point versus a worldline is closely\nanalogous to the one between Riemann and Fermi normal coordinate\nsystems.\n\nThe concept of an observer here will be taken to mean a timelike\nworldline $\\Gamma$ together with a set of hypersurfaces $\\Sigma(s)$\nthat foliate a surrounding worldtube $W$. It will be assumed that\neach of these hypersurfaces is a normal neighborhood of the point\n$\\gamma(s)$ where it intersects the central worldline. Each of them\nis therefore formed by a collection of radially-emanating geodesics\nof (usually) finite length. The most typical examples would be the\npast-directed null geodesics or the spacelike set orthogonal to\n$\\dot{\\gamma}^\\sfa = \\rmd \\gamma^\\sfa\/\\rmd s$ at $\\gamma(s)$. Other\nchoices are possible, however. Regardless, a worldline and foliation\ntogether will be referred to as an observer's reference frame.\n\n\\subsection{A family of Jacobi fields}\n\nSymmetries adapted to a particular frame can be constructed using a\none-parameter family of Jacobi fields $\\psi^a(x,\\gamma(s))$. Any\nsuch family is fixed by specifying $A_\\sfa(s)$ and $B_{\\sfa{\\mathsf{b}}}(s)$\nas defined in \\eqref{InitialData}. An optimal way of connecting\ninitial data between different points on $\\Gamma$ therefore must be\nfound. Before considering this problem, it is first useful to\ncollapse the family of Jacobi fields into an ordinary vector field\n$\\xi^a(x)$. Let $\\tau(x)$ be defined so as to identify which leaf of\nthe foliation includes an arbitrary point $x$ in the worldtube $W$.\nMore concisely, it always satisfies $x \\in \\Sigma(\\tau(x))$. The\nassumption that each hypersurface is a normal neighborhood of an\nappropriate point on $\\Gamma$ implies that $\\tau$ is always\nsingle-valued. Now set\n\\begin{equation}\n  \\xi^a (x) = \\psi^a(x, \\gamma(\\tau(x))) .\n  \\label{XiDef}\n\\end{equation}\nThe generalized affine collineations (GACs) to be defined below will\nbe of this form for a particular class of families\n$\\psi^a(x,\\gamma(s))$.\n\nOne potential application for a generalized symmetry constructed\nusing a particular frame is in the definition of quantities that\nmight be approximately conserved as one moves along $\\Gamma$. As an\nexample, consider integrals of conserved stress-energy tensors\nsimilar to (\\ref{DixMomenta}). One might define the component of\nmomentum generated by a $\\xi^a$ of the form \\eqref{XiDef} to be\n\\begin{equation}\n  \\ItP_\\xi(s) = \\int_{\\Sigma(s)} T^{a}{}_{b}\n  \\xi^b \\rmd S_a . \\label{PDef}\n\\end{equation}\nThe evolution of this quantity crucially depends on how the\nparameters $A_\\sfa(s)$ and $B_{\\sfa {\\mathsf{b}}}(s)$ in (\\ref{InitialData})\nare connected along $\\Gamma$. It is well-known that $\\ItP_\\xi$ is\nconserved if $\\xi^a$ is Killing and no matter flows across the\nboundary of the worldtube. In this case, initial data for the\none-parameter family of Jacobi fields must satisfy the Killing\ntransport (KT) equations on $\\Gamma$:\n\\numparts\n\\begin{eqnarray}\n    \\mathrm{D}A_\\sfa\/\\rmd s &= \\dot{\\gamma}^{\\mathsf{b}} B_{{\\mathsf{b}} \\sfa}\n    \\label{KTA}\n    \\\\\n    \\mathrm{D} B_{\\sfa {\\mathsf{b}}} \/ \\rmd s &= - R_{\\sfa {\\mathsf{b}}\n    {\\mathsf{c}}}{}^{\\mathsf{d}} \\dot{\\gamma}^{\\mathsf{c}} A_{\\mathsf{d}} .\n    \\label{KTB}\n\\end{eqnarray}\n\\endnumparts\nIf there exists an exact Killing vector $Y^a_{\\mathrm{K}}$ such that\n$A^\\sfa =Y^\\sfa_{\\mathrm{K}}$ and $B^{\\sfa {\\mathsf{b}}} = \\nabla^\\sfa\nY^{\\mathsf{b}}_{\\mathrm{K}}$ at a given $s = s_0$, relations like these will\nhold for all $s$. Furthermore, momenta $p^\\sfa$ and $S^{\\sfa {\\mathsf{b}}}$\nidentified using (\\ref{DixMomenta}) would satisfy\n\\begin{equation}\n  0 = (\\dot{p}_{\\sfa} - \\frac{1}{2} S^{{\\mathsf{b}} {\\mathsf{c}}} R_{{\\mathsf{b}} {\\mathsf{c}} \\mathsf{d} \\sfa}\n  \\dot{\\gamma}^{\\mathsf{c}}) Y^{\\sfa}_{\\mathrm{K}} + \\frac{1}{2} (\n  \\dot{S}_{\\sfa {\\mathsf{b}}} - 2 p_{[\\sfa} \\dot{\\gamma}_{{\\mathsf{b}}]} )\n  \\nabla^{\\sfa} Y^{\\mathsf{b}}_{\\mathrm{K}} . \\label{Papapetrou}\n\\end{equation}\nIf there were a full complement of ten Killing vectors, all possible\nversions of this expression would together be equivalent to the\nPapapetrou equations. More generally, Papapetrou's result is only an\napproximation. Any deviations can be understood using a general\n10-parameter family of possibly approximate isometries. One might\nexpect these corrections to be minimized if $A_\\sfa$ and $B_{\\sfa\n{\\mathsf{b}}}$ always satisfy the KT equations even when no exact Killing\nfields exist.\n\\begin{definition}\\label{Def:GAC}\n  Let a generalized affine collineation (GAC) $\\xi^a(x)$ associated with a\n  reference frame $\\{\\Gamma, \\Sigma\\}$ be derived from a family of\n  Jacobi fields via \\eqref{XiDef}. Individual elements of the family and their\n  first derivatives satisfy the Killing transport equations \\eqref{KTA} and \\eqref{KTB} on $\\Gamma$.\n\\end{definition}\n\nAlthough this definition was motivated by the properties of\nconserved momenta in very particular spacetimes, it also arises from\nmuch more general (if less physical) arguments. Consider all\npossible initial data for vectors built from Jacobi fields using\n\\eqref{XiDef}. It is reasonable to suppose that any GAC should be\nexactly affine on $\\Gamma$; i.e.\n\\begin{equation}\n  \\nabla_\\sfa \\LieX g_{{\\mathsf{b}} {\\mathsf{c}}} |_\\Gamma = 0.\n  \\label{GACAffine}\n\\end{equation}\nIt can also be expected that $A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$ fix\n$\\xi^a$ and its first derivatives on $\\Gamma$ just as they do for\n$\\psi^a$. Generalizing \\eqref{InitialData}, let\n\\begin{equation}\n  A^{\\sfa}(s) = \\xi^\\sfa(\\gamma(s)) , \\qquad B^{\\sfa {\\mathsf{b}}}(s) =\n  \\nabla^{\\sfa} \\xi^{\\mathsf{b}}(\\gamma(s)) . \\label{InitialDataXi}\n\\end{equation}\n\nWe start with the second of these constraints. Directly\ndifferentiating (\\ref{XiDef}) implies that\n\\begin{equation}\n  \\LieX g_{ab} = \\LieS g_{ab} + 2 \\dot{\\psi}_{(a} \\nabla_{b)} \\tau ,\n  \\label{LieXivsPsi}\n\\end{equation}\nwhere the Lie derivative with respect to $\\psi^a(x, \\tau)$ on the\nright-hand side is understood (as usual) to involve only the first\nargument of this vector field. It will be assumed that the foliation\nis always sufficiently smooth that derivatives of $\\tau$ remain\nwell-defined throughout $W$, and on $\\Gamma$ in particular.\nEvaluating \\eqref{LieXivsPsi} on the central worldline requires\nknowledge of $\\dot{\\psi}^a(\\gamma,\\gamma)$. Coincidence limits like\nthese are commonly denoted with brackets. For example,\n\\begin{equation}\n  [\\dot{\\psi}^a](\\gamma) = \\lim_{x \\rightarrow \\gamma}\n  \\frac{ \\partial}{\\partial s} \\dot{\\psi}^a(x,\\gamma(s)) .\n  \\label{CoincDef}\n\\end{equation}\nThe convention of using different fonts for indices referring to $x$\nand $\\gamma$ cannot be consistently applied in expressions like\nthis. No confusion should arise, however. Limits like\n\\eqref{CoincDef} are easily computed using Synge's rule \\cite{Synge,\nPoissonRev}. In this case,\n\\begin{eqnarray}\n  [\\dot{\\psi}^a] &= \\dot{\\gamma}^{\\mathsf{b}} [\\nabla_{\\mathsf{b}} \\psi^a] \\nonumber\n  \\\\\n   &= \\dot{\\gamma}^{\\mathsf{b}} ( \\nabla_{\\mathsf{b}} [\\psi^a] - \\delta^b_{\\mathsf{b}} [\\nabla_b \\psi^a]\n  ) \\nonumber\n  \\\\\n   &= \\mathrm{D} A^a\/\\rmd s - \\dot{\\gamma}^{\\mathsf{b}} B_{{\\mathsf{b}}}{}^{a} .\n  \\label{DotPsi}\n\\end{eqnarray}\nIt follows that (\\ref{InitialDataXi}) holds for all initial data iff\n\\eqref{KTA} is satisfied.\n\nThe other KT equation arises from enforcing \\eqref{GACAffine}.\nNoting (\\ref{JacobiAffine}) and (\\ref{KTA}), it must be true that\n\\begin{equation}\n  [\\nabla_a \\dot{\\psi}^b] = 0.\n  \\label{DelDotPsiReq}\n\\end{equation}\n$[\\ddot{\\psi}^a]$ also has to vanish, although this term is equal to\n$-\\dot{\\gamma}^b [ \\nabla_b \\dot{\\psi}^a]$. Requiring\n\\eqref{DelDotPsiReq} is therefore sufficient. Using the same type of\nprocedure as in \\eqref{DotPsi} shows that\n\\begin{equation}\n  [\\nabla_a \\dot{\\psi}_b] = \\mathrm{D} B_{ab} \/ \\rmd s+ R_{abc}{}^{d}\n  \\dot{\\gamma}^c A_d . \\label{DelDotPsi}\n\\end{equation}\nDeriving this is straightforward other than noting that\n\\eqref{Del2Affine} -- although mentioned for exact affine\ncollineations -- also holds for any Jacobi field at its origin.\nRegardless, the conclusion is that the second Killing transport\nequation \\eqref{KTB} ensures that $\\nabla_\\sfa \\LieX g_{{\\mathsf{b}}{\\mathsf{c}}}$\nvanishes everywhere on $\\Gamma$.\n\nNoting that the KT equations have the same significance for general\naffine collineations as they do for ordinary Killing fields\n\\cite{Hall}, it easily follows that\n\\begin{theorem}\\label{Thm:GACBasic}\n  The class of all generalized affine collineations associated with\n  a given reference frame forms a 20-dimensional group in four\n  spacetime dimensions. Every GAC satisfies $\\LieX \\nabla = 0$ on\n  $\\Gamma$, and all exact affine collineations are members of this\n  class.\n\\end{theorem}\nThis is closely related to theorem \\ref{Thm:JacobiBasic}. It\nstrongly supports definition \\ref{Def:GAC} and the intuitive\nidentification of GACs with approximate symmetries inside $W$.\n\nAt least in principle, finding GACs associated with a particular\nreference frame is straightforward. Suppose that $A_\\sfa(s_0)$ and\n$B_{\\sfa {\\mathsf{b}}}(s_0)$ are given as initial data at some $\\gamma_0 =\n\\gamma(s_0)$. The goal is then to determine the $\\xi^a(x)$\nsatisfying \\eqref{InitialDataXi} at the appropriate point. This is\ndone by first applying the KT equations to the given parameters\nalong $\\Gamma$ from $\\gamma_0$ to $\\gamma(\\tau(x))$. The geodesic\ndeviation equation (\\ref{Jacobi}) is then integrated between this\nlatter point and $x$ using the initial conditions\n(\\ref{InitialData}). Both of these operations simply require finding\nthe solutions to well-behaved ordinary differential equations.\nAlternatively, $\\xi^a$ could also be obtained using the explicit\nexpression \\eqref{JacobiFirstDer} together with the KT equations and\n\\eqref{XiDef}.\n\nOur prescription for generalizing arbitrary affine collineations may\nappear somewhat awkward. Killing transport equations are being\napplied along $\\Gamma$, while the Jacobi equation is used on\ngeodesics intersecting that worldline. These two procedures are not\nas different as they might appear. Trying to use Killing transport\neverywhere would generically lead to inconsistencies. Derivatives of\nthe field expected from the KT equations would not usually match the\nderivatives computed from $\\xi^a$ itself. Only the tangential\ncomponents of these derivatives can be consistently fixed by\nintegrating ordinary differential equations along a collection of\nradial geodesics. Weakening the Killing transport equations to take\nthis into account exactly reproduces the geodesic deviation\nequation. This may be seen by rewriting \\eqref{Jacobi} as a pair of\nfirst order differential equations on geodesics connecting $x$ to\n$\\gamma(\\tau(x))$. Denote the unit tangent vector to one such\ngeodesic by $u^a(l)$. Also set $\\hat{A}^a = \\psi^a$ and\n$\\hat{B}_{ab} = \\nabla_a \\psi_b$ everywhere. It is then\nstraightforward to show that \\numparts\n\\begin{eqnarray}\n  \\mathrm{D} \\hat{A}^a\/\\rmd l &= u^b \\hat{B}_{ba}\n  \\\\\n  u^a \\mathrm{D} \\hat{B}_{ab}\/\\rmd l &= - R_{abc}{}^{d} u^a u^d\n  \\hat{A}_d.\n\\end{eqnarray}\n\\endnumparts\nThe first of these equations has exactly the same form as\n\\eqref{KTA}, while the second is essentially \\eqref{KTB} contracted\nwith $u^a$. Killing and Jacobi transport are therefore very closely\nrelated operations. The latter does not uniquely propagate\n$\\hat{B}_{ab}$ from given initial data, so it is weaker. These\nremarks also clarify in what sense Jacobi fields or GACs approximate\n\\eqref{AffineDef} or \\eqref{Del2Affine}.\n\n\n\\subsection{Special cases and properties of GACs}\n\nFrom a physical perspective, momenta like \\eqref{PDef} should be\ndefinable even in the absence of any exact isometries. This is most\nconveniently done with a particular class of GACs that generalize\nonly the Killing fields. In analogy to the Killing-type Jacobi\nfields discussed in Sect. \\ref{Sect:SpecialJacobi}, suppose that\n$B_{(\\sfa {\\mathsf{b}})}$ vanishes on at least one point of $\\Gamma$. It\nimmediately follows from \\eqref{KTB} that it must actually vanish\neverywhere. The Killing-type GACs therefore form a 10-dimensional\ngroup of vector fields satisfying\n\\begin{equation}\n  \\Lie_{\\xi_\\mathrm{K}} g_{\\sfa {\\mathsf{b}}} |_\\Gamma = 0,\n  \\label{GACKilling}\n\\end{equation}\nas well as \\eqref{GACAffine}. They may be thought of as generalizing\nthe Poincar\\'{e} symmetries of flat spacetime near a given observer.\n\nIt also possible to single out GACs that are approximately\nhomothetic in the sense that\n\\begin{equation}\n  \\Lie_{\\xi_{\\mathrm{H}}} g_{\\sfa {\\mathsf{b}}} |_\\Gamma = 2 c g_{\\sfa\n  {\\mathsf{b}}}. \\label{GACHomothetic}\n\\end{equation}\nThis requires setting $B_{(\\sfa {\\mathsf{b}})} = c g_{\\sfa {\\mathsf{b}}}$. While\nalways possible, the remaining components of the initial data cannot\nbe explicitly solved for except in the case when $\\Gamma$ is a\ngeodesic. It then self-consistent to choose $B_{[\\sfa {\\mathsf{b}}]}=0$. The\nobvious way of doing this is to normalize $\\dot{\\gamma}^\\sfa$ to\nunity and set\n\\begin{equation}\n  A_\\sfa = c (s - \\bar{s} ) \\dot{\\gamma}_\\sfa\n\\end{equation}\nfor some constant $\\bar{s}$. It is easily verified that the given\nparameters satisfy the KT equations. One homothetic-type GAC\nassociated with an affinely parameterized geodesic therefore has the\nform\n\\begin{equation}\n  \\xi^a_{\\mathrm{H}} = c \\big[ \\sigma^a + (\\tau - \\bar{s}) H^{a}{}_{\\sfa}\n  \\sigma^{\\sfa}{}_{{\\mathsf{b}}} \\dot{\\gamma}^{\\mathsf{b}} \\big] .\n\\end{equation}\nAs usual, Killing-type Jacobi fields may be added to this without\nspoiling \\eqref{GACHomothetic}. It should also be emphasized that\nhomothetic-type GACs are not restricted to geodesic frames. This is\njust the case where closed-form solutions of the KT equations can be\nobtained by inspection.\n\nMany of the properties derived for Jacobi fields in Sect.\n\\ref{Sect:Symmetries} and the appendix can be carried over at least\npartially for the GACs. For example,\n\\begin{equation}\n  \\LieX \\sigma^\\sfa = \\LieS \\sigma^\\sfa  = 0  \\label{GACLie1}\n\\end{equation}\nif the arguments are of the form $(x, \\gamma(\\tau(x)))$ and $\\xi^a$\nand $\\psi^a$ are related via \\eqref{XiDef}. This may be interpreted\nas stating that spatial Fermi coordinates are preserved under flows\ngenerated by $\\xi^a$. Since the hypersurfaces $\\Sigma(s)$ can be\ndescribed as a set of geodesics intersecting $\\gamma(\\tau(x))$, it\nwill always be true that $\\sigma^a \\nabla_a \\tau =0$. This means\nthat\n\\begin{theorem}\\label{Thm:GACExactSyms}\n  Given a general GAC $\\xi^a$, $\\LieX \\sigma^\\sfa = \\LieX \\sigma^a\n  =0$ when the arguments of these equations are as in\n  \\eqref{GACLie1}. Killing-type GACs $\\xi^a_{\\mathrm{K}}$ also satisfy $\\Lie_{\\xi_{\\mathrm{K}}} \\sigma = \\Lie_{\\xi_{\\mathrm{K}}} \\sigma_\\sfa =\n  0$ with the same restriction.\n\\end{theorem}\n\nThe identity \\eqref{LieSig6} serves to constrain Lie derivatives of\nthe metric with respect to Jacobi fields. A direct analog of this\nequation for an arbitrary GAC would involve an additional term.\nDespite this, contracting the result with $\\sigma^b$ leads to the\nsimple conclusion\n\\begin{equation}\n  \\sigma^a \\sigma^b \\LieX g_{ab} = 2 \\sigma^\\sfa \\sigma^{\\mathsf{b}} B_{(\\sfa\n  {\\mathsf{b}})} .\n\\end{equation}\n``Purely radial'' components of $\\LieX g_{ab}$ therefore vanish for\nKilling-type GACs.\n\nMany other results can be carried over in similar ways. One that is\nof particular interest is the behavior of $\\LieX g_{ab}$ or\n$\\nabla_a \\LieX g_{bc}$ near $\\Gamma$. As with the Jacobi fields, it\nis possible to see how close a GAC comes to being affine as its\nreference worldline is approached. Analogs of\n\\eqref{LieGExpandJacobi} and \\eqref{LieGGradJacobi} may be obtained\nusing expansions like \\eqref{LieXivsPsi} and the identity\n\\eqref{Del2General}. Simplifying terms with the Killing transport\nequations, the lowest order correction to \\eqref{LieGExpandJacobi}\nis\n\\begin{equation}\n  \\Lie_{(\\xi-\\psi)} g_{ab} \\simeq \\frac{2}{3} \\sigma^{\\sfa}{}_{a}\n  \\sigma^{{\\mathsf{b}}}{}_{b} \\nabla_{(\\sfa} \\tau g_{{\\mathsf{b}}) {\\mathsf{f}}}\n  \\dot{\\gamma}^{\\mathsf{h}}\n  X^{\\mathsf{c}} X^{\\mathsf{d}} \\LieX R_{\\mathsf{h} {\\mathsf{c}} {\\mathsf{d}}}{}^{{\\mathsf{f}}} + \\Or(X^3).\n  \\label{LieGExpandGAC}\n\\end{equation}\nSimilarly, the first interesting change to \\eqref{LieGGradJacobi}\nhas the form\n\\begin{eqnarray}\n  \\fl \\qquad \\nabla_a \\Lie_{(\\xi-\\psi)} g_{bc} \\simeq - \\frac{4}{3} \\sigma^{\\sfa}{}_a\n  \\sigma^{{\\mathsf{b}}}{}_b \\sigma^{{\\mathsf{c}}}{}_c X^{\\mathsf{d}} \\dot{\\gamma}^{\\mathsf{f}}\n  \\big[ g_{\\mathsf{h} ({\\mathsf{b}}} \\nabla_{{\\mathsf{c}})} \\tau (\\delta^{\\mathsf{l}}_\\sfa - \\dot{\\gamma}^{\\mathsf{l}} \\nabla_\\sfa \\tau) \\LieX R_{{\\mathsf{f}} (\\mathsf{l} {\\mathsf{d}})}{}^{\\mathsf{h}} \\nonumber\n  \\\\\n  \\qquad ~  + \\frac{1}{2} \\nabla_\\sfa \\tau (R_{{\\mathsf{d}} {\\mathsf{f}}\n  ({\\mathsf{b}}}{}^{\\mathsf{h}} \\LieX g_{{\\mathsf{c}}) \\mathsf{h}} - g_{\\mathsf{h} ({\\mathsf{b}}} \\LieX R_{{\\mathsf{c}}) {\\mathsf{f}} {\\mathsf{d}}}{}^{\\mathsf{h}} ) \\big] + O(X^2).\n  \\label{LieGGradGAC}\n\\end{eqnarray}\nAs before, the magnitudes of these terms depend on how close $\\xi^a$\nis to being a symmetry of the Riemann tensor on the observer's\nworldline.\n\n\\section{Mechanics and conservation laws}\n\\label{Sect:ConsLaws}\n\nIt has already been remarked that one of the main applications of\nexact symmetries in physics is to the formulation of conservation\nlaws. These take several forms. Perhaps the most basic are those\nassociated with a spacetime's geodesics. It is sometimes possible\nfor such curves to be at least partially parameterized by a number\nof constants associated with geometric symmetries. More\ninterestingly, conservation laws can also be associated with\nextended matter distributions. Assuming only that stress-energy\ntensors satisfy\n\\begin{equation}\n  \\nabla_a T^{ab} = 0 \\label{StressCons}\n\\end{equation}\ntends to lead to the definition of slowly-varying parameters like\nthose discussed in connection with \\eqref{DixMomenta} and\n\\eqref{PDef}. The situation becomes much more interesting in full\ngeneral relativity. Einstein's equation implies stress-energy\nconservation, although it also connects symmetries of the geometry\nto those of the matter distribution (and vice versa). This allows\nthe introduction of exact conservation laws in arbitrary spacetimes.\n\n\\subsection{Geodesics}\n\nIt is well-known that any Killing fields that may exist provide\nfirst integrals of the geodesic equation. These can be used both to\nderive and parameterize the geodesics of a given spacetime. While\nless commonly discussed, similar quantities can also be associated\nwith other kinds of symmetries \\cite{GeodesicConsts,KatzinGeo}.\nUnlike in the Killing vector case, the presence of more general\ncollineations sometimes implies the existence of interesting\nconserved quantities that are not linear in the geodesic's\nfour-velocity. The curve's affine parameter can also appear\nexplicitly. As a direct calculation will easily verify, two\nconstants associated with an exact affine collineation\n$Y_{\\mathrm{A}}^a$ are \\numparts\\label{GeoConsts}\n\\begin{eqnarray}\n    C_1 &= \\dot{y}^a \\dot{y}^b \\Lie_{Y_{\\mathrm{A}}} g_{ab}\n    \\label{GeoConst1}\n    \\\\\n    C_2 & = \\dot{y}_a Y^a_{\\mathrm{A}} - \\frac{1}{2} l C_1 .\n    \\label{GeoConst2}\n\\end{eqnarray}\n\\endnumparts\nThese quantities remain fixed along any affinely-parameterized\ngeodesic $y(l)$. The first becomes degenerate if $Y^a_{\\mathrm{A}}$\nis Killing. $C_2$ then reduces to the standard conserved quantity\nassociated with a Killing field. Other constants can sometimes be\nwritten down by combining $Y^a_{\\mathrm{A}}$ with an exact Killing\ntensor \\cite{GeodesicConsts, GeoExamples}. Such constructions will\nnot be discussed here.\n\nConsider instead expressions like those just given with\n$Y_{\\mathrm{A}}^a(x)$ replaced by some Jacobi field\n$\\psi^a(x,\\gamma)$. We then have\n\\begin{eqnarray}\n  \\dot{C}_1 = \\frac{\\rmd C_1}{\\rmd l} = -\\frac{2}{l} \\frac{\\rmd C_2}{ \\rmd l} = \\dot{y}^a \\dot{y}^b \\dot{y}^c \\nabla_{(a} \\LieS\n  g_{bc)} .\n\\end{eqnarray}\nThe tangent vectors are proportional to $\\sigma^a(y,\\gamma)$ for the\nspecial case of geodesics passing through $\\gamma$. It then follows\nfrom \\eqref{LieSig9} that both $C_1$ and $C_2$ remain conserved\nalong all such trajectories. Each Jacobi field generates exact\ngeodesic constants in this way. In terms of the initial data\n$A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$, these have the values\n\\begin{eqnarray}\n  C_1  &= 2 \\dot{y}^\\sfa \\dot{y}^{\\mathsf{b}} B_{(\\sfa {\\mathsf{b}})}, \\qquad\n  C_2 &= \\dot{y}^\\sfa A_\\sfa \\label{GeoConstInit}\n\\end{eqnarray}\nwhen the parameter $l$ is chosen to vanish at $\\gamma$. It is clear\nfrom this that $B_{[\\sfa {\\mathsf{b}}]}$ is irrelevant. Multiple Jacobi\nfields may therefore generate the same constants on a particular\ncurve.\n\nExact affine collineations generalize these results by also applying\nto non-radial geodesics. Expansions like \\eqref{LieGGradJacobi} can\nbe used to derive how close general Jacobi fields come to this\nideal. To lowest nonvanishing order,\n\\begin{equation}\n  \\dot{C}_1 \\simeq - \\frac{4}{3} (\\dot{y}^a \\sigma^{\\sfa}{}_{a}) (\\dot{y}^b \\sigma^{{\\mathsf{b}}}{}_{b}) (\\dot{y}^c\n   \\sigma^{{\\mathsf{c}}}{}_{c}) X^{\\mathsf{d}} R_{{\\mathsf{d}} {\\mathsf{b}}\n  {\\mathsf{c}}}{}^{{\\mathsf{f}}} \\LieS g_{\\sfa {\\mathsf{f}}} + \\Or(X^2) . \\label{Cdot}\n\\end{equation}\nThis term vanishes if $B_{(\\sfa {\\mathsf{b}})}=0$, so the Killing-type\nJacobi fields usually provide more accurate ``conservation laws''\nfor arbitrary geodesics near $\\gamma$. In these cases,\n$\\dot{C}_1(l)$ scales like $(X\/\\mathcal{R})^2\/\\mathcal{R}$, where\n$\\mathcal{R}$ is a curvature radius. This is a worst-case estimate.\n$C_1$ and $C_2$ will probably vary much more slowly if there is a\nphysical sense in which the system is approximately symmetric. Rates\nof change for $C_1$ or $C_2$ can also be constrained using\nidentities like \\eqref{LieSig7}. This effectively restricts how much\nthese parameters can vary as a geodesic moves away from $\\gamma$.\nMore of their changes tend to occur as $y(l)$ moves across rather\nthan with the radial geodesics.\n\nParameters like $C_1$ and $C_2$ can also be defined with respect to\na GAC $\\xi^a$. These should remain approximately conserved for\ngeodesics near an observer's worldline rather than geodesics near a\npoint. They are exact constants for curves passing through $\\Gamma$\nalong the reference foliation. This can be seen by using\n\\eqref{LieXivsPsi} to show that\n\\begin{equation}\n  \\dot{y}^a \\dot{y}^b \\dot{y}^c \\nabla_a \\LieX g_{bc} =   \\dot{y}^a \\dot{y}^b \\dot{y}^c \\nabla_{a} \\LieS g_{bc}\n\\end{equation}\nwhen $\\dot{y}^a \\nabla_a \\tau =0$. Values of $C_1$ and $C_2$ here\nare the same as in \\eqref{GeoConstInit} if the quantities in that\nequation are evaluated at the point where $y$ intersects $\\Gamma$.\nThis might be useful in a coordinate system constructed from some\ncollection of GACs. Alternatively, it can be viewed as a\ngeneralization of standard results near a given observer.\n\nFurther methods of parameterizing geodesics may be found by\ngeneralizing the constants associated with higher-rank Killing\ntensors. Given any exact Killing tensor $K_{a_1 \\cdots a_n} =\nK_{(a_1 \\cdots a_n)}$, the scalar\n\\begin{equation}\n  C_K = K_{a_1 \\cdots a_n} \\dot{y}^{a_1} \\cdots \\dot{y}^{a_n}\n  \\label{CK}\n\\end{equation}\nis conserved along all geodesics $y(l)$. An analog of this quantity\nfor the approximate Killing tensor \\eqref{KTApprox1} is exactly\n$C_1$ defined above. Something more interesting can be generated by\nsubstituting \\eqref{KTApprox2} into \\eqref{CK}. In the second-rank\ncase, one may define\n\\begin{equation}\n  C_K = (\\dot{y}_a \\psi^a_{\\mathrm{K}}) (\\dot{y}_b \\bar{\\psi}^b_{\\mathrm{K}}) \\label{CKEx}\n\\end{equation}\nfor some Killing-type Jacobi fields $\\psi^a_{\\mathrm{K}}$ and\n$\\bar{\\psi}^a_{\\mathrm{K}}$. This is easily generalized to involve\nan arbitrary number of products, although it is only interesting to\nconsider linearly independent collections of Jacobi fields. The\nmaximum number of useful products is therefore ten. All of these can\nbe generated just from individual terms of the form $\\dot{y}_a\n\\psi^a_{\\mathrm{K}}$. These are interpreted as approximate constants\nassociated with objects that are nearly first-rank Killing tensors\n(i.e. Killing vectors). Everything of interest here can therefore be\nderived from the behavior of\n\\begin{equation}\n  C_3 = \\dot{y}_a \\psi^a_{\\mathrm{K}} = C_2 + \\frac{1}{2} l C_1.\n  \\label{C3}\n\\end{equation}\nAlthough this depends only on the two approximate constants defined\nbefore, it may be interpreted as an additional useful parameter. It\nhas the interesting property that\n\\begin{equation}\n  \\dot{C}_3 = \\frac{1}{2} C_1 . \\label{C3Dot}\n\\end{equation}\nTime derivatives do not appear on the right hand side. Consider the\nspecial case of a geodesic that passes through $\\gamma$. Since the\nJacobi field was assumed to be Killing at its origin, $C_1 = 0$.\nThis is true everywhere, so $C_3$ also remains fixed along the\nentire geodesic. It actually coincides with $C_2$ in this case.\n\nDifferences arise when considering non-radial geodesics. It was\nremarked above that there was a sense in which $C_1$ and $C_2$ only\nvaried due to non-radial components of $\\dot{y}^a$. This type of\nstatement can be made much more precise for $C_3$. Let\n\\begin{equation}\n  \\dot{y}^a(l) = u_{||} (l) \\sigma^a(y(l),\\gamma) + u^a_\\bot(l) ,\n\\end{equation}\nwhere $u^{[a}_\\bot \\sigma^{b]} =0$. Now \\eqref{LieSig6},\n\\eqref{GeoConst1}, and \\eqref{C3Dot} show that\n\\begin{equation}\n  \\dot{C}_3 = \\frac{1}{2} u_\\bot^a u_\\bot^b \\LieS g_{ab} .\n  \\label{C3DotBeta}\n\\end{equation}\nThis result is exact for all geodesics $y(l)$. There are many cases\nwhere $u_\\bot^a$ becomes vanishingly small as $l \\rightarrow \\pm\n\\infty$ (when the geodesic exists for these parameter values), so\n\\eqref{C3DotBeta} provides a strong restriction on how much $C_3$\ncan vary in any given situation. Very near $\\gamma$,\n\\eqref{LieGExpandJacobi} can be used to show that\n\\begin{equation}\n  \\dot{C}_3 \\simeq - \\frac{1}{6} (u_\\bot^a \\sigma^{\\sfa}{}_{a}) (u_\\bot^b \\sigma^{{\\mathsf{b}}}{}_{b})\n  X^{\\mathsf{c}} X^{\\mathsf{d}} \\LieS R_{\\sfa {\\mathsf{c}} {\\mathsf{b}} {\\mathsf{d}}} + \\Or(X^3) .\n\\end{equation}\nThe lowest order contributions here scale like\n$(X\/\\mathcal{R})^2\/\\mathcal{R}$. This is similar to the result\nexpected for $\\dot{C}_1$ when computed using a Killing-type Jacobi\nfield.\n\n\\subsection{Extended matter\ndistributions}\\label{Sect:ExtendedMatter}\n\nFrom a physical perspective, it is often more interesting to\nconsider possibly approximate integrals of the equations of motion\ndescribing an extended matter distribution rather than a pointlike\ntest particle. Suppose that this matter is modeled by a conserved\nstress-energy tensor $T^{ab}$. Contracting it with any exact Killing\nvectors that may exist yields conserved currents. These are\nequivalent to some subset of the typical laws of linear and angular\nmomentum conservation known in flat spacetime. More generally,\n\\eqref{StressCons} shows that\n\\begin{equation}\n  \\nabla_a ( T^{a}{}_{b} Y^b ) = \\frac{1}{2} T^{ab} \\Lie_Y g_{ab} .\n\\end{equation}\nThis holds for any vector field $Y^a$, although it is convenient to\nassume that it is a Killing-type GAC. The source term on the\nright-hand side may then be considered small near $\\Gamma$. This\ntherefore serves as an approximate conservation law. As long as\nthere is no matter flow through $\\partial \\Sigma$, quantities like\n\\eqref{PDef} might be expected to vary slowly in time.\n\n\nStress-energy conservation can be seen as a consequence of the\ndiffeomorphism invariance of a system's underlying action.\nConstructions that are based on it are therefore useful in many\ntheories of gravity besides general relativity. They can also hold\nfor test bodies in fixed background geometries. This generality is\nquite restrictive. Much more can be said if the full Einstein\nequation is assumed to hold. Symmetries in the geometry are then\nrelated to symmetries in the matter fields. The presence of an exact\nKilling field $Y^a_{\\mathrm{K}}$ that also satisfies\n$\\Lie_{Y_\\mathrm{K}} T_{ab} = 0$ allows many interesting results to\nbe proven regarding the momenta $p_\\sfa$ and $S_{\\sfa {\\mathsf{b}}}$ defined\nin \\eqref{DixMomenta}. For example, the net force and torque on a\nbody can be written explicitly in terms of the Killing field and its\nfirst derivative at a point. If $Y^a_{\\mathrm{K}}$ is timelike, a\nbody's center-of-mass can be shown to follow one of its orbits.\nFurthermore, mass centers always lie on the central axis of\ncylindrically symmetric spacetimes \\cite{SchattStreub1,\nSchattStreub2}.\n\nMomenta defined in terms of $T^{a}{}_{b} \\xi^b$ are useful for many\npurposes, although they are not conserved in the absence of exact\nKilling fields. Determining how they vary over time requires\ndetailed knowledge of a body's internal structure. Alternative\ndefinitions for the linear and angular momenta of an extended body\narise when using the full Einstein equation rather than just\n\\eqref{StressCons}. Taking the trace of Ricci's identity and\nrearranging terms shows that\n\\begin{equation}\n  R^{a}{}_{b} Y^b = \\frac{1}{2} ( g^{ac} g^{bd} - g^{ab} g^{cd} )\n  \\nabla_b \\Lie_Y g_{cd} + \\nabla_b \\nabla^{[a} Y^{b]} .\n  \\label{RicciId}\n\\end{equation}\nThis holds for any vector field $Y^a$. Note that the second term on\nthe right-hand side is always conserved. It follows that\n\\begin{eqnarray}\n  \\nabla_a [ 2 (T^{a}{}_{b} Y^b -\\frac{1}{2} Y^a T^{b}{}_{b}) + j^a_Y] = 0, \\label{AffineConsLaw}\n\\end{eqnarray}\nwhere the ``gravitational current'' $j^a_Y$ associated with $Y^a$\nhas been defined by\n\\begin{equation}\n  j^a_Y = \\frac{1}{8 \\pi} ( g^{ab} g^{cd} - g^{ac} g^{bd} )\n  \\nabla_b \\Lie_Y g_{cd} .\n  \\label{GravCurrent}\n\\end{equation}\nIt clearly vanishes if $Y^a$ is an exact affine collineation. This\nis not the only case where the current's contribution to\n\\eqref{AffineConsLaw} disappears. Using the Bianchi identity,\n\\begin{equation}\n  \\nabla_a j^a_Y = - \\frac{1}{8\\pi} g^{ab} \\Lie_Y R_{ab}.\n  \\label{divJ}\n\\end{equation}\nAny vector field satisfying $g^{ab} \\Lie_Y R_{ab} =0$ will therefore\ngenerate conserved matter currents involving only $T^{a}{}_{b} Y^b -\nY^a T^{b}{}_b\/2$. That such ``contracted Ricci collineations''\ngenerate conservation laws for matter distributions has been noted\nbefore in \\cite{KatzinLevine, Collinson}.\n\nThe viewpoint here will be to apply \\eqref{RicciId} and\n\\eqref{AffineConsLaw} with $Y^a$ replaced by some approximate\nsymmetry.\n\\begin{definition}\n    Fix some family of Jacobi fields $\\psi^a(x,\\gamma(s))$ that generates a GAC\n    via \\eqref{XiDef}. Define the generalized Komar momentum\n    $\\ItP^*_\\psi$ associated with these fields by\n    \\begin{equation}\n        \\ItP^*_\\psi(s) = \\frac{1}{8\\pi} \\oint_{\\partial \\Sigma(s)} \\nabla^{[a}\n        \\psi^{b]} \\rmd S_{ab} . \\label{PStarDef}\n    \\end{equation}\n\\end{definition}\nAs the name suggests, this has the same form and interpretation as a\nKomar integral. It is convenient to assume that $s$ is a fixed\nparameter for the purpose of evaluating the derivative in\n\\eqref{PStarDef}. Directly using a GAC in place of $\\psi^a$ would\nadd a dependence on the reference frame. Note that no such\ndistinctions had to be made for the $\\ItP_\\xi$ defined in\n\\eqref{PDef}. Applying Stokes' theorem together with \\eqref{RicciId}\nand \\eqref{GravCurrent} shows that\n\\begin{equation}\n  \\ItP^*_\\psi =  \\int_\\Sigma \\big[ 2 (T^{a}{}_{b} \\psi^b - \\frac{1}{2}\n  \\psi^a T^{b}{}_{b} ) + j^a_\\psi \\big] \\rmd S_a . \\label{PStar2}\n\\end{equation}\nThere is a well-defined sense in which changes in this quantity are\ndetermined by a combination of ``gravitational wave'' and matter\nfluxes across the boundary $\\partial \\Sigma$. In this\ninterpretation, the amount of $\\ItP^*_\\psi$ carried away from a\nsystem via gravitational waves vanishes if the GAC associated with\n$\\psi^a$ is an affine collineation.\n\nThe 20-parameter family of scalars $\\ItP^*_\\psi$ is intended to\ndefine the linear and angular momenta of an extended body. This is\nat least the interpretation for the 10-parameter subset satisfying\n$B_{(\\sfa {\\mathsf{b}})}=0$. As in \\eqref{DixMomenta}, it is possible to\nwrite these momenta in the more conventional form of tensor fields\non $\\Gamma$. Let\n\\begin{equation}\n  \\ItP^*_\\psi = p_\\sfa^* A^\\sfa + \\frac{1}{2} S^*_{\\sfa {\\mathsf{b}}}\n  B^{\\sfa {\\mathsf{b}}}\\label{KomarMoment}\n\\end{equation}\nfor all $A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$. As written, the generalized\nangular momentum $S_{\\sfa{\\mathsf{b}}}^*$ needn't have any particular index\nsymmetries. Non-Killing Jacobi fields generate the symmetric\ncomponents of this tensor, although such generality isn't necessary.\nVarying among all combinations of initial data completely recovers\n$p^*_\\sfa$ and $S^*_{\\sfa {\\mathsf{b}}}$. Direct expressions can also be\nobtained with the use of \\eqref{JacobiFirstDer}. Continuing to work\nwith the less explicit form \\eqref{KomarMoment}, rates of change of\nthe tensor momenta may easily be extracted from $\\dot{\\ItP}^*_\\psi$.\nUsing the KT equations \\eqref{KTA} and \\eqref{KTB},\n\\begin{equation}\n  \\dot{\\ItP}^*_\\psi = ( \\dot{p}^*_\\sfa - \\frac{1}{2} S^*_{{\\mathsf{b}} {\\mathsf{c}}}\n  R^{{\\mathsf{b}} {\\mathsf{c}}}{}_{{\\mathsf{d}} \\sfa} \\dot{\\gamma}^{\\mathsf{d}} ) A^\\sfa +\n  \\frac{1}{2} ( \\dot{S}^*_{\\sfa {\\mathsf{b}}} + 2 \\dot{\\gamma}_\\sfa p^*_{\\mathsf{b}}) B^{\\sfa {\\mathsf{b}}}.\n\\end{equation}\nThis is closely analogous to \\eqref{Papapetrou}. The left-hand side\nis parameterized entirely by $A_\\sfa$ and $B_{\\sfa {\\mathsf{b}}}$, so\nvarying these quantities determines all of the corrections to the\nPapapetrou equations.\n\nTwo definitions have now been suggested for the momenta of an\nextended body. The first -- summarized by \\eqref{DixMomenta} and\n\\eqref{PDef} -- is closely related to the one given by Dixon\n\\cite{Dix70a, Dix74, Dix79}. It is well-adapted to the construction\nof multipole moments for $T^{ab}$ that intrinsically take into\naccount stress-energy conservation. Mass centers defined from these\nmomenta are known to have most of the properties one might expect\n\\cite{SchattStreub1,CM}. The boundary of the worldtube $W$ isn't\nimportant as long as it lies outside of the matter distribution\nunder discussion. There is no vacuum momentum under this definition.\nUnfortunately, there does not appear to be any exact analog of\nGauss' law either. The momenta of a matter distribution (and changes\nto it) must be computed by integrating over 3-volumes. The\ngeneralized Komar integrals defined by \\eqref{PStarDef} have\ncomplementary characteristics. Their main advantage is in having a\ndirect interpretation analogous to Gauss' law. Changes in the\ncomponent of an isolated body's momentum generated by $\\psi^a$ only\ndepends on the gravitational flux $j^a_\\psi$ passing through the\nsurface $\\partial W$. The mass and angular momenta expected from\nthis definition also agree with commonly-accepted notions at least\nin appropriately symmetric spacetimes. It is potentially problematic\nthat $\\ItP^*_\\psi$ includes what is effectively a vacuum energy.\nThese scalars usually depend on the spatial extent of $W$ even when\nits boundary lies far outside of any matter distribution. This can\nmake it difficult to neatly separate the properties of disjoint\nmatter distributions, although similar situations are found even in\nordinary electromagnetism. It might be conceptually simpler to\nextend $\\partial W$ to infinity, although it is unlikely that all of\nthe bitensors used here would remain well-defined. There might also\nbe convergence problems. Related concepts presented in\n\\cite{EnergyConference} could be more useful for defining momenta\nover very large regions.\n\nSome insight into the behavior of the $\\ItP^*_\\psi$ defined here can\nbe gained by computing it for very small spheres. To be specific,\nlet $C(r,s)$ be the closed 2-surface on $\\Sigma(s)$ satisfying\n$X^\\sfa X_\\sfa = r^2$ for some $r>0$. This is effectively a sphere\nof proper radius $r$ centered at the point $\\gamma(s)$. It follows\nfrom \\eqref{GACAffine} and \\eqref{GravCurrent} that $j^a_\\psi$ is\nnegligible for small radii in the presence of matter. The momentum\ninside $C$ is approximately\n\\begin{equation}\n  P^*_\\xi \\simeq \\frac{8\\pi}{3} r^3 (T^{\\sfa}{}_{{\\mathsf{b}}} A^{\\mathsf{b}} -\n  \\frac{1}{2} A^{\\sfa} T^{{\\mathsf{b}}}{}_{{\\mathsf{b}}}) \\nabla_\\sfa \\tau + \\Or(r^4) .\n\\end{equation}\nIt is perhaps more interesting to consider regions that are locally\ndevoid of matter. These can be understood from the behavior of the\ngravitational current. Its approximate behavior near $\\Gamma$ is\neasily calculated from \\eqref{LieGGradJacobi} and\n\\eqref{GravCurrent}. To lowest nontrivial order,\n\\begin{eqnarray}\n  \\fl \\qquad \\qquad j^a_\\psi \\simeq - \\frac{1}{8 \\pi} \\sigma^{a}{}_{\\sfa} X^{\\mathsf{b}} \\big[\n  \\LieS R^{\\sfa}{}_{{\\mathsf{b}}} + \\frac{1}{3} \\big( g^{\\sfa {\\mathsf{c}}} R^{{\\mathsf{d}}}{}_{{\\mathsf{b}}} + 2 R_{{\\mathsf{b}}}{}^{{\\mathsf{c}} \\sfa {\\mathsf{d}}} \\big) \\LieS\n  g_{{\\mathsf{c}} {\\mathsf{d}}} \\big] + \\Or(X^2). \\label{GravCurrentEst}\n\\end{eqnarray}\nThis vanishes in vacuum for Killing-type Jacobi fields. A little\nmore calculation finds the same conclusion at order $X^2$ as well.\nThis contrasts sharply with other quasilocal notions of vacuum\nmomentum in general relativity. An extensive review of these\nconcepts may be found in \\cite{EnergyRev}. As remarked there, the\nenergy contained in small spheres has been calculated using several\ndifferent definitions. The generic result is that it is proportional\nto the Bel-Robinson tensor, and scales like $r^5$. If this were true\nfor the definition suggested here, terms quadratic in the curvature\nwould appear at order $X^2$ in the current. These are not found.\nVacuum momenta should really only be associated with Killing-type\nsymmetries, so the relevant currents defined here decrease at least\nas fast as $r^6$ as $r \\rightarrow 0$. Other definitions in the\nliterature find more energy in very small spheres. It is not clear\nhow to interpret this, although it might have interesting\nconsequences for the use of near zones and related concepts\nconnected to the mechanics of compact bodies.\n\n\n\\section{An example: gravitational plane waves}\\label{Sect:Example}\n\nThe discussion so far has mainly focused on the behavior of\ngeneralized symmetries near the point or worldline used to construct\nthem. With the exception of general identities like \\eqref{LieSig6},\nit has not been clear what happens to these vector fields far away\nfrom their origins. It is therefore useful to consider an example.\nGiven \\eqref{JacobiFirstDer}, the Jacobi fields can all be\ncalculated simply by differentiating the world function. This makes\nit convenient to consider spacetimes where $\\sigma$ is known\nexactly. Essentially the only examples of this type are\n\\textit{pp}-waves or assorted cosmological models (see\n\\cite{WorldFunction} and references cited therein).\n\nIn the interest of understanding the generalized Komar momenta\n\\eqref{PStarDef} in a vacuum spacetime, only \\textit{pp}-waves will\nbe considered here. Coordinates may be introduced such that the\nmetric satisfies\n\\begin{equation}\n  \\rmd s^2 = - 2 \\rmd u \\rmd v + a(u) \\rmd x^2 + b(u) \\rmd y^2 .\n  \\label{ppGeneral}\n\\end{equation}\nIt can then be shown that one-half of the geodesic distance between\npoints with coordinates $(u,v,x,y)$ and $(\\mathsf{u}, \\mathsf{v},\n\\mathsf{x}, \\mathsf{y})$ is given by \\cite{Gunther,Friedlander}\n\\begin{equation}\n  \\sigma = \\frac{1}{2} [ \\alpha (u,\\mathsf{u}) (x-\\mathsf{x})^2 + \\beta (u,\\mathsf{u}) (y-\\mathsf{y})^2 ] -\n  (u-\\mathsf{u}) (v-\\mathsf{v}), \\label{SigmaExact}\n\\end{equation}\nwhere\n\\begin{equation}\n  \\alpha(u,\\mathsf{u}) = \\frac{u-\\mathsf{u}}{\\int_{\\mathsf{u}}^{u} a^{-1}(w) \\rmd w} ; \\quad \\beta(u,\\mathsf{u}) = \\frac{u-\\mathsf{u}}{\\int_{\\mathsf{u}}^{u} b^{-1}(w) \\rmd\n  w}.\n\\end{equation}\nIt is clear by inspection that $\\partial\/\\partial x$,\n$\\partial\/\\partial y$, and $\\partial\/\\partial v$ are exact Killing\nvectors. They are not the only ones. All nontrivial spacetimes in\nthis class admit between five and seven linearly-independent Killing\nfields.\n\nMost \\textit{pp}-waves are effectively ``null dust'' solutions of\nEinstein's equation, although there are vacuum examples as well. One\nof these is given by\n\\begin{equation}\n  a(u) = \\cos^2 (\\lambda u) ; \\quad b(u) = \\cosh^2 (\\lambda u)\n , \\label{GravWave}\n\\end{equation}\nwhere we assume that $|\\lambda u|< \\pi\/2$ in order to avoid the two\ncoordinate singularities. This represents a simple plane-fronted\ngravitational wave with amplitude $\\lambda$. The only non-vanishing\ncomponents of the curvature are\n\\begin{equation}\n  C_{uxu}{}^{x} = C_{uyu}{}^{y} = \\lambda^2.\n\\end{equation}\nIt is trivial to modify this spacetime to be flat for (say) $u<0$\n\\cite{PlaneWaves}, although impulsive waves of this type will not be\ndiscussed here.\n\nGiven \\eqref{SigmaExact}, it is straightforward to explicitly\ncompute the Jacobi propagators $H^{a}{}_{{\\mathsf{b}}}\n\\sigma^{{\\mathsf{b}}}{}_{\\sfa}$ and $H^{a}{}_{\\sfa} \\sigma_{\\mathsf{b}}$ using\n\\eqref{HDef}. This can be done for any choice of $a(u)$ and $b(u)$,\nalthough we will specialize to the case defined by \\eqref{GravWave}.\nThe results are not particularly enlightening to write down in\ndetail, although they have some interesting consequences. First, all\nJacobi fields are found to be exactly Killing if their first\nderivatives vanish at the base point (denoted as usual with\nsans-serif font). The Jacobi fields $H^{a}{}_{[\\mathsf{x}}\n\\sigma_{\\mathsf{v}]}$ and $H^{a}{}_{[\\mathsf{y}}\n\\sigma_{\\mathsf{v}]}$ are also Killing. This identifies six\nindependent Killing fields. It also implies that the linear\ngravitational momentum $p^*_\\sfa $ defined in \\eqref{KomarMoment}\nmust vanish. This can be taken to imply (unsurprisingly) that the\ngravitational wave has zero rest mass: $|p^*|^2=0$.\n\nThere remain four non-Killing Jacobi fields with skew-symmetric\n$B_{\\sfa {\\mathsf{b}}}$. The one associated with spatial rotations in the\n$x-y$ plane is relatively simple to write down when $\\mathsf{u}=0$:\n\\begin{eqnarray}\n  \\fl \\qquad \\psi_{\\mathsf{xy}}^a = 2 H^{a}{}_{\\mathsf{[x}} \\sigma_{\\mathsf{y}]} =\n  -(y-\\mathsf{y})\n  \\left( \\frac{ \\tan(\\lambda u) }{ \\tanh(\\lambda u) } \\right)\n  \\frac{\\partial}{\\partial x} + (x - \\mathsf{x}) \\left( \\frac{ \\tanh(\\lambda u) }{\n  \\tan(\\lambda u) } \\right) \\frac{\\partial}{\\partial y}\n  \\nonumber\n  \\\\\n  \\qquad \\qquad ~  + \\lambda ( x -\\mathsf{x})\n  (y - \\mathsf{y}) \\left( \\frac{ \\tanh(\\lambda u) - \\tan (\\lambda u) }{ \\tan(\\lambda u) \\tanh(\\lambda u) } \\right) \\frac{\\partial}{\\partial\n  v} .\n\\end{eqnarray}\nThis clearly reduces to its expected form when $u \\rightarrow\n\\mathsf{u}$. The degree to which it succeeds in being a genuine\nKilling field may be estimated by noting that\n\\begin{equation}\n    \\fl \\qquad |\\mathcal{L}_{\\psi_{\\mathsf{xy}}} g_{ab}|^2 = \\left( \\frac{\\cos(2 \\lambda u) + \\cosh(2 \\lambda u) -2 }{\\sqrt{2} \\sin(\\lambda u) \\sinh(\\lambda\n    u)} \\right)^2 \\simeq \\frac{8}{9} (\\lambda u)^4 + \\Or(u^8)\n    \\label{LieEstimate1}\n\\end{equation}\nwhen $\\mathsf{u}=0$. The quadratic growth estimate here is typically\nquite good even near the coordinate singularities. There is little\nqualitative change in the nature of this expression if $\\mathsf{u}\n\\neq 0$. More interestingly, the gravitational current\n\\eqref{GravCurrent} associated with $\\psi^a_{\\mathsf{xy}}$ always\nvanishes. This suggests that an observer would not be compelled to\nascribe any $xy$ component of angular momentum to gravitational\nwaves with the given form. $H^{a}{}_{[\\mathsf{u}}\n\\sigma_{\\mathsf{v}]}$ has similar properties. It satisfies an\nequation almost identical to \\eqref{LieEstimate1}, and the\ngravitational current generated by it always vanishes.\n\nMore interesting are the remaining two Killing-type Jacobi fields\n$\\psi_{\\mathsf{xu}}^a = 2 H^{a}{}_{[\\mathsf{x}}\n\\sigma_{\\mathsf{u}]}$ and $\\psi_{\\mathsf{yu}}^a = 2\nH^{a}{}_{[\\mathsf{y}} \\sigma_{\\mathsf{u}]}$. Specializing again to\nthe case $\\mathsf{u}=0$,\n\\begin{eqnarray}\n  \\fl \\quad |\\mathcal{L}_{\\psi_{\\mathsf{xu}}} g_{ab}|^2 \\simeq |\\mathcal{L}_{\\psi_{\\mathsf{yu}}} g_{ab}|^2 \\simeq \\frac{2}{3}\n  \\lambda^4 u^2 \\Big(\n  [ (x - \\mathsf{x})^2+(y - \\mathsf{y})^2] + \\frac{2}{3} (v-\\mathsf{v}) u \\Big) + \\Or(u^4) .\n\\end{eqnarray}\nUnlike the expansion in \\eqref{LieEstimate1}, this approximation\nfails long before $|\\lambda u| \\rightarrow \\pi\/2$. In general, the\ntwo magnitudes on the left-hand side have distinct behaviors that\nstrongly depend on $\\mathsf{u}$. Oscillations generically arise as\n$u$ is varied, for example. See Fig. 1.\n\n\\begin{figure}\n    \\begin{center}\n\n    \\includegraphics[width=0.8\\textwidth]{LieEx.eps}\n\n    \\caption{Plots of $|\\mathcal{L}_{\\psi_{\\mathsf{xu}}} g_{ab}|^2$ for\n    a gravitational plane wave described by \\eqref{ppGeneral} and\n    \\eqref{GravWave}. The origin is assumed to be at $\\mathsf{u}=0$.\n    Both solid curves assume that $v-\\mathsf{v} = 0$. The dashed ones\n    use $\\lambda(v-\\mathsf{v})=1\/2$ instead. Both thicker curves set\n    $x-\\mathsf{x}=0$ and $\\lambda(y-\\mathsf{y})=1\/4$. The thinner ones use\n    $\\lambda(x-\\mathsf{x})=1$ and $y-\\mathsf{y}=0$. Plots for\n    $|\\mathcal{L}_{\\psi_{\\mathsf{yu}}} g_{ab}|^2$ look very similar\n    unless $\\mathsf{u} \\neq 0$.}\n    \\end{center}\n\\end{figure}\n\nThere are gravitational currents associated with both\n$\\psi^{a}_{\\mathsf{xu}}$ and $\\psi^{a}_{\\mathsf{yu}}$. Applying\n\\eqref{GravCurrent} and expanding near $\\mathsf{u}=0$,\n\\begin{equation}\n  j^a_{\\psi_{\\mathsf{xu}}} = \\frac{\\lambda^6}{180 \\pi} u^4\n  \\left[ u \\frac{\\partial}{\\partial x} + (x - \\mathsf{x}) \\frac{\\partial}{\\partial v} \\right] + \\Or (u^6) .\n\\end{equation}\nSimilarly,\n\\begin{equation}\n  j^a_{\\psi_{\\mathsf{yu}}} = - \\frac{\\lambda^6}{180 \\pi} u^4\n  \\left[ u \\frac{\\partial}{\\partial y} + (y - \\mathsf{y}) \\frac{\\partial}{\\partial v} \\right] + \\Or(u^6).\n\\end{equation}\nThese expansions are qualitatively accurate throughout the region of\ninterest. It is now clear from \\eqref{PStar2} and\n\\eqref{KomarMoment} that the only non-vanishing gravitational\nmomenta (associated with Killing-type GACs) are\n\\begin{equation}\n  S^*_{\\mathsf{x u}} = \\int_\\Sigma j^a_{\\psi_{xu}} \\rmd S_a ; \\qquad  S^*_{\\mathsf{y u}}\n  = \\int_\\Sigma j^a_{\\psi_{yu}} \\rmd S_a .\n\\end{equation}\nThe rates at which these quantities change depends on the relevant\nfluxes through $\\partial \\Sigma$. Regardless, the magnitude of the\nangular momentum tensor always vanishes. Intuitively, these\nstatements might be taken to mean that the gravitational wave has a\n``mass dipole moment'' equal to its one non-vanishing component of\nordinary angular momentum.\n\nThe results here could be straightforwardly extended to much more\ngeneral \\textit{pp}-wave (and other) spacetimes. The most\ninteresting point is perhaps the calculation of explicit\ngravitational currents in a vacuum spacetime. In these cases, the\ngeneral results obtained in Sect. \\ref{Sect:ExtendedMatter} only\nstate that $j^a$ will decrease no slower than $X^3$ as $X\n\\rightarrow 0$. The example here scales like $X^5$. Although this\nconclusion would probably not be preserved in more complicated\nspacetimes, it shows that momenta not arising from stress-energy\ntensors can sometimes be ignored in remarkably large regions.\n\n\\section{Conclusions}\n\nTwo different notions of approximate affine collineations have been\nintroduced. One has the physical interpretation of capturing\nsymmetry principles in a normal neighborhood of a point, while the\nother is adapted to the measurements of a particular observer. Flows\ngenerated by both of these objects leave $\\sigma^\\sfa(x,\\gamma)$\ninvariant. This has the simple interpretation that Jacobi fields\npreserve Riemann normal coordinates. GACs do the same for the\nspatial components of Fermi normal coordinate systems. These objects\nalways exist, and each forms a 20-dimensional group. Individual\nelements may be interpreted using the values of the field and its\nfirst derivatives at the appropriate base point. The only caveat to\nthis is that a GAC which might initially appear to be purely\ntranslational could slowly acquire some rotational and boost-type\ncomponents. This mixing is essential in order to ensure that the\nfields nearly satisfy \\eqref{AffineDef} near the observer's\nworldline.\n\nThe relevance of these definitions ultimately lies in their\napplications. The approximate symmetries introduced here have been\nused to write down analogs of the typical conservation laws applying\nto geodesics in spacetimes admitting affine collineations. Some of\nthe resulting parameters are exact constants of motion, while others\nare only expected to vary slowly near the preferred point or\nobserver. Regardless, they may be used to classify and derive\ngeodesics in certain regions. Similar results have also been\ndiscussed in connection with extended matter distributions. This led\nto natural notions for the linear and angular momenta of a spacetime\nvolume as viewed in a particular frame. There seems to be some\ndisagreement with other quasilocal notions of gravitational momenta,\nso it is not clear how the definition here should be interpreted. It\nis unknown if it has any positivity or related characteristics.\n\nConcepts discussed here might also be applied to simplify\nperturbation theory off of some background geometry possessing an\nexact affine collineation. It could be useful, for example, to\nuniquely construct GACs with respect to a center-of-mass worldline\nthat coincides with exact timelike or axial Killing fields in the\nunperturbed geometry. Center-of-mass trajectories might also be\nestimated using notions of approximate stationarity. More\nconcretely, an analysis of the quantities $\\ItP_\\xi$ defined in\n\\eqref{PDef} can be shown to provide significant insights into the\neffects of self-forces and self-torques on isolated bodies. Details\nare presented elsewhere \\cite{HarteFuture}.\n\n\\ack\n\nI am grateful for many helpful discussions and comments from Robert\nWald and Samuel Gralla. This work was supported by NSF grant\nPHY04-56619 to the University of Chicago.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{#1} #2}\n\t\\vspace{-2pt}\n}\n\n\n\\title{Fair Division of Time: Multi-layered Cake Cutting}\n\n\\author{\nHadi Hosseini$^1$\\and\nAyumi Igarashi$^2$\\and\nAndrew Searns$^1$\n\\affiliations\n$^1$Rochester Institute of Technology, US\\\\\nNational Institute of Informatics, Japan\n\\emails\nhhvcs@rit.edu,\nayumi\\_igarashi@nii.ac.jp,\nabs2157@rit.edu\n}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nWe initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each time interval. Several real-life applications exhibit such restrictions on overlapping pieces; for example, assigning time intervals over multiple facilities and resources or assigning shifts to medical professionals. We investigate the existence and computation of envy-free and proportional allocations. We show that envy-free allocations that are both feasible and contiguous are guaranteed to exist for up to three agents with two types of preferences, when the number of layers is two. We also show that envy-free feasible allocations where each agent receives a polynomially bounded number of intervals exist for any number of agents and layers under mild conditions on agents' preferences. We further devise an algorithm for computing proportional allocations for any number of agents and layers. \n\\end{abstract}\n\n\\section{Introduction}\nConsider a group of students who wish to use multiple college facilities such as a conference room and an exercise room over different periods of time.\nEach student has a preference over what facility to use at different time of the day: Alice prefers to set her meetings in the morning and exercise in the afternoon, whereas Bob prefers to start the day with exercising for a couple of hours and meet with his teammates in the conference room for the rest of the day.\n\nThe fair division literature has extensively studied the problem of dividing a heterogeneous divisible resource (aka a \\textit{cake}) among several agents who may have different preference over the various pieces of the cake~\\citep{Steinhaus48,Robertson98,Brams06}.\nThese studies have resulted in a plethora of axiomatic and existence results~\\citep{barbanel2005geometry,moulin2004fair} as well as computational solutions~\\citep{procaccia2013cake,aziz2016discretefour} under a variety of assumptions, and were successfully implemented in practice (see~\\citep{procaccia_moulin_2016,Brams96} for an overview).\nIn the case of Alice and Bob, each facility represents a layer of the cake in a \\textit{multi-layered cake cutting} problem, and the question is how to allocate the time intervals (usage right) of the facilities according to their preferences in a fair manner.\n\nOne naive approach is to treat each cake independently and solve the problem through well-established cake-cutting techniques by performing a fair division on each layer separately.\nHowever, this approach has major drawbacks: First, the final outcome, although fair on each layer, may not necessarily be fair overall. Second, the allocation may not be feasible, i.e., it may assign two overlapping pieces (time intervals) to a single agent. \nIn our example, Alice cannot simultaneously utilize the exercise room and the conference room at the same time if she receives overlapping  intervals.\nSeveral other application domains exhibit similar structures over resources: assigning nurses to various wards and shifts, doctors to operation rooms, and research equipment to groups, to name a few.\n\nIn multi-layared cake cutting, each layer represents a divisible resource. Each agent has additive preferences over every disjoint (non-overlapping) intervals. A division of a multi-layered cake is \\emph{feasible} if no agent's share contains overlapping intervals, and is contiguous if each allocated piece of a layer is contiguous. \nThere has been some recent work on dividing multiple cakes among agents~\\citep{cloutier2010two,lebert2013envy}. Yet, none of the previous work considered the division of multiple resources under feasibility and contiguity constraints. \nIn this paper, we thus ask the following question: \n\\begin{quote}\n\\textit{What fairness guarantees can be achieved under feasibility and contiguity constraints for various number of agents and layers?}\n\\end{quote}\n\n\\subsection{Our Results}\nWe initiate the study of the multi-layered cake cutting problem for allocating divisible resources, under contiguity and feasibility requirements. Our focus is on two fairness notions, \\textit{envy-freeness} and \\textit{proportionality}. Envy-freeness (EF) requires that each agent believes no other agent's share is better than its share of the cake. Proportionality (Prop) among $n$ agents requires that each agent receives a share that is valued at least $\\frac{1}{n}$ of the value of the entire cake.\nFor efficiency, we consider \\textit{complete} divisions with no leftover pieces.\n\nFocusing on envy-free divisions, we show the existence of envy-free and complete allocations that are both feasible and contiguous for two-layered cakes and up to three agents with at most two types of preferences. These cases are particularly appealing since many applications often deal with dividing a small number of resources among few agents (e.g. assigning meeting rooms). Turning our attention to the case when the contiguity requirement is dropped, we then show that envy-free feasible allocations exist for any number $n$ of agents and any number $m$ of layers with $m \\leq n$, under mild conditions on agents' preferences.\nWe further show that proportional complete allocations that are both feasible and contiguous exist when the number of layers isa power of two. \nSubsequently, we show that although this result cannot be immediately extended to any number of agents and layers, a proportional complete allocation that is feasible exists when the number of layers is at most the number of agents, and can be computed efficiently.\n\n\n\\begin{table}[t]\n\\small \n\\centering\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\nAgents ($n$)            & Layers ($m$)                         & EF & Prop &  \\\\ \\midrule\n2               & 2                         &  \\cmark (Thm. \\ref{thm:EF:two} ) &   \\cmark (Thm. \\ref{thm:exponential})    &  \\\\\n3               & 2                         &  \\cmark (Thm. \\ref{thm:EF:any}$^\\diamondsuit\\dagger$) &    \\cmark (Thm. \\ref{thm:exponential})  &  \\\\\nany $n\\geq m$              & $2^{a}$,   \\scriptsize{$a\\in\\mathbb{Z}_{+}$}   &   \\cmark (Thm. \\ref{thm:EF:any}$^\\diamondsuit\\dagger$)  &   \\cmark (Thm. \\ref{thm:exponential})   &  \\\\ \nany $n\\geq m$  & any $m$ &  \\cmark (Thm. \\ref{thm:EF:any}$^\\diamondsuit\\dagger$)  &   \\cmark (Thm. \\ref{thm:prop:feasible:any}$^\\diamondsuit$)   &  \\\\ \\bottomrule\n\\end{tabular}\n\\caption{The overview of our results. $\\dagger$ assumes continuity of value density functions. $\\diamondsuit$ indicates that existence holds without contiguity requirement. Note that when $m> n$, no complete and feasible (non-overlapping) solution exists.}\n\\end{table}\n\n\\subsection{Related Work}\nIn recent years, cake cutting has received significant attention in artificial intelligence and economics as a metaphor for algorithmic approaches in achieving fairness in allocation of resources \\citep{procaccia2013cake,branzei2019communication,kurokawa2013cut,aziz2016discrete}. \nRecent studies have focused on the fair division of resources when agents have requirements over multiple resources that must be simultaneously allocated in order to carry out certain tasks (e.g. CPU and RAM) \\citep{Ghodsi:2011:DRF:1972457.1972490,gutman2012fair,parkes2015beyond}. \nThe most relevant work to ours is the envy-free multi-cake fair division that considers dividing multiple cakes among agents with linked preferences over the cakes. Here, agents can simultaneously benefit from all allocated pieces with no constraints. They show that envy-free divisions with only few cuts exist for two agents and many cakes, as well as three agents and two cakes~\\citep{cloutier2010two,lebert2013envy,nyman2020fair}. In contrast, a multi-layered cake cutting requires non-overlapping pieces. Thus, \\cite{cloutier2010two}'s generalized envy-freeness notion on multiple cakes does not immediately imply envy-freeness in our setting and no longer induces a feasible division.\n\n\\section{Our Model}\nOur setting includes a set of {\\em agents} denoted by $N=[n]$, a set of {\\em layers} denoted by $L=[m]$, where for a natural number $s \\in \\mathbb{N}$, $[s]=\\{1,2,\\ldots,s\\}$.\nGiven two real numbers $x,y \\in \\mathbb{R}$, we write $[x,y]=\\{\\, z \\in \\mathbb{R} \\mid x \\le z \\le y \\,\\}$ to denote an interval. We denote by $\\mathbb{R}_{+}$ (respectively $\\mathbb{Z}_{+}$) the set of non-negative reals (respectively, integers) including $0$. \nA {\\em piece} of cake is a finite set of disjoint subintervals of $[0,1]$. We say that a subinterval of $[0,1]$ is a {\\em contiguous piece} of cake. An {\\em $m$-layered cake} is denoted by $\\mathcal{C}=(C_j)_{j \\in L}$ where $C_j \\subseteq [0,1]$ is a contiguous piece for $j \\in L$. We refer to each $j \\in L$ as $j$-th {\\em layer} and $C_j$ as $j$-th {\\em layered cake}. \n\nEach agent $i$ is endowed with a non-negative {\\em integrable density function} $v_{ij}:C_j \\rightarrow \\mathbb{R}_{+}$. For a given piece of cake $X$ of $j$-th layer, $V_{ij}(X)$ denotes the value assigned to it by agent $i$, i.e., $V_{ij}(X)=\\sum_{I \\in X}\\int_{x \\in I} v_{ij}(x) dx$. These functions are assumed to be {\\em normalized} over layers: for each $i \\in N$, $\\sum_{j \\in L}V_{ij}(C_j)=1$. A {\\em layered piece} is a sequence $\\mathcal{X}=(X_j)_{j \\in L}$ of pieces of each layer $j \\in L$; a layered piece is said to be {\\em contiguous} if each $X_j$ is a contiguous piece of each layer. \nWe assume valuation functions are \\emph{additive} on layers and write $V_{i}(\\mathcal{X})=\\sum_{j \\in L}V_{ij}(X_{j})$.\n\nA layered contiguous piece is said to be {\\em non-overlapping} if no two pieces from different layers overlap, i.e, for any pair of distinct layers $j,j' \\in L$ and for any $I \\in X_j$ and $I' \\in X_{j'}$, $I \\cap I'=\\emptyset$. For two layered pieces $\\mathcal{X}$ and $\\mathcal{X}'$, we say that agent $i$ {\\em weakly prefers} $\\mathcal{X}$ to $\\mathcal{X}'$ if $V_i(\\mathcal{X}) \\geq V_i(\\mathcal{X}')$. \n\nA {\\em multi-allocation} $\\mathcal{A}=(\\mathcal{A}_1,\\mathcal{A}_2,\\ldots,\\mathcal{A}_n)$ is a partition of the $m$-layered cake $\\mathcal{C}$ where each $\\mathcal{A}_i=(A_{ij})_{j \\in L}$ is a layered piece of the cake allocated to agent $i$; we refer to each $\\mathcal{A}_i$ as a {\\em bundle} of $i$. For a multi-allocation $\\mathcal{A}$ and $i \\in N$, we write $V_{i}(\\mathcal{A}_i)=\\sum_{j \\in L}V_{ij}(A_{ij})$ to denote the value of agent $i$ for $\\mathcal{A}_i$. A {\\em multi-allocation} $\\mathcal{A}$ is said to be \n\\begin{itemize}\n\\item {\\em contiguous} if each $\\mathcal{A}_i$ for $i \\in N$ is contiguous; \n\\item {\\em feasible} if each $\\mathcal{A}_i$ for $i \\in N$ is non-overlapping.\n\\end{itemize}\n\nWe focus on \\emph{complete} multi-allocations where the entire cake must be allocated.\nNotice that some layers may be disjoint (see Figure \\ref{fig:server}), and the number of agents must exceed the number of layers, i.e. $n \\geq m$; otherwise the multi-allocation will contain overlapping pieces. We illustrate our model in the following example. \n\n\\begin{example}[Resource sharing]\nSuppose that there are three meeting rooms $r_1$, $r_2$, and $r_3$ with different capacities, and three researchers Alice, Bob, and Charlie. The first room is available all day, the second and the third rooms are only available in the morning and late afternoon, respectively (see Fig.~\\ref{fig:server}). Each researcher has a preference over the access time to the shared rooms. For example, Alice wants to have a group meeting in the larger room in the morning and then have an individual meeting in the smaller one in the afternoon. \n\\end{example}\n\n\\begin{figure}[hbt]\n\\centering\n\\begin{tikzpicture}[scale=0.5, transform shape]\n\\draw[fill=red!10, thick] (0,1.3) rectangle (10,2.3);\n\\draw[fill=blue!10, thick] (0,0) rectangle (4,1); \n\\draw[thick] (5,-1.3) rectangle (10,-0.3);\n\n\\node at (-0.5,1.8) {\\huge $r_1$};\n\\node at (-0.5,0.5) {\\huge $r_2$};\n\\node at (-0.5,-0.8) {\\huge $r_3$};\n\n\\draw[thick,->] (-0.5,-2) -- (10.5,-2);\n\\node at (11.4,-2) {\\Huge time};\n\n\\end{tikzpicture}\n\\caption{Example of a multi-layered cake. There are three meeting rooms $r_1$, $r_2$, and $r_3$ with different capacities, shared among several research groups.}\n\\label{fig:server}\n\\end{figure}\n\n\\paragraph{Fairness.}\nA multi-allocation is said to be {\\em envy-free} if no agent {\\em envies} the others, i.e., $V_{i}(\\mathcal{A}_i) \\ge V_{i}(\\mathcal{A}_{i'})$ for any pair of agents $i,i' \\in N$. A multi-allocation is said to be {\\em proportional} if each agent gets his {\\em proportional fair share}, i.e., $V_{i}(\\mathcal{A}_i) \\ge \\frac{1}{n}$ for any $i \\in N$. The following implication, which is well-known for the standard setting, holds in our setting as well.\n\n\\begin{lemma}\\label{lem:propEF}\nAn envy-free complete multi-allocation satisfies proportionality. \n\\end{lemma}\n\\begin{proof}\nConsider an envy-free complete multi-allocation $\\mathcal{A}_i=(A_{ij})_{j \\in L}$ and an agent $i \\in N$. By envy-freeness, we have that $V_{i}(\\mathcal{A}_i) \\geq V_{i}(\\mathcal{A}_j)$ for any $j \\in N$. Summing over $j \\in N$, we get $V_{i}(\\mathcal{A}_i) \\geq \\frac{1}{n}\\sum_{j \\in N}V_{i}(\\mathcal{A}_j)=\\frac{1}{n}$ by additivity. \n\\end{proof}\n\n\\paragraph{The $m$-layered cuts.}\nIn order to cut the layered cake while satisfying the non-overlapping constraint, we define a particular approach for partitioning the entire cake into diagonal pieces. Consider the $m$-layered cake $\\mathcal{C}$ where $m$ is an even number. For each point $x$ of the interval $[0,1]$, we define\n\\begin{itemize}\n\\item $LR(x,\\mathcal{C})=(\\bigcup^{\\frac{m}{2}}_{j=1}C_j \\cap [0,x]) \\cup (\\bigcup^{m}_{j=\\frac{m}{2}+1}C_j \\cap [x,1])$; \n\\item $RL(x,\\mathcal{C})=(\\bigcup^{\\frac{m}{2}}_{j=1}C_j \\cap [x,1]) \\cup (\\bigcup^{m}_{j=\\frac{m}{2}+1}C_j \\cap [0,x])$. \n\\end{itemize}\n$LR(x,\\mathcal{C})$ consists of the top-half subintervals of points left of $x$ and the lower-half subintervals of points right of $x$; similarly, $RL(x,\\mathcal{C})$ consists of the top-half subintervals of points right of $x$ and the lower-half subintervals of points left of $x$ (Fig. \\ref{fig:LR:RL}). We abuse the notation and write $LR(x)=LR(x,\\mathcal{C})$ and $RL(x)=RL(x,\\mathcal{C})$ if $\\mathcal{C}$ is clear from the context. \n\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}[scale=0.5, transform shape]\n\n\\draw[fill=blue!10, thick] (0,0) rectangle (6,2); \n\\draw[thick] (0,-2) rectangle (6,0);\n\\draw[dotted] (0,1) -- (6,1);\n\\draw[dotted] (0,-1) -- (6,-1);\n\n\\node[thick] at (0,2.5) {\\Large $x=0$};\n\\node at (3,-1) {\\huge $LR(x)$};\n\\node at (3,1) {\\huge $RL(x)$};\n\n\\node at (-0.8,1.5) {\\Large $j=1$};\n\\node at (-0.8,0.5) {\\Large $j=2$};\n\\node at (-0.8,-0.5) {\\Large $j=3$};\n\\node at (-0.8,-1.5) {\\Large $j=4$};\n\n\\begin{scope}[xshift=8cm]\n\n\\draw[thick] (0,0) rectangle (2,2);\n\\draw[fill=blue!10, thick] (0,-2) rectangle (2,0); \n\\draw[fill=blue!10, thick] (2,0) rectangle (6,2); \n\\draw[thick] (2,-2) rectangle (6,0);\n\\draw[dotted] (0,1) -- (6,1);\n\\draw[dotted] (0,-1) -- (6,-1);\n\n\\node[thick] at (2.0,2.5) {\\bf \\Large $x=\\frac{2}{5}$};\n\\node at (1,1) {\\huge $LR(x)$};\n\\node at (4,-1) {\\huge $LR(x)$};\n\\node at (4,1) {\\huge $RL(x)$};\n\\node at (1,-1) {\\huge $RL(x)$};\n\n\\node at (-0.8,1.5) {\\Large $j=1$};\n\\node at (-0.8,0.5) {\\Large $j=2$};\n\\node at (-0.8,-0.5) {\\Large $j=3$};\n\\node at (-0.8,-1.5) {\\Large $j=4$};\n\n\\end{scope}\n \n\\end{tikzpicture}\n\\caption{Examples of the partitions induced by $x=0$ and $x=\\frac{2}{5}$ for a {\\bf four-layered} cake.}\n\\label{fig:LR:RL}\n\\end{figure}\n\n\n\\paragraph{Computational model.}\nFollowing the standard {\\em Robertson-Webb Model} \\citep{Robertson98}, we introduce two types of queries: those for a cake on each layer (called a {\\em short knife}) and those for the entire cake (called a {\\em long knife}). \n\n\\paragraph{Short knife.} Short eval query: given an interval $[x,y]$ of the $j$-th layered cake $C_j$, $eval_j(i,x,y)$ asks agent $i$ for its value $[x,y]$, i.e., $V_{ij}([x,y])$.\n Short cut query: given a point $x$ and $r \\in [0,1]$, $cut_j(i,x,r)$ asks agent $i$ for the minimum point $y$ such that $V_{ij}([x,y])=r$.\n\n\\paragraph{Long knife.} Long eval query: given a point $x$, $eval(i,x)$ asks agent $i$ for its value $LR(x)$, i.e., $V_{i}(LR(x))$. \nLong cut query: given $r \\in [0,1]$, $cut(i,r)$ asks agent $i$ for the minimum point $x$ such that $V_{i}(LR(x))=r$ if such point $x$ exists.\n\n\n\\section{Existence of a switching point}\nWe start by showing the existence of a point $x$ that equally divides the entire cake into two pairs of diagonal pieces, both for the individuals and for the majority; these will serve as a fundamental property in our problem. \nWe say that $x \\in [0,1]$ is a {\\em switching point} for agent $i$ over a layered cake $\\mathcal{C}$ if $V_i(LR(x))=V_i(RL(x))$. \n\n\\begin{lemma}\\label{lem:switching}\nSuppose that the number $m$ of layers is even. Take any $i \\in N$. Let $r\\in \\mathbb{R}$ be such that $($i$)$ $V_i(LR(0)) \\geq r$ and $V_i(RL(0)) \\leq r$, or $($ii$)$ $V_i(LR(0)) \\leq r$ and $V_i(RL(0)) \\geq r$. \nThen, there exists a point $x \\in [0,1]$ such that $i$ values $LR(x)$ exactly at $r$, i.e. $V_i(LR(x)) = r$. In particular, a switching point for $i$ always exists. \n\\end{lemma}\n\\begin{proof}\nSuppose without loss of generality that $V_i(LR(0)) \\geq r$ and $V_i(RL(0)) \\leq r$. Consider the function $f(x)=V_i(LR(x))$ for $x \\in [0,1]$. Recall that $f(x)$ is a continuous function written as the sum of continuous functions:\n$\nf(x)=\\sum^{\\frac{m}{2}}_{j=1} V_{ij}(C_j \\cap [0,x])+ \\sum^{m}_{j=\\frac{m}{2}+1} V_{ij}(C_j \\cap [x,1]). \n$\nSince $f(0) \\geq r$ and $f(1) \\leq r$, there is a point $x \\in [0,1]$ with $f(x)=r$ by the intermediate value theorem, which proves the claim. Further, by taking $r=\\frac{1}{2}$, the point $x$ where $V_i(LR(x))=\\frac{1}{2}$ is a switching point for agent $i$. \n\\end{proof}\n\nWe will generalize the notion of a switching point from the individual level to the majority. For layered contiguous pieces $\\mathcal{I}$ and $\\mathcal{I}'$, we say that the majority weakly prefer $\\mathcal{I}$ to $\\mathcal{I}'$ (denoted by $\\mathcal{I} \\mathop{\\stackrel{m}{\\succeq}} \\mathcal{I}'$) if there exists $S \\subseteq N$ such that $|S| \\geq \\ceil{\\frac{n}{2}}$ and each $i \\in S$ weakly prefers $\\mathcal{I}$ to $\\mathcal{I}'$. We say that $x \\in [0,1]$ is a {\\em majority switching point} over $\\mathcal{C}$ if $LR(x) \\mathop{\\stackrel{m}{\\succeq}} RL(x)$ and $RL(x) \\mathop{\\stackrel{m}{\\succeq}} LR(x)$.\nThe following lemma guarantees the existence of a majority switching point, for any even number of layers and any number of agents. \n\n\\begin{lemma}\\label{lem:majority:switching}\nSuppose that the number of layers, $m$, is even. Then, there exists a majority switching point for any number $n \\geq m$ of agents. \n\\end{lemma}\n\\begin{proof}\nSuppose without loss of generality that the majority of agents weakly prefer $LR(0)$ to $RL(0)$. Since $LR(0)=RL(1)$ and $RL(0)=LR(1)$, this means that by the time when the long knife reaches the right-most point, i.e., $x=1$, the majority preference switches. \n\nFormally, consider the following set of points $x \\in [0,1]$ where the majority weakly prefer $LR(x)$ to $RL(x)$:  \n\\[\nM:= \\{\\, x \\in [0,1] \\mid LR(x) \\mathop{\\stackrel{m}{\\succeq}} RL(x)\\,\\}. \n\\]\nWe will first show that $M$ is a compact set. Clearly, $M$ is bounded. To show that $M$ is closed, consider an infinite sequence as follows $X=\\{x_k\\}_{k=1,2, \\ldots} \\subseteq M$ that converges to $x^*$. For each $k=1,2,\\ldots$, we denote by $S_k$ the set of agents who weakly prefer $LR(x_k)$ to $RL(x_k)$; by definition, $|S_k| \\geq \\ceil{\\frac{n}{2}}$. Since there are finitely many subsets of agents, there is one subset $S_k \\subseteqq N$ that appears infinitely often; let $S^*$ be such subset and $\\{x^*_k\\}_{k=1,2, \\ldots}$ be an infinite sub-sequence of $X$ such that for each $k$, each agent in $S^*$ weakly prefers $LR(x^*_k)$ to $RL(x^*_k)$. Since the valuations $V_i$ for $i \\in S^*$ are continuous, each agent $i \\in S^*$ weakly prefers $LR(x^*)$ to $RL(x^*)$ at the limit $x^*$, which implies that $x^* \\in M$  and hence $M$ is closed. Now since $M$ is a compact set, the supremum $t^*=\\sup M$  belongs to $M$. By the maximality of $t^*$, at least $\\ceil{\\frac{n}{2}}$ agents weakly prefer $RL(t^*)$ to $LR(t^*)$. Since $t^* \\in M$, at least $\\ceil{\\frac{n}{2}}$ agents weakly prefer $LR(t^*)$ to $RL(t^*)$ as well. Thus, $t^*$ corresponds to a majority switching point. \n\\end{proof}\n\n\\section{Envy-free multi-layered cake cutting}\nFirst, we will look into the problem of obtaining complete envy-free multi-allocations, while satisfying non-overlapping constraints. When there is only one layer, it is known that an envy-free contiguous allocation exists for any number of agents under mild assumptions on agents' preferences \\citep{Stromquist1980,Su1999}.\nGiven the contiguity and feasibility constraints, the question is whether it is possible to guarantee an envy-free division in the multi-layered cake-cutting model. \n\n\\subsection{Two agents and two layers}\nWe answer the above question positively for a simple, yet important, case of two agents and two layers. The standard protocol that achieves envy-freeness for two agents is known as the {\\em cut-and-choose} protocol: Alice divides the entire cake into two pieces of equal value. Bob selects his preferred piece over the two pieces, leaving the remainder for Alice. \n\nWe extend this protocol to the multi-layered cake cutting using the notion of a switching point. Alice first divides the layered cake into two {\\em diagonal pieces}: one that includes the top left and lower right parts and another that includes the top right and lower left parts of the cake. Our version of the cut-and-choose protocol is specified as follows:\n\n\\vspace{5pt}\n\n\\noindent\\fbox{%\n\t\\parbox{0.985\\linewidth}\n\t   {%\n\t\t\\textbf{Cut-and-choose protocol for $n=2$ agents} over a two-layered cake $\\mathcal{C}$: \\\\\n\t\t\\textit{Step 1.} Alice selects her switching point $x$ over $\\mathcal{C}$.\\\\\n\t\n\t\t\\textit{Step 2.} Bob chooses a weakly preferred layered contiguous piece among $LR(x)$ and $RL(x)$. \\\\\n\t\t\\textit{Step 3.} Alice receives the remaining piece.\n}%\n}\n\\begin{figure}[htb]\n\\centering\n\\begin{tikzpicture}[scale=0.7, transform shape]\n\\draw[thick] (0,0) rectangle (3,1);\n\\draw[fill=blue!10, thick] (3,0) rectangle (10,1); \n\\draw[fill=blue!10, thick] (0,-1) rectangle (3,0); \n\\draw[thick] (3,-1) rectangle (10,0); \n\\node at (3.0,1.3) {\\large $x$};\n\\node at (1.5,0.5) {\\large $LR(x)$};\n\\node at (6.5,-0.5) {\\large $LR(x)$};\n\\node at (1.5,-0.5) {\\large $RL(x)$};\n\\node at (6.5,0.5) {\\large $RL(x)$};\n\\end{tikzpicture}\n\\caption{Cut-and-Choose for two-layered cake}\n\\label{fig:EF:two}\n\\end{figure}\n\n\n\\begin{theorem}\\label{thm:EF:two}\nThe cut-and-choose protocol yields a complete envy-free multi-allocation that is feasible and contiguous for two agents and a two-layered cake using $O(1)$ number of long eval and cut queries.\n\\end{theorem}\n\\begin{proof}\nIt is immediate to see that the protocol returns a complete multi-allocation where each agent is assigned to a non-overlapping layered contiguous piece. The resulting allocation satisfies envy-freeness: Bob does not envy Alice since he chooses a preferred piece among $LR(x)$ and $RL(x)$. Alice does not envy Bob by the definition of a switching point. \n\\end{proof}\n\nAs we noted in Section $2$, the existence result for two agents does not extend beyond two layers: if there are at least three layers, there is no feasible multi-allocation that completely allocates the cake to two agents. \n\n\\subsection{Three agents and two layers}\nWe move on to the case of three agents and two layers. We will design a variant of Stromquist's protocol that achieves envy-freeness for one-layered cake \\citep{Stromquist1980}: The referee moves two knives: a short knife and a long knife. The short knife points to the point $y$ and moves from left to right over the top layer, gradually increasing the left-most top piece (denoted by $Y$). The long knife keeps pointing to the point $x$, which can partition the remaining cake, denoted by $\\mathcal{C}^{-y}$, into two diagonal pieces $LR(x)$ and $RL(x)$ in an envy-free manner. Each agent shouts when the left-most top piece $Y$ becomes at least as highly valuable as the preferred piece among $LR(x)$ and $RL(x)$. Some agent, say $s$, shouts eventually (before the left-most top piece becomes the top layer), assuming that there is at least one agent who weakly prefers the top layer to the bottom layer. We note that $x$ may be positioned left to $y$; see Figure \\ref{fig:EF:three} for some possibilities of the long knife's locations. \n\nWe will show that the above protocol works, for a special case when there are at most two types of preferences: In such cases, the majority switching points coincide with the switching points of an agent with the majority preference.\n\n\\begin{lemma}\\label{lem:switching:identical}\nSuppose that $m=2$, $n=3$, and there are two different agents $i,j \\in N$ with the same valuation $V$. Then, $x$ is a majority switching point over $\\mathcal{C}$ if and only if $x$ is a switching point for $i$. \n\\end{lemma}\n\\begin{proof}\nSuppose that agents $i,j \\in N$ have the same valuation $V$. Suppose that $x$ is a majority switching point over $\\mathcal{C}$. Then, at least two agents weakly prefer $LR(x)$ to $RL(x)$, meaning that at least one of the two agents $i$ and $j$ weakly prefers $LR(x)$ to $RL(x)$, which means that both agents weakly prefers $LR(x)$ to $RL(x)$ since $i$ and $j$'s valuations are identical. Similarly, both $i$ and $j$ weakly prefer $RL(x)$ to $LR(x)$. Thus, $x$ is a switching point for $i$. \nThe converse direction is immediate. \n\\end{proof}\n\nAn implication of the above lemma is that when performing Stromquist's protocol, one can point out to a switching point of an individual, instead of a majority one. This allows the value of each piece to change continuously. For a given two-layered cake $\\mathcal{C}$, we write $\\mathcal{C}^{-y}=(C^{-y}_1,C_2)$ as a two-layered cake obtained from $\\mathcal{C}$ where the first segment $[0,y]$ of the top layer is removed, i.e., $C^{-y}_1 = C_1 \\setminus [0,y]$. For each majority switching point $x$ over $\\mathcal{C}^{-y}$, we select three different agents $\\ell(x)$, $m(x)$, and $r(x)$ as follows: \n\\begin{itemize}\n\\item $\\ell(x)$ is an agent who weakly prefers $LR(x,\\mathcal{C}^{-y})$ to $RL(x,\\mathcal{C}^{-y})$;\n\\item $m(x)$ is an agent who is indifferent between $LR(x,\\mathcal{C}^{-y})$ and $RL(x,\\mathcal{C}^{-y})$; and \n\\item $r(x)$ and agent who weakly prefers $RL(x,\\mathcal{C}^{-y})$ to $LR(x,\\mathcal{C}^{-y})$. \n\\end{itemize}\n\n\n\\begin{theorem}\\label{thm:twolayers3agents}\nSuppose that $m=2$ and $n=3$. If there are two different agents with the same valuation, an envy-free complete multi-allocation that is feasible and contiguous exists. \n\\end{theorem}\n\\begin{proof}\nAssume w.l.o.g. that at least one agent prefers the top layer over the bottom layer. This means that such agent weakly prefers the top layer to any of the pieces $LR(z,\\mathcal{C}^{-y})$ and $RL(z,\\mathcal{C}^{-y})$ when $y=1$. Suppose that $i \\in N$ is one of the two different agents with the same valuations. We design the following protocol for three agents over a two-layered cake: \n\n\\vspace{3pt}\n\\noindent\\fbox{%\n\t\\parbox{0.985\\linewidth}{%\n\t\t\\textbf{Moving-knife protocol for $n=3$ agents} over a two-layered cake $\\mathcal{C}$: w.l.o.g. assume that at least one agent weakly prefers the top layer $(j=1)$ over the bottom layer $(j=2)$\\\\\n\t\t\\textit{Step 1.} The referee continuously moves a short knife from the left-most point $(y=0)$ to the right-most point $(y=1)$ over the top layer, while continuously moving a long knife pointing to a switching point over $\\mathcal{C}^{-y}$ for $i$. \n        Let $y$ be the position of the short knife and $Y$ be the top layer piece to its left. Let $x$ be the position of the long knife.\\\\\t\t\n\t\t\\textit{Step 2.} The referee stops moving the short knife when some agent $s$ {\\em shouts}, i.e., $Y$ becomes at least as highly valuable as the preferred piece among $LR(x,\\mathcal{C}^{-y})$ and $RL(x,\\mathcal{C}^{-y})$. \\\\\n\t\t\\textit{Step 3.} We allocate the shouter $s$ to the left-most top piece $Y$ and partitions the rest into $LR(x,\\mathcal{C}^{-y})$ and $RL(x,\\mathcal{C}^{-y})$. \n\\begin{itemize}\n\\item If $s=\\ell(x)$, then we allocate $LR(x,\\mathcal{C}^{-y})$ to $m(x)$ and $RL(x,\\mathcal{C}^{-y})$ to $r(x)$. \n\\item If $s=m(x)$, then we allocate $LR(x,\\mathcal{C}^{-y})$ to $\\ell(x)$ and $RL(x,\\mathcal{C}^{-y})$ to $r(x)$. \n\\item If $s=r(x)$, then we allocate $LR(x,\\mathcal{C}^{-y})$ to $\\ell(x)$ and $RL(x,\\mathcal{C}^{-y})$ to $m(x)$. \n\\end{itemize}\n\t}%\n}\n\\vspace{3pt}\n\nBy our assumption, some agent eventually shouts and thus the protocol returns an allocation $\\mathcal{A}$. Clearly, $\\mathcal{A}$ is feasible, contiguous, and complete. Also, it is easy to see that the shouter $s$ who receives a bundle $Y$ does not envy the other two agents. The agents $i \\neq s$ do not envy $s$ because the referee continuously moves both a short and a long knife. Finally, the agents $i \\neq s$ do not envy each other by the definition of a majority switching point and by Lemma \\ref{lem:switching:identical}. \n\\end{proof}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[scale=0.55, transform shape]\n\\draw[fill=red!10, thick] (0,0) rectangle (4,1);\n\\draw[fill=blue!10, thick] (0,-1) rectangle (3,0); \n\\draw[fill=blue!10, thick] (4,0) rectangle (10,1); \n\\draw[thick] (3,-1) rectangle (10,0);\n\n\\node at (2.0,0.5) {$Y$};\n\\node at (1.5,-0.5) {$RL(x)$};\n\\node at (6.5,0.5) {$RL(x)$};\n\\node at (6.0,-0.5) {$LR(x)$};\n\n\\node at (3.0,1.3) {$x$};\n\\node at (4.0,1.3) {$y$};\n\n\\begin{scope}[yshift=-3cm]\n\\draw[fill=red!10, thick] (0,0) rectangle (4,1);\n\\draw[thick] (4,0) rectangle (10,1);\n\\draw[fill=blue!10, thick] (6,0) rectangle (10,1); \n\\draw[fill=blue!10, thick] (0,-1) rectangle (6,0); \n\\draw[thick] (6,-1) rectangle (10,0);\n\n\\node at (2.0,0.5) {$Y$};\n\\node at (6.0,1.3) {$x$};\n\\node at (4.0,1.3) {$y$};\n\n\\node at (3.0,-0.5) {$RL(x)$};\n\\node at (8.0,0.5) {$RL(x)$};\n\\node at (5.0,0.5) {$LR(x)$};\n\\node at (8.0,-0.5) {$LR(x)$};\n\\end{scope}\n \n\\end{tikzpicture}\n\\caption{Moving knife protocol for three agents over a two-layered cake. Note that the position of $x$ may appear before $y$. \n}\n\\label{fig:EF:three}\n\\end{figure}\n\nIn the general case, the existence question of contiguous and feasible envy-free multi-allocations deems to be challenging due to the non-monotonicity of valuations over diagonal pieces.\\footnote{See Section \\ref{sec:discussion} for an extensive discussion.} \nIn the next subsection, we thus turn our attention to the case when the contiguity requirement is relaxed.\n\n\\subsection{Non-connected pieces}\nHaving seen that an envy-free multi-allocation that is both feasible and contiguous exists for a special case, we will consider the case when the contiguity requirement is dropped, namely, agents may receive a collection of sub-intervals of each layer. \nWe will show the existence of an envy-free multi-allocation that is feasible and uses at most poly$(n)$ number of cuts within each layer, assuming that each density function $v_{ij}$ is continuous. In what follows, we will reduce the problem to finding a `perfect' allocation of a one-layered cake. An allocation of a single-layered cake is called {\\em perfect} if each agent values every allocated piece exactly at his proportional fair share $\\frac{1}{n}$. It is known that such allocation consisting of at most poly$(n)$ number of contiguous pieces exists whenever agents' value density functions are continuous \\citep{Alon1987}. It is not surprising that the existence of a perfect allocation implies the existence of an envy-free allocation over a single-layered cake. We show that this result also implies the existence of envy-free allocations over a multi-layered cake. \n\n\\begin{theorem}\\label{thm:EF:any}\nSuppose that $m \\leq n$ and each $v_{i,j}$ for $i \\in N$ and $j \\in L$ is continuous. Then, an envy-free complete feasible multi-allocation $\\mathcal{A}=(\\mathcal{A}_i)_{i \\in N}$ where each piece $A_{ij}$ for agent $i \\in N$ and layer $j \\in L$ contains at most poly$(n)$ number of contiguous pieces exists. \n\\end{theorem}\n\\begin{proof}\nWe assume without loss of generality that each layer $C_j$ is the whole interval $[0,1]$ by just putting zero valuations on the part outside $C_j$. \nNow, we construct an instance of a single-layered cake $I=[0,1]$ as follows. First, create a dummy agent $i_j$ for each agent $i \\in N$ and each layer $j \\in L$. Each dummy agent $i_j$ has a valuation $v'_{i_j}$ determined by agent $i$'s valuation for the $j$-th layered cake, i.e., for each sub-interval $X \\subseteq I$, $v'_{i_j}(X)=v_{i,j}(X)$. \nWe will show that a perfect allocation of the artificial cake among the $mn$ dummy agents induces an envy-free multi-allocation of the original layered cake. \n\nSpecifically, take a perfect allocation $(X_1,X_2,\\ldots,X_{mn})$ of this instance where each $X_t$ for $t \\in [mn]$ contains at most poly$(n)$ number of contiguous pieces, which is guaranteed to exist \\citep{Alon1987}. We then group each consecutive $m$ sequence of pieces together: namely, let $Y_{h}=\\bigcup^{im}_{t=(i-1)m+1}X_t$ for each $h \\in [n]$. By the definition of a perfect allocation, we have\n\\begin{align}\\label{eq}\n&v_{ij}(Y_h)= \\frac{v_{ij}(C_j)}{n},\n\\end{align}\nfor any $h \\in [n]$. Now, we partition each layer into $n$ pieces using the partition $(Y_1,Y_2,\\ldots,Y_n)$ of the artificial cake and allocate to the agents so that each agent  receives exactly one piece $Y_h$ for each layer. Formally, consider a permutation $\\sigma_j:[n] \\rightarrow [n]$ where\n\\[\n\\sigma_j(i)\n=i+j -1 \\pmod{m}. \n\\]\nConstruct an multi-allocation $\\mathcal{A}=(\\mathcal{A}_i)_{i \\in N}$ where each agent $i \\in N$ is assigned to $A_{ij}=Y_{\\sigma_j(i)}$ for each layer $j \\in L$. By our construction, each $A_{ij}$ contains at most poly$(n)$ number of contiguous pieces. Also, each layered piece $\\mathcal{A}_i$ is non-overlapping as $(Y_1,Y_2,\\ldots,Y_n)$ is a partition of the interval $[0,1]$. By \\eqref{eq}, it is immediate to see that $\\mathcal{A}$ is envy-free. \n\\end{proof}\n\n\n\\section{Proportional multi-layered cake cutting}\nFocusing on a less demanding fairness notion, it turns out that a complete proportional multi-allocation that is both feasible and contiguous exists, for a wider class of instances, i.e., when the number $m$ of layers is some power of two, and the number $n$ of agents is at least $m$. Notably, we show that the problem can be decomposed into smaller instances when the number of agents is at least the number of layers and the number of layers is a power of two. Building up on the {\\em base case} of two layers, our algorithm recursively calls the same algorithm to decide on how to allocate the cake of the sub-problems. We further show that if we relax the contiguity requirement, a proportional feasible multi-allocation can be computed efficiently whenever $m \\leq n$. \n\n\\subsection{Connected pieces}\nIn this subsection, we will show that a proportional complete multi-allocation exists for any $n \\geq m$ when $m$ is some power of $2$. We start by presenting two auxiliary lemmata. We define a {\\em merge} of two disjoint contiguous pieces $I_j$ and $I_{j'}$ of layers $j$ and $j'$ as replacing the $j$-th layered cake with the union $I_j \\cup I_{j'}$ and removing $j'$-th layered cake. The {\\em merge} of a finite sequence of mutually disjoint contiguous pieces $(I_1,\\ldots,I_k)$ can be defined inductively: merge $(I_1,\\ldots,I_{k-1})$ and then apply the merge operation to the resulting outcome and $I_k$. \nNow we observe that if there are two disjoint layers, one can safely merge these layers and reduce the problem size. \n\n\\begin{lemma}\\label{lem:merge}\nSuppose that $C_j$ and $C_{j'}$ are two disjoint layers of a layered cake $\\mathcal{C}$, and $\\mathcal{C}'$ is obtained from $\\mathcal{C}$ by merging $C_j$ and $C_{j'}$. Then, each non-overlapping contiguous layered piece of $\\mathcal{C}'$ is a non-overlapping contiguous layered piece of the original cake $\\mathcal{C}$. \n\\end{lemma}\n\\begin{proof}\nSuppose that $C_j$ and $C_{j'}$ are two disjoint layers of a layered cake $\\mathcal{C}=(C_{t})_{t \\in L}$, and the layered cake $\\mathcal{C}'=(C'_{t})_{t \\in L \\setminus \\{j'\\}}$ is obtained from $\\mathcal{C}$ by merging $C_j$ and $C_{j'}$. Let $\\mathcal{X}'=(X'_{t})_{t \\in L \\setminus \\{j'\\}}$ be a non-overlapping contiguous piece of $\\mathcal{C}'$. Consider the corresponding layered piece $\\mathcal{X}=(X_{t})_{t \\in L}$ of the original cake $\\mathcal{C}$ where $X_t=X'_{t}$ for $t \\in L \\setminus \\{j,j'\\}$ and $X_t = X'_t \\cap C_{t}$ for $t \\in \\{j,j'\\}$. It is immediate to see that $\\mathcal{X}'$ is non-overlapping and contiguous, since $C_j$ and $C_{j'}$ are disjoint. \n\\end{proof}\n\nThe above lemma can be generalized further: Let $\\mathcal{C}$ be a $2m$-layered cake and $x \\in [0,1]$. We define a {\\em merge} of $LR(x)=(S_j)_{j \\in L}$ by merging the pair $(S_j,S_{j+m})$ for each $j \\in [m]$. A {\\em merge} of $RL(x)$ can be defined analogously. Such operation still preserves both feasibility and contiguity. \n\n\\begin{corollary}\\label{cor:merge}\nLet $\\mathcal{C}$ be a $2m$-layered cake and $x \\in [0,1]$. Suppose that $\\mathcal{C}'$ is a $m$-layered cake obtained by merging $LR(x,\\mathcal{C})$ or $RL(x,\\mathcal{C})$. Then, each non-overlapping contiguous layered piece of $\\mathcal{C}'$ is a non-overlapping contiguous layered piece of the original cake $\\mathcal{C}$. \n\\end{corollary}\n\\begin{proof}\nSuppose that $\\mathcal{C}'$ is a $m$-layered cake obtained by merging $LR(x,\\mathcal{C})=(S_j)_{j \\in L}$ of a $2m$-layered cake $\\mathcal{C}$. By Lemma \\ref{lem:merge}, a non-overlapping contiguous layered piece of the cake obtained from each merge of the pair $(S_j,S_{j+m})$ for $j \\in [m]$ still corresponds to a non-overlapping and contiguous piece of the original cake. Thus, the claim holds. An analogous argument applies to the case when we merge $RL(x,\\mathcal{C})$. \n\\end{proof}\n\nWe are now ready to prove that a proportional complete multi-allocation exists for any $n=m$ when $m$ is some power of $2$. In essence, the existence of a majority switching point, as proved in Lemma \\ref{lem:majority:switching}, allows us to divide the problem into two instances. We will repeat this procedure until the number of layers of the subproblem becomes $2$, for which we know the existence of a proportional, feasible, contiguous multi-allocation by Theorem \\ref{thm:EF:two}.  \n\n\\begin{theorem}\\label{thm:prop:base}\nA proportional complete multi-allocation that is feasible and contiguous exists, for any number $m$ of layers and any number $n= m$ of agents where $m=2^a$ for some $a \\in \\mathbb{Z}_{+}$. \n\\end{theorem}\n\\begin{proof}\nWe design the following recursive algorithm $\\mathcal{D}$ that takes a subset $N'$ of agents with $|N'| \\geq 2$, a $|L'|$-layered cake $\\mathcal{C}'$, and a valuation profile $(V_{i})_{i \\in N'}$, and returns a proportional complete multi-allocation of the cake to the agents which is feasible. Suppose that $m=n$. If $m=n=1$, then we allocate the entire cake to the single agent. If $m=n=2$, we run the cut-and-choose algorithm as described in the proof of Theorem \\ref{thm:EF:two}. \nNow consider the case when $m=n=2^a$ for some integers $a \\geq 1$. Then the algorithm finds a majority switching point $x$ over $\\mathcal{C}'$. We let $\\mathcal{I}_1=LR(x)$ and $\\mathcal{I}_2=RL(x)$. By definition of a majority switching point and the fact that $n$ is even, we can partition the set of agents $N'$ into $N_1$ and $N_2$ where $N_1$ is the set of agents who weakly prefer $\\mathcal{I}_1$ to $\\mathcal{I}_2$, $N_2$ be the set of agents who weakly prefer $\\mathcal{I}_2$ to $\\mathcal{I}_1$, and $|N_k| = \\frac{|N'|}{2}$ for each $k=1,2$. \nWe apply $\\mathcal{D}$ to the merge of $\\mathcal{I}_k$ with the agent set $N_k$ for each $k=1,2$, respectively. \n\nWe will show that by induction on the exponential $a$, that the complete multi-allocation $\\mathcal{A}$ returned by $\\mathcal{D}$ satisfies proportionality as well as feasibility and contiguity. This is clearly true when $m=n=2$ due to Lemma \\ref{lem:propEF} and Theorem \\ref{thm:EF:two}. Suppose that the claim holds for $m=n=2^a$ with $1 \\leq a \\leq k-1$; we will prove it for $a=k$. Suppose that the algorithm divides the input cake $\\mathcal{C}'$ via a majority switching point $x$ into $\\mathcal{I}_1=LR(x)$ and $\\mathcal{I}_2=RL(x)$. Suppose that $(N_1,N_2)$ is a partition of the agents where $N_1$ is the set of agents who weakly prefer $\\mathcal{I}_1$ to $\\mathcal{I}_2$, $N_2$ is the set of agents who weakly prefer $\\mathcal{I}_2$ to $\\mathcal{I}_1$, and $|N_k| = \\frac{|N'|}{2}$ for each $k=1,2$. Observe that each agent $i \\in N_1$ weakly prefers $\\mathcal{I}_1$ to $\\mathcal{I}_2$ and thus $V_i(\\mathcal{I}_1) \\geq \\frac{1}{2}V_i(\\mathcal{C}')$. Similarly, $V_i(\\mathcal{I}_2) \\geq \\frac{1}{2}V_i(\\mathcal{C}')$ for each $i \\in N_2$. Thus, by the induction hypothesis, each agent $i$ has value at least $\\frac{1}{|N'|}V_i(\\mathcal{C}')$ for its allocated piece $\\mathcal{A}_{i}$. Further, by Corollary \\ref{cor:merge}, each non-overlapping contiguous layered piece of the merge of $\\mathcal{I}_1$ (respectively, $\\mathcal{I}_2$) is a contiguous non-overlapping layered piece of the original cake $\\mathcal{C}$. By the induction hypothesis, the algorithm outputs a multi-allocation of each merge that is contiguous. Thus, the algorithm returns a proportional complete multi-allocation that is feasible and contiguous.\n\\end{proof}\n\nWe will generalize the above theorems to the case when the number of agents is strictly greater than the number of layers. Intuitively, when $n>m$, then there is at least one layer whose sub-piece can be `safely' allocated to some agent without violating the non-overlapping constraint. \n\n\\begin{theorem}\\label{thm:exponential}\nA proportional complete multi-allocation that is feasible and contiguous exists, for any number $m$ of layers and any number $n \\geq m$ of agents where $m=2^a$ for some $a \\in \\mathbb{Z}_{+}$. \n\\end{theorem}\n\\begin{proof}\nWe design the following recursive algorithm $\\mathcal{D}$ that takes a subset $N'$ of agents with $|N'| \\geq 2$, a $|L'|$-layered cake $\\mathcal{C}'$, and a valuation profile $(V_{i})_{i \\in N'}$, and returns a proportional complete multi-allocation of the layered cake to the agents  which is feasible. For $n=m$, we apply the algorithm described in the proof of Theorem \\ref{thm:prop:base}. Suppose that $n>m$. The algorithm first identifies a layer $C_j$ whose entire valuation is at least $\\frac{1}{n}$ for some agent; assume w.l.o.g. that $j=1$. We move a knife from left to right over the top cake $C_1$ until some agent $i$ {\\em shouts}, i.e., agent $i$ finds the left contiguous piece $Y$ at least as highly valued as his proportional fair share $\\frac{1}{n}$. The algorithm $\\mathcal{D}$ then gives the piece to the shouter. To decide on the allocation of the remaining items, we apply $\\mathcal{D}$ to the reduced instance $(N' \\setminus \\{i\\},(C'_j)_{j \\in L},(V_{i'})_{i' \\in N' \\setminus \\{i\\}})$ where $C'_j=C_j \\setminus Y$ for $j=1$ and $C'_j=C_j$ for $j \\neq 1$. \n\nWe will prove by induction on $|N'|$ that the complete multi-allocation $\\mathcal{A}=(\\mathcal{A}_1,\\mathcal{A}_2,\\ldots,\\mathcal{A}_n)$ returned by $\\mathcal{D}$ satisfies proportionality as well as feasibility and contiguity. This is clearly true when $m=|N'|$, due to Theorem  \\ref{thm:prop:base}. Suppose that the claim holds for $|N'|$ with $m \\leq |N'| \\leq k-1$; we will prove it for $|N'|=k$. Suppose agent $i$ is the shouter who gets the left contiguous piece $Y$. Clearly, agent $i$ receives her proportional share under $\\mathcal{A}$. Observe that all remaining agents have the value at least $\\frac{|N'|-1}{|N'|}V_i(\\mathcal{C}')$ for the remaining cake. Thus, by the induction hypothesis, each agent $i' \\neq i$ has value at least $\\frac{1}{|N'|}V_i(\\mathcal{C}')$ for its allocated piece $\\mathcal{A}_{i'}$. The feasibility and contiguity of $\\mathcal{A}$ are immediate by the induction hypothesis. This completes the proof. \n\\end{proof}\n\n\n\\subsection{Non-connected pieces}\nSince envy-freeness implies proportionality, Theorem \\ref{thm:EF:any} in the previous section implies the existence of a proportional feasible multi-allocation when agents' value density functions are continuous. We strengthen this result, by showing that such desirable allocation exists for a more general case and by providing an efficient algorithm for finding one. \n\n\\begin{theorem}\\label{thm:prop:feasible}\nA proportional complete multi-allocation that is feasible exists when $m =n$ and can be computed using $O(nm^2)$ number of short eval queries and $O(nm)$ number of long eval and cut queries. Further, each bundle of the resulting multi-allocation includes at most two contiguous pieces within each layer. \n\\end{theorem}\n\nBelow, we show that each agent can divide the entire cake into $n$ equally valued layered pieces. A multi-allocation $\\mathcal{A}$ is {\\em equitable} if for each agent $i \\in N$, $V_{i}(\\mathcal{A}_i)=\\frac{1}{n}$. We design a recursive algorithm that iteratively finds two layers for which one has value at most $\\frac{1}{m}$ and at least $\\frac{1}{m}$ and removes a pair of diagonal pieces of value exactly $\\frac{1}{m}$ from the two layers.\n\n\\begin{lemma}\\label{lem:equitable}\nFor any number $m$ of layers and any number $n = m$ of agents with the identical valuations, an equitable complete multi-allocation that is feasible and contiguous exists and can be found using $O(nm^2)$ number of short eval queries and $O(nm)$ number of long cut queries.  \n\\end{lemma}\n\\begin{proof}\nWe denote by $V=V_i$ the valuation function for each agent $i \\in N$. \nConsider the following recursive algorithm $\\mathcal{D}$ that takes a subset $N'$ of agents with $|N'| \\geq 1$, a $|L'|$-layered cake $\\mathcal{C}'$, and a valuation profile $(V_{i})_{i \\in N'}$, and returns an equitable complete multi-allocation of the layered cake to the agents. When $|L'|=|N'|=1$, then the algorithm allocates the entire cake to the single agent. \nSuppose that $|L'|=|N'| \\geq 2$. \nThe algorithm first finds a layer $j$ whose entire value is at most $\\frac{1}{m}$ and another layer $j'$ whose entire value is at least $\\frac{1}{m}$. The algorithm $\\mathcal{D}$ then finds a point $x \\in [0,1]$ where $V(S_{j} \\cup S_{j'})=\\frac{1}{m}$ for $S_j=C_j \\cap [0,x]$ and $S_{j'}=C_j \\cap [x,1]$; such point exists due to Lemma \\ref{lem:switching}. We allocate $S_{j} \\cup S_{j'}$ to one agent and apply $\\mathcal{D}$ to the remaining cake $\\mathcal{C}''$ with $|N'|-1$ agents where $\\mathcal{C}''$ is obtained from merging the remaining $j$-th layered cake $C_j \\setminus S_j$ and the $j'$-th layered cake $C_{j'} \\setminus S_{j'}$. The correctness of the algorithm as well as the bound on the query complexity are immediate. \n\\end{proof}\n\n\\begin{figure*}[hbt]\n\\centering\n\\begin{tikzpicture}[scale=0.6, transform shape]\n\n\\draw[thick] (0,0) rectangle (3,1);\n\\draw[fill=red!10, thick] (3,0) rectangle (6,1);\n\\draw[fill=blue!10, thick] (0,-1) rectangle (2,0); \n\\draw[fill=red!10, thick] (2,-1) rectangle (3,0);\n\\draw[thick] (3,-1) rectangle (6,0);\n\\draw[fill=red!10, thick] (0,-2) rectangle (2,-1);\n\\draw[fill=blue!10, thick] (2,-2) rectangle (6,-1); \n\n\n\\node at (4.5,0.5) {$I_{11}$};\n\\node at (2.5,-0.5) {$I_{12}$};\n\\node at (1,-1.5) {$I_{13}$};\n\n\\node at (1.5,0.5) {$I_{21}$};\n\\node at (4.5,-0.5) {$I_{22}$};\n\n\\node at (1,-0.5) {$I_{32}$};\n\\node at (4,-1.5) {$I_{33}$};\n\n\\draw[ultra thick,->] (6.5,-0.5)--(7.5,-0.5);\n\n\\begin{scope}[xshift=8cm,yshift=-0.5cm]\n\\node[fill=red!10,draw, circle](I1) at (1,1) {$\\mathcal{I}_1$};\n\\node[draw, circle](I2) at (3,1) {$\\mathcal{I}_2$};\n\\node[fill=blue!10,draw, circle](I3) at (5,1) {$\\mathcal{I}_3$};\n\n\\node[fill=gray!10,draw, circle](a1) at (1,-1) {$1$};\n\\node[fill=gray!10,draw, circle](a2) at (3,-1) {$2$};\n\\node[fill=gray!10,draw, circle](a3) at (5,-1) {$3$};\n\n\\draw[-,red, >=latex,thick] (I1)--(a1);\n\\draw[-, >=latex,thick] (I2)--(a1);\n\\draw[-,>=latex,thick] (I3)--(a1);\n\\draw[-, >=latex,thick] (I2)--(a2);\n\\draw[-, >=latex,thick] (I2)--(a3);\n\n\\draw[ultra thick,->] (6.5,0)--(7.5,0);\n\\end{scope}\n\n\\begin{scope}[xshift=16cm,yshift=0.5cm]\n\\draw[thick] (0,-1) rectangle (3,0);\n\n\\draw[fill=blue!10, thick] (0,-2) rectangle (2,-1); \n\\draw[thick] (3,-1) rectangle (6,0);\n\\draw[fill=blue!10, thick] (2,-2) rectangle (6,-1); \n\n\\node at (1.5,-0.5) {$I_{21}$};\n\\node at (4.5,-0.5) {$I_{22}$};\n\n\\node at (1,-1.5) {$I_{32}$};\n\\node at (4,-1.5) {$I_{33}$};\n\n\\draw[thick,dotted] (4,0.5)--(4,-2.5);\n\\end{scope}\n \n\\end{tikzpicture}\n\\caption{Protocol for proportionality for three agents and three layers. Agent $1$ divides the entire cake into three equally valued layered pieces $\\mathcal{I}_1$, $\\mathcal{I}_2$, and $\\mathcal{I}_3$ (the left-most picture). Here, $\\mathcal{I}_i=(I_{ij})_{j=1,2,3}$ for each $i =1,2,3$ where $I_{23}=I_{31}=\\emptyset$. In the middle picture, agent $1$ is adjacent to every piece, meaning that he has value at least proportional fair share for every piece; on the other hand, the other agents are adjacent to the second piece only. The maximum envy-free matching is an edge between $\\mathcal{I}_1$ and agent $1$ (red edge), so the algorithm allocates $\\mathcal{I}_1$ to agent $1$ and merges $\\mathcal{I}_2$, and $\\mathcal{I}_3$. Then it applies the cut-and-choose among the remaining agents (the right-most picture).}\n\\label{fig:PROP:three}\n\\end{figure*}\n\nEquipped with Lemma \\ref{lem:equitable}, we will prove Theorem \\ref{thm:prop:feasible} by recursively computing an {\\em envy-free matching} between $n$ agents and $n$ layered pieces where one agent has proportional fair share for every piece. Specifically, given a bipartite graph $G$ with one side being the set of agents and the other side being the set of items, an envy-free matching $M$ of $G$ is a matching where no unmatched agent {\\em envies} some matched agent, i.e., no unmatched agent is adjacent to any matched item in $G$. The problem of finding an envy-free matching of maximum size can be solved in polynomial time \\citep{EladErel,GAN2019}. Further, using Hall's type condition, it can be easily shown that if there is one agent who is adjacent to every item and the number of agents is at most the number of items, then there is a non-empty envy-free matching (Corollary $1.4$ $($c$)$ of \\citet{EladErel}).  \n\n\\begin{proof}[Proof of Thm. \\ref{thm:prop:feasible}]\nIn order to obtain a proportional feasible multi-allocation, we will recursively compute a non-empty envy-free matching:\nFor each iteration, let one agent partition $n$-equally valued layered pieces, find a maximum envy-free matching between agents and pieces, and assign the matched agents to the matched pieces. By Corollary $1.4$ $($c$)$ of \\citet{EladErel}, the envy-free matching computed at each step is non-empty; thus, at least one agent is matched and we will apply the same procedure to the unmatched agents and pieces. The formal description is given as follows. See Figure \\ref{fig:PROP:three} for an illustration for $m=n=3$. \n\n\\vspace{5pt}\n\\noindent\\fbox{%\n\t\\parbox{0.985\\linewidth}{%\n\t\t\\textbf{A protocol for proportional feasible multi-allocations for $n = m$ agents} over a $m$-layered cake $\\mathcal{C}$: \\\\\n\t\t\\textit{Step 1.} One agent partitions the cake into $n$ non-overlapping layered contiguous pieces $\\mathcal{I}_1,\\mathcal{I}_2, \\ldots, \\mathcal{I}_n$ which she considers of equal value, using the algorithm in the proof of Lemma \\ref{lem:equitable}.\\\\\n\t\t\\textit{Step 2.} Construct a bipartite graph $G$ with the agents being one side and the pieces being on the other side, where there is an edge between agent $i$ and $\\mathcal{I}_h$ if agent $i$ has value at least his proportional far share for $\\mathcal{I}_h$. \\\\\n\t\t\\textit{Step 3.} Compute a maximum-size envy-free matching $M$ of $G$. Assign matched agent to the corresponding piece. \\\\\n\t\t\\textit{Step 4.} For each unmatched piece $\\mathcal{I}_h$, merge all the disjoint contiguous pieces in $\\mathcal{I}_h$, and create a $m-\\ell$-layered cake $\\mathcal{C}'$ consisting of each merge of $\\mathcal{I}_h$ where $\\ell$ is the number of matched pieces. Apply the same protocol to $\\mathcal{C}'$ among the remaining unmatched agents. \n\t}%\n}\n\\vspace{5pt}\n\nWe will show by the induction on the number of agents $n=m$ that the resulting multi-allocation $\\mathcal{A}$ is proportional and feasible. The claim clearly holds for $n=m=1$. Suppose that the claim holds for $n$ with $m=n \\leq k-1$; we will prove it for $m=n=k$. We will first show that the resulting multi-allocation $\\mathcal{A}$ is proportional. Clearly, the agents who get matched has proportional fair share for his bundle. Further, each $i$ of the remaining unmatched $n-\\ell$ agents have value less than $\\frac{1}{n}$ for each matched piece and thus has at least $1-\\frac{\\ell}{n}$ for the remaining unmatched pieces; thus, $V_i(\\mathcal{A}_i)$ is at least $\\frac{1}{n}$. It can be easily verified that each bundle $\\mathcal{A}_i$ is non-overlapping. Each iteration requires $O(m^2)$ number of short eval queries and $O(m)$ number of long cut queries for the cutter, and $O(m^2)$ number of short eval queries for each agent. Further, the number of iterations is at most $n$, which proves the bound on the query complexity. \nThis completes the proof.\n\\end{proof}\n\nSimilarly to the proof for Theorem \\ref{thm:exponential}, we can generalize the above theorem to the case when the number of agents is strictly greater than the number of layers. \n\n\\begin{theorem}\\label{thm:prop:feasible:any}\nA proportional complete multi-allocation that is feasible exists and can be computed using $O(nm^2)$ number of short eval queries and $O(nm)$ number of long cut queries, for any number $m$ of layers and any number $n \\geq m$ of agents. \n\\end{theorem}\n\nIt remains open whether a proportional contiguous multi-allocation exists when the number of layers is three. A part of the reason is that our algorithm for finding an equitable multi-allocation (Lemma \\ref{lem:equitable}) may not return a `balanced' partition: The number of pieces contained in each layered piece may not be the same when the number of layers is odd. For example, one layered piece may contain pieces from three different layers while the other two parts may contain pieces from two different layers, as depicted in Figure \\ref{fig:PROP:three}. \n\n\n\\section{Discussion} \\label{sec:discussion}\nWe initiated the study of multi-layered cake cutting, demonstrating the rich and intriguing mathematical feature of the problem. There are several exciting questions left open for future work. Below, we list some of them. \n\n\\begin{itemize}\n\\item {\\bf Existence of fair allocations}:\nWe have seen that an envy-free contiguous and feasible multi-allocation of a two-layered cake exists for two or three agents with at most two types of preferences. An interesting open problem is whether such allocation also exists for any number of agents over a two-layered cake. One might expect that the Simmon-Su's technique \\citep{Su1999} using Sperner's Lemma can be adopted to our setting by considering all possible diagonal pieces. However, this approach may not work because multi-layered cake-cutting necessarily exhibits non-monotonicity in that the value of a pair of diagonal pieces may decrease when the knife moves from left to right. For proportionality, one intriguing future direction is extending our existence result for $m=2^a$ to any $m$. This requires careful consideration of contiguity and feasibility, which are often at odds with completeness.\n\n\\item {\\bf Query complexity of fair allocations}:\nThe query complexity of finding an envy-free feasible multi-allocation is open in the multi-layered cake-cutting problem. In particular, it would be challenging to extend the celebrated result of \\citet{aziz2016discrete} -- who showed the existence of a bounded protocol for computing an envy-free allocation of a single-layered cake with any number $n$ of agents --  to our setting. We expect that a direct translation may not work, due to the intricate nature of the feasibility constraint. With respect to proportionality, our existence proof implies that if there is a way to compute a majority switching point efficiently, one can compute a proportional contiguous feasible multi-allocation for special cases when the number of layers is a power of two. It is open whether such cutting point can be computed using a bounded number of queries.\n\n\\item {\\bf Approximate fairness}: \nIn the presence of contiguity requirement, it is known that no finite protocol computes an envy-free allocation even for three agents and a single-layered cake \\citep{Stromquist2008}. However, several positive results are known when the aim is to approximately bound the envy between agents \\citep{Deng2012,Goldberg2020,Arunachaleswaran19}.\nPursuing a similar direction in the context of multi-layered cake cutting would be an interesting research topic. \n\n\\item {\\bf Efficiency requirement}: \nBesides fairness criteria, another basic desideratum is economic efficiency. In the context of a single-layered cake cutting, several works studied the relation between fairness and efficiency \\citep{CohlerLPP11,Bei12, AumannDomb2010, AumannDH13}. \nThe question of what welfare guarantee can be achieved together with fairness is open in our model. In particular, it would be interesting to investigate the compatibility of the fairness notions with an efficiency requirement, under feasibility constraints. \n\n\n\\end{itemize}\n\n\\section*{Acknowledgement}\nHadi Hosseini was supported by the National Science Foundation (grant IIS-1850076).\nAyumi Igarashi was supported by the KAKENHI Grant-in-Aid for JSPS Fellows no. 18J00997 and JST, ACT-X. The authors would like to thank Yuki Tamura for introducing the problem to us and for the fruitful discussions. The authors also acknowledge the helpful comments by the GAIW and IJCAI reviewers. \nWe are grateful to an anonymous GAIW reviewer for the proof of the existence of an envy-free multi-allocation in the general case. \n\n\n\\bibliographystyle{named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is well known that volumetric photonic crystals have the efficacy of controlling  propagating  waves, e.g.~\\cite{joannopoulos2008molding}, both in three and two dimensions, e.g. \\cite{Kraus}. \nRecently, in parallel with thorough and general investigations of time-varying electromagnetic systems (see, e.g., Refs.~\\cite{XUCHENPRAPPLIED, SAJJADarXiv, groupEnergy}), three-dimensional (volumetric) temporal photonic crystals whose material properties are uniform in space but changing in time have attracted significant attention~\\cite{zurita2009reflection}. This is due to the intriguing effects that they have on propagating plane waves. \n\nIt appears that it is fundamentally important to explore properties of temporal metasurfaces which are 2D material sheets or boundaries with time-varying properties. In this case, the waves are surface waves, bound to the sheet or boundary. The eigenmode problem for waves along spatially uniform but time-modulated boundaries is one of the important canonical problems in electromagnetics of time-varying structures.\n\nIn this work, we solve this problem and discuss dispersion properties of surface waves over time-modulated reactive boundaries. As a simple canonical case, we consider a planar infinite boundary that is modeled by a surface capacitance which is spatially uniform over the surface and temporally modulated by an arbitrary periodical function. As a particular realization, one can consider, for example, a high-impedance surface of the type introduced by D. Sievenpiper \\cite{sievenpiper1999high}, where the capacitance between patches is modulated using varactors.\n\n\nIn addition to derivation of the dispersion relation for surface waves over such time-varying boundary, we give numerical examples of dispersion plots. Also, we explain the conditions for appearing stop bands for propagation constants and reveal phenomena of exponential field growth, reflection amplification, and radiation of space waves from those time-modulated boundaries.\n\n\n\n\\section{Theory}\n\n\\begin{figure}\n\\includegraphics[width=.95\\linewidth]{geom.pdf}\n\\caption{Geometry of the problem: A TE-polarized surface wave over a time-modulated capacitive boundary. }\n\\label{fig:geom}\n\\end{figure}\n\nLet us consider a spatially uniform and time-varying reactive boundary. Here, as an example, we assume a time-varying capacitive one which is represented by $C(t)$. As is well known, such a boundary supports surface waves which have the transverse-electric (TE) polarization with respect to the propagation direction. In other words, in free space, the electric field is perpendicular to the propagation plane, as shown in Fig.~\\ref{fig:geom}. For a stationary boundary, the dispersion equation defines a relation between the frequency $\\omega$ and the propagation constant along the surface $\\beta$. Since the capacitive boundary periodically changes in time, the eigenmode $\\beta$ will contain components at frequencies $\\omega_n=\\omega+n\\omega_{\\rm{M}}$, $n=0,\\pm 1,\\pm 2,\\dots$. Here, $\\omega_{\\rm{M}}$ is the fundamental modulation frequency. The electric field is expressed as \n\\begin{equation}\n\\mathbf{E}=\\sum_{n=-\\infty}^{+\\infty}E_n\\exp(j\\omega_nt)\\mathbf{a}_y,\n\\end{equation}\nin which \n\\begin{equation}\nE_n=A_n\\exp(j\\beta z)\\exp(-\\alpha_nx).\n\\end{equation}\nHere, $A_n$ is the amplitude of the wave corresponding to each frequency harmonic,  and $\\alpha_n$ denotes the attenuation constant along the normal direction, for each harmonic.  The plane-wave dispersion equation for free space above the boundary sets the following relation for each harmonic: \n\\begin{equation}\n\\beta^2=\\alpha_n^2+\\omega_n^2\\epsilon_0\\mu_0 .\n\\label{eq:betaalpha}\n\\end{equation}\nIt is worth mentioning that since we investigate surface waves, $\\alpha_n$ must be a real value.\n\nSimilarly to the electric field, the tangential component of the magnetic field which is directed along the $z$-axis is given by \n\\begin{equation}\n\\mathbf{H}_{\\rm{t}}=\\sum_{n=-\\infty}^{+\\infty}H_n\\exp(j\\omega_nt)\\mathbf{a}_z,\n\\end{equation}\nwhere\n\\begin{equation}\nH_n=B_n\\exp(j\\beta z)\\exp(-\\alpha_nx).\n\\end{equation}\nThe Maxwell  equations relate the amplitudes of the electric field and the tangential component of the magnetic field to each other. By applying Eq.~\\eqref{eq:betaalpha} and doing some algebraic manipulations, we obtain a matrix relation  $\\mathbf{M}\\cdot \\mathbf{A}=\\mathbf{B}$. Here, $\\mathbf{M}$ is a matrix that has $2N+1$ rows and columns, and it is a function of $\\beta$ and $\\omega_n$. The matrices $\\mathbf{A}$ and $\\mathbf{B}$ have only one column and $2N+1$ rows, and they are representing the amplitudes.  \n\nIn analogy with the circuit theory, where we explicitly express the relation between the electric current flowing through the time-varying capacitance $i(t)$ and the voltage over it $v(t)$ as \n\\begin{equation}\n\\int i(t)dt=C(t)v(t),\n\\end{equation}\nwe simply write the relation between the tangential components of the electric and magnetic fields. In fact, this is the boundary condition in the dynamic scenario. By imposing the boundary condition, we obtain another matrix equation as $\\mathbf{Y}\\cdot \\mathbf{A}=\\mathbf{B}$. The matrix $\\mathbf{Y}$ is a function of $\\omega_n$ and the Fourier coefficients of the periodic function $C(t)$ which is expressed by the Fourier series in the exponential form. Similar modeling method has been used in our recent paper \\cite{wang2020nonreciprocity}.\n\nNow, we have two matrix equations $\\mathbf{M}\\cdot \\mathbf{A}=\\mathbf{B}$ and\\break $\\mathbf{Y}\\cdot \\mathbf{A}=\\mathbf{B}$. Therefore, we conclude that $\\big[\\mathbf{Y}-\\mathbf{M}\\big]\\cdot\\mathbf{A}=0$.  The determinant of the whole matrix in the square brackets must be zero in order to allow nonzero solutions for the electric field. Consequently,  relation \n\\begin{equation}\n\\det\\Big[\\mathbf{Y}-\\mathbf{M}\\Big]=0 \\label{eq: dispersion}\n\\end{equation}\ndetermines the dispersion of the surface waves above  time-varying capacitive boundaries. In the following, we give some particular examples and numerically investigate the dispersion curves.\n\n\n\\section{Numerical Examples and Discussion}\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{dispersion_one_tune.pdf}\n\t\\caption{Dispersion plot for modulation with one harmonic tune. The figure is plotted for $C_0=1$~pF and $\\omega_{\\rm M}=3$~GHz. \n\t} \n\t\\label{fig: dispersion_one_tunes}\n\\end{figure}\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{dispersion_two_tune.pdf}\n\t\\caption{Dispersion plot for modulation with two harmonic tunes:  $C(t)=C_0[1+0.3\\cos(\\omega_{\\rm M}t)+0.2\\cos(2\\omega_{\\rm M}t)]$. \n\t} \n\t\\label{fig: dispersion_two_tunes}\n\\end{figure}\n\n\n\nFirst, we consider a capacitive boundary that is modulated in time harmonically, assuming, as an example,  $C(t)=C_0[1+0.3\\cos(\\omega_{\\rm M}t)]$. \nTo obtain the dispersion diagram, we specify the value of $\\beta$, and solve the dispersion equation Eq.~(\\ref{eq: dispersion})  for the corresponding  eigenfrequencies. The corresponding dispersion curves are shown in  Fig.~\\ref{fig: dispersion_one_tunes}. One  can see  that for a fixed value of the propagation constant $\\beta$, there are many solutions for the eigenfrequencies $\\omega$, meaning that an eigenmode contains components at many frequencies. This property means that we can excite the mode by external sources at  many possible frequencies. \nThe excitation frequency can even be above the light line (purple dot) which excites higher-order frequency harmonics below the light line (green dot).  This indicates that, in time-varying structures, it is possible to launch surface waves with an incident plane wave, as is also reported in a recent work \\cite{galiffi2020wood}.\nMost importantly, temporal modulation opens up a band gap in $k$-space, which is a dual phenomenon of spatial periodic structures where the band gap is at the frequency axis. Similar properties of bulk media have been reported in \\cite{zurita2009reflection,lustig2018topological}.\nIncreasing the modulation depth, one  can widen the band gap. Interestingly, the number of band gaps corresponds to the number of modulation tones. Figure~\\ref{fig: dispersion_two_tunes} shows that by  adding a second modulation tone, a second band gap opens up. Positions and widths of the gaps can be tuned by varying the modulation spectrum. These band gaps provide great possibilities to control the propagation of surface waves on a metasurface plane. \n\n\\begin{figure}[!h]\n\\includegraphics[width=.95\\linewidth]{amplification.pdf}\n\\caption{(a) Surface wave on a time-invariant boundary. (b) Amplified surface wave on a time-varying boundary ($\\Delta t=5T_0$).  }\n\\label{fig:amplification}\n\\end{figure}\n\nNext, we examine the wave behavior when the excited surface modes are inside a band gap. For a specified wavenumber $\\beta$ in the band gap, the solved eigenfrequencies are complex numbers $\\omega=\\omega^\\prime\\pm j\\omega^{\\prime\\prime}$, meaning that  the wave can be exponentially attenuated or amplified in time. Next, we use COMSOL to numerically simulate the time-varying structure in this regime. \nFigure~\\ref{fig:amplification}(a) illustrates   a constant-amplitude surface wave propagating along an unmodulated capacitive boundary. Then, the temporal modulation of the surface reactance is suddenly  switched on.  Evidently, as it can be seen in Fig.~\\ref{fig:amplification}(b, ) the surface mode is significantly amplified after modulating with a duration of $\\Delta t=5T_0$, where $T_0$ is the time period at the excitation frequency.  In addition, there are higher-order harmonics generated in free space, forming a standing wave pattern along the surface, due to symmetry of the structure. \n\n\n\n\n\\section{Conclusions}\nHere, we have presented the dispersion equation and example dispersion plots for a temporally modulated electromagnetic boundary. The results show that  time modulation induces band gaps in the two-dimensional $k$-space, which provides opportunities to control surface wave propagation. In the presentation, we will discuss in detail interesting properties of surface waves launched  inside a band gap.  \nLet us note that one can repeat the above path to achieve the dispersion equation associated with  time-varying inductive boundaries. The difference is that the wave has a transverse-magnetic polarization (i.e., magnetic field is perpendicular to the propagation plane). \n\n\n\\section{Acknowledgment}\nThe authors thank Dr. V.~Asadchy for useful discussions and valuable comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Highlights}\n\\begin{itemize}\n\\item Privacy-preserving multi-agent reinforcement learning is used to coordinate residential energy  \n\\item Learning from optimisations improves coordination scalability in stochastic environments\n\\item Marginal reward signals further enhance cooperation relative to previous approaches\n\\item The curse of dimensionality is mitigated by the use of fixed-size Q-tables\n\\item Case studies with large real-life datasets yield 33.7\\% local and global cost reductions \n\\end{itemize}\n\n\\begin{multicols}{2}\n\n\\section{Introduction}\nThis paper addresses the scalability issue of distributed domestic energy flexibility coordination in a cost-efficient and privacy-preserving manner.  A novel class of coordination strategies using optimisation-based multi-agent reinforcement learning (MARL\\footnote{A full nomenclature is available in \\Cref{app:nomenclature}}) with fixed Q-table size is proposed for household-level decision-making, tackling the challenge of scalability for simultaneously learning independent agents under partial observability in a stochastic environment \\citep{Matignon2012}. Multiple versions of the novel strategy are assessed to maximise the statistical expectation of system-wide benefits, including local battery costs, grid costs and greenhouse gas emissions. \n\nWidespread electrification of primary energy provision and decarbonisation of the power sector are two vital prerequisites for limiting anthropogenic global warming to 1.5$^o$C above pre-industrial levels. To reduce risks of climate-related impacts on health, livelihood, security and economic growth, intermittent renewable power supplies could be required to supply 70\\% to 85\\% of electricity by 2050 \\citep{IPCC2015}. However, this poses the challenges of the intermittency and limited controllability of resources \\citep{Bose2019}.  Therefore, a robust, decarbonised power system will rely on two structural features: decentralisation and demand response (DR) \\citep{Leautier2019}. The coordination of distributed flexible energy resources can help reduce costs for transmission, storage,  peaking plants and capacity reserves,  improve grid stability, align demand with decarbonised energy provision, promote energy independence and security, and lower household energy bills \\citep{Vazquez-Canteli2019, Pumphrey2020}.\n\nResidential sites constitute a significant share of potential DR, representing for example 38.5\\% of the 2019 UK electricity demand, and 56.4\\% of energy consumption if including transport and heat, which are both undergoing electrification \\citep{BEIS2021}. Increasing ownership of EVs and PV panels has been facilitated by regulatory changes, with many countries committing to internal combustion car phase-outs in the near future, and by plummeting costs, with an 82\\% and 87\\% levelised cost drop between 2010 and 2019 for EVs and PV panels \\citep{Agency2018,BloomberNEF2019}. This potential is so far underexploited, as DR primarily focuses on larger well-known industrial and commercial actors that require less coordination and data management \\citep{CharlesRiverAssociates2017}, with most customers still limited to trade with utility companies \\citep{Chen2019}. The primary hurdles to unlocking residential flexibility are the high capital cost of communication and control infrastructure as the domestic potential is highly fragmented \\citep{Leautier2019}, concerns about privacy and hindrance of activities \\citep{Bugden2019,Pumphrey2020}, and computational challenges for real-time control at scale \\citep{Moret2019}.  \n\nTraditionally, convex optimisation would be used to maximise global coordination objectives in convex problems with variables known ahead of time. Techniques such as least-squares and linear programming have been well-studied for over a century \\citep{Boyd2009}. However, residential energy coordination presents challenges to its application. Firstly,  optimisations that are centralised are hindered by privacy, acceptance, and communication constraints, and present exponential time complexity at the scale of millions of homes \\citep{Dasgupta2016}. Secondly, standard optimisation methods cannot be used without full knowledge of the system's inputs and dynamics \\citep{Recht2018}. In residential energy, agents only have partial observability of the system due to both the stochasticity and uncertainty of environment variables such as individual residential consumption and generation profiles, and to the privacy and infrastructure cost constraints that hinder communication between agents during implementation \\citep{FrancoisLavet2017}. Not relying on shared information may also improve the robustness of the solutions to failure of other agents, communication delays, and unreliable information, and improve adaptability to changing environments \\citep{Sen1994}. Finally, the real-life complex electricity grid environment may not be amenable to a convex model representation. Due to the heterogeneity of users and behaviours needing different parameters and models,  the large-scale use of model-based controllers is cumbersome \\citep{Ruelens2017}. A model-free approach instead avoids modelling non-trivial interactions of parameters, including private information \\citep{Dasgupta2016}.  \n\nGiven these challenges to residential energy flexibility coordination, and the specific constraints of the problem at play which renders traditional approaches unsuitable, we seek to develop a novel coordination mechanism which satisfies the following criteria,  as tested in real-life scenarios:\n\\begin{itemize}\n\\item Computational scalability: minimal and constant computation burden during implementation as the system size increases; \n\\item Performance scalability: no drop in coordination performance as the system size increases, measured in savings obtained per hour and per agent;\n\\item Acceptability: local control of appliances, no communication of personal data, thermal discomfort, or hindrance\/delay of activities.\n\\end{itemize}\n\nThe rest of this paper is organised as follows. In \\Cref{sec:gapanalysis} we motivate the novel MARL approach with a literature review and a gap analysis. In \\Cref{system}, a system model is presented that includes household-level modelling of EVs, space heating, flexible loads and PV generation.  \\Cref{RLSection} lays out the MARL methodology, with various methodological options for independent agents to learn to cooperate. In \\Cref{data}, the input data used to populate the model is presented. In \\Cref{results}, the performance of different MARL strategies is compared to lower and upper bounds in case studies. Finally, we conclude in \\Cref{conclusion}.\n\n\n\\section{MARL-based energy coordination: literature review and gap analysis}\\label{sec:gapanalysis}\nReinforcement learning (RL) can overcome the constraints faced by centralised convex optimisation for residential energy coordination, by allowing for decentralised and model-free decision-making based on partial knowledge. RL is an artificial intelligence (AI) framework for goal-oriented agents\\footnote{Here agents are independent computer systems acting on behalf of prosumers \\citep{Wooldridge2002}. Prosumers are proactive consumers with distributed energy resources actively managing their consumption, production and storage of energy \\citep{Morstyn2018_Federated}. } to learn sequential decision-making by interacting with an uncertain environment \\citep{Sutton1998}.  As an increasing wealth of data is collected in local electricity systems, RL is of growing interest for the real-time coordination of distributed energy resources (DERs) \\citep{Antonopoulos2020,Vazquez-Canteli2019}.  Instead of optimising based on inherently uncertain data, RL more realistically searches for statistically optimal sequential decisions given partial observation and uncertainty, with no \\emph{a priori} knowledge \\citep{Recht2018}.  Approximate learning methods may be more computationally scalable, more efficient in exploring high-dimensional state spaces and therefore more scalable than exact global optimisation with exponential time complexity \\citep{Schellenberg2020, Dasgupta2016}. \n\nAs classified in \\citep{CharbonnierReview}, numerous RL-based coordination methods have been proposed in the literature for residential energy coordination, though with remaining limitations in terms of scalability and privacy protection. On the one hand, in RL-based direct control strategies, a central controller directly controls individual units, and households directly forfeit their data and control to a central RL-based scheduler \\citep{ONeill2010}. While most existing AI-based DR research thus assumes fully observable tasks \\citep{Antonopoulos2020}, direct controllability of resources from different owners with different objectives and resources and subject to privacy, comfort and security concerns is challenging \\citep{Darby2020}. Moreover, centralised policies do not scale due to the curse of dimensionality as the state and action spaces grow exponentially with the system size \\citep{Powell2011}.  On the other hand, RL-based indirect control strategies consider decision-making at the prosumer level, entering the realm of MARL. This can be achieved using different communication structures, with either centralised, bilateral, or no sharing of personal information, as presented below.\n\nFirstly, agents may share information with a central entity,  which in turn broadcasts signals based on a complete picture of the coordination problem. For example, the central entity may send unidirectional price signals to customers based on information such as prosumers' costs, constraints and day-ahead forecasts. RL can inform both the dynamic price signal \\citep{Lu2019, Kim2016}, and the prosumer response to price signals \\citep{Kim2016,Babar2018}. The central entity may also collect competitive bids and set trades and match prosumers centrally, where RL algorithms are used to refine individual bidding strategies \\citep{Vaya2014, Ye2020, Dauer2013,Sun2015,Kim2020} or to dictate the auction market clearing \\citep{Chen2019,Claessens2013}. Units may also use RL to cooperate towards common objectives with the mediation of a central entity that redistributes centralised personal information \\citep{Zhang2017,Dusparic2015,Dusparic2013,Hurtado2018}. However, information centralisation also raises costs, security, privacy and scalability of computation issues. Biased information may lead to inefficient or even infeasible decisions \\citep{Morstyn2020_P2P}. \n\nSecondly, RL-based coordination has been proposed where prosumers only communicate information bilaterally without a central authority. For example, in \\citep{Taylor2014} agents use transfer learning with distributed W-learning to achieve local and system objectives. Bilateral peer-to-peer communication offers autonomy and expression of individual preferences, though with remaining risks around privacy and bounded rationality \\citep{Herbert1982}. There is greater robustness to communication failures compared situations with a single point of failure. However, as the system size increases,  the number of communication iterations until algorithmic convergence increases, requiring adequate computational resources and limited communication network latency for feasibility \\citep{Guerrero2020}. The safe way of implementing distributed transactions to ensure data protection is an ongoing subject of research \\citep{CharbonnierReview}. \n\nFinally,  in RL-based implicit coordination strategies, prosumers rely solely on local information to make decisions. For example, in \\citep{Cao2019, Yang2019}, competitive agents in isolation maximise their profits in RL-based energy arbitrage, though they do not consider the impacts of individual actions on the rest of the system, with potential negative impacts for the grid. For example, a concern is that all loads receive the same incentive, the natural diversity on which the grid relies may be diminished \\citep{Crozier2018_Mitigating}, and the peak potentially merely displaced, with overloads on upstream transformers. Implicit cooperation, which keeps personal information at the local level while encouraging cooperation towards global objectives, has been thus far under-researched beyond frequency control.  In \\citep{Rozada2020}, agents learn the optimal way of acting and interacting with the environment to restore frequency using local information only. This is a promising approach for decentralised control. However, the applicability in more complex scenarios with residential electric vehicles and smart heating load scheduling problems has not been considered. Moreover, the convergence slows down for increasing number of agents, and scalability beyond 8 agents has not been investigated. Indeed, fundamental challenges to the coordination of simultaneously learning independent agents at scale under partial observability in a stochastic environment have been identified when using traditional RL algorithms [1]: independent learners may reach individual policy equilibriums that are incompatible with a global Pareto optimal, the non-stationarity of the environment due to other concurrently learning agents affects convergence, and the stochasticity of the environment prevents agents from discriminating between their own contribution to global rewards and noise from other agents or the environment. Novel methods are therefore needed to develop this approach.\n\nWe seek to bridge this gap, using implicit coordination to unlock the so-far largely untapped value from residential energy flexibility to provide both individual and system benefits. We propose a new class of MARL-based implicit cooperation strategies for residential DR, to make the best use of the flexibility offered by increasingly accessible assets such as photovoltaic (PV) panels, electric vehicle (EV) batteries, smart heating and flexible loads. Agents learn RL policies using a data-based, model-free statistical approach by exploring a shared environment and interacting with decentralised partially observable Markov decision processes (Dec-POMDPs), either through random exploration or learning from convex optimisation results. In the first rehearsal phase \\citep{Kraemer2016} with full understanding of the system, they learn to cooperate to reach system-wide benefits by assessing the global impact of their individual actions, searching for trade-offs between local, grid and social objectives. The pre-learned policies are then used to make decisions under uncertainty given limited local information only.\n\nThis approach satifies the computational scalability, coordination scalability and acceptance criteria set out in this paper.\n\nFirstly,  the real-time control method is computationally scalable thanks to fixed-size Q-tables which avoid the curse of dimensionality, and there is only minimal, constant local computation required to implement the pre-learned policies during implementation. No further communication is required for implementation. This increases robustness to communication issues and data inaccuracy relative to when relying on centralised and bilateral communication, and cuts the costs of household computation and two-way communication infrastructure. \n\nSecondly, we address the outstanding MARL coordination performance scalability issue for agents with partial observability in a stochastic environment seeking to maximise rewards which also depend on other concurrently learning agents \\citep{Busoniu2008,Matignon2012}. The case studies in this paper show that allowing agents to learn from omniscient, stable, and consistent optimisation solutions can successfully act as an equilibrium-selection mechanism, while the use of marginal rewards improves learnability\\footnote{\\say{the sensitivity of an agent's utility to its own actions as opposed to actions of others, which is often low in fully cooperative Markov games} \\citep{Matignon2012}} by isolating individual contributions to global rewards. This novel methodological combination offers significant improvements on MARL scalability and convergence issues, with high coordination performance maintained as the number of agents increases, where that of standard MARL drops at scale. \n\nFinally,  this method tackles acceptability issues, with no interference in personal comfort nor communication of personal data. \n\nThe specific novel contributions of this paper are (a) a novel class of decentralised flexibility coordination strategies, MARL-based implicit cooperation, with no communication and fixed-size Q-tables to mitigate the curse of dimensionality; (b) a novel MARL exploration strategy for agents under partial observability to learn from omniscient, convex optimisations prior to implementation for convergence to robust cooperation at scale; and (c) the design and testing with large banks of real-world data of combinations of reward definitions, exploration strategies and multi-agent learning frameworks for assessing individual impacts on global energy, grid and storage costs. Methodologies are identified which outperform a baseline with increasing numbers of agents despite uncertainty.\n\n\\section{Local system description}\\label{system}\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.7\\linewidth]{energybalance.pdf}\n\n\\end{center}\n\\caption{Local system model. Red dotted lines denote energy balances.}\n\\label{fig:EnergyBalance}\n\\end{figure*}\n\nIn this section, the variables, objective function and constraints of the problem are described. This sets the frame for the application of the RL algorithms presented in \\Cref{RLSection}.\n\n\\subsection{Variables}\\label{variables}\nWe consider a set of time steps $t \\in \\mathcal{T} = \\{t_0,...,t_\\textrm{end}\\}$ and a set of prosumers $i \\in \\mathcal{P} = \\{1,...,n\\}$. Decision variables are \\emph{italicised} and input data are written in roman. Energy units are used unless specified otherwise. Participants have an EV, a PV panel, electric space heating and generic flexible loads.\n\nThe EV at-home availability $\\upmu_i^t$ (1 if available, 0 otherwise), EV demand for required trips $\\textrm{d}_{\\textrm{EV},i}^t$, household electric demand $\\textrm{d}_i^{t}$,  PV production $\\textrm{p}_{\\textrm{PV},i}^t$, external temperature $\\textrm{T}_{\\textrm{e}}^t$ and solar heat flow rate $\\upphi^t$ are specified as inputs for $t \\in \\mathcal{T}$ and $i \\in \\mathcal{P}$.\n\nThe local decisions by prosumers are the energy flows in and out of the battery $b_{\\textrm{in},i}^t$ and $b_{\\textrm{out},i}^t$, the electric heating consumption $h_i^t$ and the prosumer consumption $c_i^t$. These have both local and system impacts (\\Cref{fig:EnergyBalance}). Local impacts include battery energy levels $E_i^t$,  losses $\\epsilon_{\\textrm{ch},i}^t$ and $\\epsilon_{\\textrm{dis},i}^t$,  prosumer import $p_i^t$, building mass temperature $T_{\\textrm{m},i}^t$ and indoor air temperature $T_{\\textrm{air},i}^t$. System impacts arise through the costs of total grid import $g^t$ and distribution network trading. Distribution network losses and reactive power flows are not included.  \n\n\\subsection{Objective function}\\label{objfunc}\n\nProsumers cooperate to minimise system costs consisting of grid ($c_\\textrm{g}^t$), distribution ($c_\\textrm{d}^t$) and storage ($c_\\textrm{s}^t$) costs. This objective function will be maximised both in convex optimisations off-line -- to provide an upper bound for the achievable objective function, and in some cases to provide information to the learners during the simulated learning phase -- and in the learning of MARL policies for decentralised online implementation. \n\n\\begin{equation}\n\\max F = \\sum_{\\forall t \\in \\mathcal{T}}{\\hat{F}_t} = \\sum_{\\forall t \\in \\mathcal{T}}{- (c_\\textrm{g}^t + c_\\textrm{d}^t + c_\\textrm{s}^t )}\n\\end{equation}\n\n\\begin{equation}\nc_\\textrm{g}^t = \\textrm{C}_\\textrm{g}^t \\left( g^t + \\epsilon_g \\right)\n\\end{equation}\nWhere losses incurred by imports and exports from and to the main grid are approximated as\n\\begin{equation}\n\\epsilon_g = \\frac{\\textrm{R}}{\\textrm{V}^2}\\left(g^t\\right)^2\n\\end{equation}\n\nThe grid cost coefficient $\\textrm{C}_\\textrm{g}^t$ is the sum of the grid electricity price and the product of the carbon intensity of the generation mix at time $t$ and the Social Cost of Carbon which reflects the long-term societal cost of emitting greenhouse gases \\citep{ParryM}. The impacts of local decisions on upstream energy prices are neglected.  Grid losses are approximated using the nominal root mean square grid voltage $\\textrm{V}$ and the average resistance between the main grid and the distribution network $\\textrm{R}$ \\citep{Multiclass}, based on the assumption of small network voltage drops and relatively low reactive power flows \\citep{Coffrin2012}. The second-order dependency disincentivises large power imports and exports, which helps ensure interactions of transmission and distribution networks do not reduce system stability.\n\n\\begin{equation}\nc_\\textrm{d}^t = \\textrm{C}_\\textrm{d}\\sum_{i \\in \\mathcal{P}}{\\max\\left(- p_i^t,0\\right)}\n\\end{equation}\n\\noindent Distribution costs $c_\\textrm{d}^t$ are proportional to the distribution charge $\\textrm{C}_\\textrm{d}$ on exports. The resulting price spread between individual imports and exports decreases risks of network constraints violation by incentivising the use of local flexibility first \\citep{Morstyn2020_IntegratingP2P}. Distribution network losses due to power flows between prosumers are neglected so there is no second-order dependency.\n\\begin{equation}\nc_\\textrm{s}^t = \\textrm{C}_\\textrm{s}\\sum_{i \\in \\mathcal{P}}{\\left(b_{\\textrm{in},i}^t + b_{\\textrm{out},i}^t\\right)}\n\\end{equation}\n\n\\noindent Storage battery depreciation costs $c_\\textrm{s}^t$ are assumed to be proportional to throughput using the depreciation coefficient $\\textrm{C}_\\textrm{s}$, assuming a uniform energy throughput degradation rate \\citep{DufoLopez2014}.\n\n\\subsection{Constraints}\n\n\nLet $\\textrm{E}_0$,  $\\underline{\\textrm{E}}$ and $\\overline{\\textrm{E}}$ be the initial, minimum and maximum battery energy levels, $\\upeta_\\textrm{ch}$ and $\\upeta_\\textrm{dis}$ the charge and discharge efficiencies, and $\\overline{\\textrm{b}_\\textrm{in}}$ the maximum charge per time step. Demand $\\textrm{d}_{i,k}^{t_\\textrm{D}}$ is met by the sum of loads consumed $\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ at time $t_\\textrm{C}$ by prosumer $i$ for load of type $k$ (fixed or flexible) demanded at $t_\\textrm{D}$. The flexibility boolean $\\textrm{f}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ indicates if time $t_\\textrm{C}$ lies within the acceptable range to meet $\\textrm{d}_{i,k}^{t_\\textrm{D}}$. A Crank-Nicholson scheme \\citep{ISO2007} is employed to model heating, with $\\upkappa$ a 2x5 matrix of temperature coefficients, and $\\underline{\\textrm{T}}_i^t$ and $\\overline{\\textrm{T}}_i^t$ lower and upper temperature bounds. System constraints for steps $\\forall \\ t \\in \\mathcal{T}$ and prosumers $\\forall \\ i \\in \\mathcal{P}$ are:\n\n\\begin{itemize}\n\\item Prosumer and substation energy balance (see \\Cref{fig:EnergyBalance})\n\\begin{equation}\n\tp_i^t = c_i^t + h_i^t + \\frac{b_{\\textrm{in},i}^t}{\\upeta_\\textrm{ch}} - {\\upeta_\\textrm{dis}} b_{\\textrm{out},i}^t - \\textrm{p}_{\\textrm{PV},i}^t \n\\end{equation}\n\\begin{equation}\n\\sum_{i \\in \\mathcal{P}}{p_i^t} = g^t\n\\end{equation}\n\\item Battery energy balance \n\\begin{equation}\n\tE_i^{t+1} = E_i^t  + b_{\\textrm{in},i}^t - b_{\\textrm{out},i}^t - \\textrm{d}_{\\textrm{EV},i}^t \n\\end{equation}\n\\item Battery charge and discharge constraints\n\\begin{equation}\n\t \\textrm{E}_0 = E_i^{t_0}  = E_i^{t_\\textrm{end}} + b_{\\textrm{in},i}^{t_{\\textrm{end}}} - b_{\\textrm{out},i}^{t_{\\textrm{end}}} - \\textrm{d}_{\\textrm{EV},i}^{t_{\\textrm{end}}} \n\\end{equation}\n\\begin{equation}\n\t\\upmu_i^t\\underline{\\textrm{E}}_i \\leq E_i^t \\leq \\overline{\\textrm{E}}_i\n\\end{equation}\n\\begin{equation}\n\tb_{\\textrm{in},i}^t \\leq \\upmu_i^t \\overline{\\textrm{b}_\\textrm{in}}\n\\end{equation}\n\\begin{equation}\n\tb_{\\textrm{out},i}^t \\leq \\upmu_i^t \\overline{\\textrm{E}}_i\n\\end{equation}\n\n\\item Consumption flexibility --- the demand of type $k$ at time $t_\\textrm{D}$ by prosumer $i$ must be met by the sum of partial consumptions $\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ at times $t_\\textrm{C}...t_\\textrm{C}+\\textrm{n}_\\textrm{flex}$ within the time frame $\\textrm{n}_\\textrm{flex}$ specified by the flexibility of each type of demand in matrix $\\textrm{f}_{i,k,t_\\textrm{C},t_\\textrm{D}}$\n\\begin{equation}\\label{eq:demandmet}\n\t\\sum_{t_\\textrm{C}\\in\\mathcal{T}}{\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}} \\textrm{f}_{i,k,t_\\textrm{C},t_\\textrm{D}}} = \\textrm{d}_{i,k}^{t_\\textrm{D}} \n\\end{equation}\n\\item Consumption --- the total consumption at time $t_\\textrm{C}$ is the sum of all partial consumptions $\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}$ meeting parts of demands from current and previous time steps $t_\\textrm{D}$:\n\\begin{equation}\\label{eq:totalcons}\n\t\\sum_{t_\\textrm{D}\\in\\mathcal{N}}{\\hat{c}_{i,k,t_\\textrm{C},t_\\textrm{D}}}= c_{i,k}^{t_\\textrm{C}} \n\\end{equation}\n\n\\item Heating --- the workings to obtain this equation are included in \\Cref{app:heating}:\n\\begin{equation}\\label{eq:main_heating}\n\\begin{bmatrix}\nT_{\\textrm{m},i}^{t+1}\\\\\nT_{\\textrm{air},i}^{t+1} \n\\end{bmatrix}\n = \\upkappa \n \\begin{bmatrix}\n1,\nT_{\\textrm{m},i}^{t},\n\\textrm{T}_{\\textrm{e}}^t,\n\\upphi^t,\nh_i^t\n\\end{bmatrix}^\\intercal\n\\end{equation}\n\\begin{equation}\n\\underline{\\textrm{T}}_i^t \\leq T_{\\textrm{air},i}^t \\leq \\overline{\\textrm{T}}_i^t\n\\end{equation}\n\\item Non-negativity constraints \n\\begin{equation}\n\tc_i^t, h_i^t,E_i^t, b_{\\textrm{in},i}^t, b_{\\textrm{out},i}^t, \\hat{c}_{i,l,t_\\textrm{C},t_\\textrm{D}} \\geq 0\n\\end{equation}\n\\end{itemize}\n\nWhile the proposed framework could accommodate the use of idiosyncratic satisfaction functions to perform trade-offs between flexibility use and users' comfort, no such trade-offs are considered in this paper, with comfort requirements for temperature and EV usage always being met. Field evaluations have shown that programmes that do not maintain thermal comfort are consistently overridden, increasing overall energy use and costs \\citep{Sachs2012}, while interference in consumption patterns and temperature set-points cause dissatisfaction \\citep{Vazquez-Canteli2019}.  Meeting fixed domestic loads,  ensuring sufficient charge for EV trips, and maintaining comfortable temperatures are therefore set constraints. \n\n\\section{Reinforcement learning methodology}\\label{RLSection}\nThe MARL approach is now presented in which independent prosumers learn to make individual decisions which together maximise the statistical expectation of the objective function in \\Cref{system}.  \n\nAt time step $t \\in \\mathcal{T}$, each agent is in a state $s_i^t \\in \\mathcal{S}$ corresponding to accessible observations (here the time-varying grid cost), and selects an action $a_i^t \\in \\mathcal{A}$ as defined in \\Cref{sec:agentdecision}. This action dictates the decision variables in \\Cref{variables} $b_{\\textrm{in},i}^t$, $b_{\\textrm{out},i}^t$, $h_i^t$ and $c_i^t$. The environment then produces a reward $r^t \\in \\mathcal{R}$ which corresponds to the share $\\hat{F}_t$ of the system objective function presented in \\Cref{objfunc} and agents transition to a state $s_i^{t+1}$. Agents learn individual policies $\\pi_i$ by interacting with the environment using individual, decentralised fixed-size Q-tables.\n\nWe first introduce the Q-learning methodology.  Then, the mapping between the RL agent action and the decision variables in \\Cref{variables} is presented. Finally, we propose variations on the learning method, with different experience sources, multi-agent structures and reward definitions.\n\n\\subsection{Q-Learning}\nWhile any reinforcement learning methodology could be used with the framework proposed in this paper, here we focus on Q-learning, a model-free, off-policy RL methodology.  Its simplicity and proof of convergence make it suited to developing novel learning methodologies in newly defined environments \\citep{Vazquez-Canteli2019}. State-actions values $Q(s,a)$ represent the expected value of all future rewards $r_t$ $\\forall \\ t \\in \\mathcal{T}$ when taking action $a$ in state $s$ according to policy $\\pi$:\n\\begin{equation}\nQ(s,a) \\triangleq E^{\\pi}{[r_{t} + \\gamma r_{t+1} + \\gamma^2r_{t+2}...|s_t = s, a_t = a ]}\n\\end{equation}\n\n\\noindent where $\\gamma$ is the discount factor setting the relative importance of future rewards. Estimates are refined incrementally as\n\\begin{equation}\n\\hat{Q}(s,a)\\leftarrow \\hat{Q}(s,a) + \\alpha\\delta\n\\end{equation}\nwhere $\\delta$ is the temporal-difference error,\n\\begin{equation}\n\\delta = \\left(r_t +\\gamma \\hat{V}(s^\\textrm{next})-\\hat{Q}(s,a)\\right)\n\\end{equation}\n$\\hat{V}$ is the state-value function estimate,\n \\begin{equation}\n\\hat{V}(s) = \\max_{a^* \\in \\mathcal{A}(s)}{\\hat{Q}(s,a^*)}\n\\end{equation}\nand $\\alpha$ is the learning rate. In this work we use hysteretic learners, i.e. chiefly optimistic learners that use an increase rate superior to the decrease rate in order to reduce oscillations in the learned policy due to actions chosen by other agents \\citep{Matignon2012, Matignon2007}. For $\\beta < 1$:\n\\begin{equation}\n  \\alpha  =\n    \\begin{cases}\n      \\alpha_0 & \\text{if $\\delta > 0$}\\\\\n      \\alpha_0\\beta & \\text{otherwise}\\\\\n    \\end{cases}       \n\\end{equation}\n\nAgents follow an $\\epsilon$-greedy policy to balance exploration of different state-action pairs and knowledge exploitation. The greedy action with highest estimated rewards is selected with probability $1-\\epsilon$ and random actions otherwise. \n\\begin{equation}\n\\label{eqn:greedy}\n  a^*  =\n    \\begin{cases}\n      \\argmax_{\\ a^* \\in \\mathcal{A}}\\hat{Q}(s,a^*) & \\text{if $x \\sim U(0,1) > \\epsilon$}\\\\\n      a \\sim p(a) = \\frac{1}{|\\mathcal{A}|} \\ \\forall \\ a \\ \\in \\mathcal{A}& \\text{otherwise}\\\\\n    \\end{cases}       \n\\end{equation}\n\nHenceforth, we refer to the estimates $\\hat{Q}$ and $\\hat{V}$ as $Q$ and $V$ to reduce the amount of notation.\n\n\\subsection{Agent state}\nThe agent state is defined by the time-dependent grid cost coefficient $\\textrm{C}_\\textrm{g}^t$, i.e. the sum of the grid electricity price and the product of the carbon intensity of the generation mix at time $t$ and the social cost of carbon.\n\nTo convert the RL policy action into local decisions, the agent also requires information on their current PV generation, battery level, flexible loads and indoor air temperature, as described below in \\Cref{sec:agentdecision}.\n\n\\subsection{Agent action}\\label{sec:agentdecision}\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{inkscape_mu_4.pdf}\n\n\\end{center}\n\\caption{Decision variable $\\psi$. Sections 1-5 denote the trade-off regimes described in \\Cref{sec:agentdecision}. At each step, the fixed requirements for loads, heat and upcoming EV trips are first met. The $\\psi$ decision then applies to the remaining flexibility, from maximal energy exports (full use of flexibility) at $\\psi = 0$, to maximal energy imports (no use of flexibility) at $\\psi = 1$. $\\textrm{d}_\\textrm{tot}$ and $\\textrm{d}_\\textrm{fixed}$ are the sum of household and heating loads with and without their flexible component. If fixed loads cannot be fully met by PV energy,  the residual is met by storage and imports (2). If there is additional PV energy after meeting all loads, it can be stored or exported (4).}\n\\label{fig:mu}\n\\end{figure*}\n\nLarge action spaces compound the curse of dimensionality in Q-learning and waste exploration resources \\citep{Powell2011}. At each time step, the decision variables in \\Cref{system} controlling the flows in and out of the battery$b_{\\textrm{in},i}^t$ and $b_{\\textrm{out},i}^t$, the electric heating consumption $h_i^t$ and the prosumer consumption $c_i^t$ for household $i$ are therefore synthesised into a single variable $\\psi \\in [0,1]$ controlling the use of available local flexibility. \\Cref{fig:mu} shows how consumption (for domestic loads and heat), imports and storage change with $\\psi$.\n\nAt each step, the fixed requirements for loads, heat and upcoming EV trips are first met. The $\\psi$ decision then applies to the remaining flexibility. In conditions deemed optimal for energy exports $\\psi = 0$, all initial storage and residual PV generation is exported and flexible loads are delayed. On the other end, a \\emph{passive} agent does not utilise its flexibility and uses the \\emph{default} action $\\psi = 1$, maximising imports with EVs charged when plugged in and no flexible loads delayed.  Intermediate imports trade-offs are mapped on \\Cref{fig:mu}:\n \n\\begin{enumerate}\n\\item From exporting all to none of the initial storage $E_i^t$\n\\item From meeting fixed loads $\\textrm{d}_{i,\\textrm{fixed}}^t$ with the energy stored to importing the required amount \n\\item From no to maximum flexible consumption $\\textrm{d}_{i,\\textrm{tot}}^t$ \n\\item From exporting to storing PV energy $\\textrm{p}_{\\textrm{PV},i}^t$ remaining after meeting loads\n\\item From importing no additional energy to filling up the battery to capacity $\\overline{\\textrm{E}}_i$\n\\end{enumerate}\n\nCostlier actions incurring battery depreciation, losses and export costs are towards either $\\psi$ extreme, only used in highly beneficial situations (convex local costs function in the lower plot of \\Cref{fig:mu}). Ranking actions consistently ensures agents do not waste resources trialling sub-optimal combinations of decisions. For example, it is more cost-efficient to first absorb energy imports by consuming flexible loads, and only use the battery (incurring costs) if imports are large.  \n\nNote that although this action space is continuous, it can be discretised into intervals for implementation in Q-learning. \n\n\\subsection{Variations of the learning method}\\label{methodologies}\nDifferent experience sources, reward definitions and MARL structures are proposed within the MARL approach. The performance of these combinations of algorithmic possibilities will be assessed in \\Cref{results} to inform effective model design.\\\\\n\n\\subsubsection{Experience sources}\nIn data-driven strategies, the learning is determined by the collected experience.\n\\begin{itemize}\n\\item \\textbf{Environment exploration}. Traditionally, agents collect experience by interacting with an environment \\citep{Sutton1998}.\n\n\\item \\textbf{Optimisations}. A novel approach collects experience from optimisations. Learning from entities with more knowledge or using knowledge more effectively than randomly exploring agents has previously been proposed, as with agents \\say{mimicking} humans playing video games \\citep{Grandmaster}. Similarly, agents learn from convex \\say{omniscient} optimisations on historical data with perfect knowledge of current and future variables.  This experience is then used under partial observability and control for stable coordination between prosumers at scale. Note in this case that, although the MARL learning and implementation are model-free, a model of the system is used to run the convex optimisation and produce experience to learn from.  A standard convex optimiser uses the same data that would be used to populate the environment explorations but solves over the whole day-horizon with perfect knowledge of all variables using the problem description in \\Cref{system}. Then, at each time step, the system variables are translated into equivalent RL $\\{s_t,a_t,r_t, s_{t+1}\\}$ tuples for each agent, which are used to update the policies in the same way as for standard Q-learning as presented below.\\\\\n\\end{itemize}\n \n\\subsubsection{MARL structures} Both the centralised and decentralised structures proposed use fixed-size $|\\mathcal{S}| \\times |\\mathcal{A}|$ Q-tables corresponding to individual state-action pairs. The size of a global Q-table referencing all possible combinations of states and actions would grow exponentially with the number of agents. This would limit scalability due to memory limitations and exploration time requirements. Moreover, as strategies proposed in this paper are privacy-preserving, only local state-action pairs are used for individual action selection, wasting the level of detail of a global Q-table.\n\n\\begin{itemize}\n\\item \\textbf{Distributed learning}. Each agent $i$ learns its $Q_i$ table with its own experience. No information is shared between agents. \n\n\\item \\textbf{Centralised learning}. A single table $Q_\\textrm{c}$ uses experience from all agents during pre-learning. All agents use the centrally learned policy for decentralised implementation.\n\\end{itemize}\n\n\\subsubsection{Reward definitions}\nThe reward definition is central to learning as its maximisation forms the basis for incrementally altering the policy \\citep{Sutton1998}.  Assessing the impact of individual actions on global rewards accurately is key to the effective coordination of a large number of prosumers. In the following,  the Q-tables $Q^0$, $Q^\\textrm{diff}$,$Q^\\textrm{A}$ and $Q^\\textrm{count}$ may be either agent-specific $Q_i$ or centralised $Q_\\textrm{c}$ based on the MARL structure. We proposed four variations of the Q-table update rule for each experience step tuple collected $(s_i^t, a_i^t, r^t,s_i^{t+1})$.\n \\begin{equation}\nQ(s_i^t, a_i^t)\\leftarrow Q(s_i^t,a_i^t) + \\alpha \\delta\n\\end{equation}\n\\begin{itemize}\n\\item \\textbf{Total reward}. The instantaneous total system reward $r^t = \\hat{F}_t$ is used to update the Q-table $Q^0$.\n\\begin{equation}\n\\delta = r^t + \\gamma V^0(s_i^{t+1}) - Q^0(s_i^t, a_i^t)\n\\end{equation}\n\\item \\textbf{Marginal reward}. The difference in total instant rewards $r^t$ between that if agent $i$ selects the greedy action and that if it selects the default action is used to update $Q^\\textrm{diff}$ \\citep{Wolpert2002}. The default action $a_\\textrm{default}$ corresponds to $\\psi = 1$, where no flexibility is used. The default reward $r^t_{a_{i}=a_\\textrm{default}}$, where all agents perform their greedy action apart from agent $i$ which performs the default action, is obtained by an additional simulation.\n\\begin{equation}\n\\delta = \\left(r^t - r^t_{a_{i}=a_\\textrm{default}}\\right) + \\gamma V^\\textrm{diff}(s_i^{t+1}) - Q^\\textrm{diff}(s_i^t, a_i^t)\n\\end{equation}\n\\item \\textbf{Advantage reward}. The post difference between $Q^0$ values when $i$ performs the greedy and the default action is used. This corresponds to the estimated increase in rewards not just instantaneously but over all future states, analogously to in \\citep{Foerster2018}. No additional simulations are required as the Q-table values are refined over the normal course of explorations.\n\\begin{equation}\n\\delta = \\left(Q^0(s_i^t, a_i^t) - Q^0(s_i^t, a_{a_i=a_\\textrm{default}})\\right) - Q^\\textrm{A}(s_i^t, a_i^t)\n\\end{equation}\n\\item \\textbf{Count}. The Q-table stores the number of times each state-action pair is selected by the optimiser. \n \\begin{equation}\n\\alpha\\delta = 1\n\\end{equation}\n\\end{itemize}\n\n\\section{Input Data}\\label{data}\n\n\\begin{table*}[!t]\n   \\newcommand{\\rule[-10pt]{0pt}{20pt}}{\\rule[-10pt]{0pt}{20pt}}\n   \\newcommand{\\rule[-5pt]{0pt}{30pt}}{\\rule[-5pt]{0pt}{30pt}}\n\\begin{tabularx}{\\textwidth}{ c|m{5cm}|m{5cm}}\n\\toprule\n& Normalised profile  &  Scaling factor \\\\ \n\\hline\n   PV \\rule[-10pt]{0pt}{20pt} &  Randomly selected from current month bank $b_{t+1}=(m)$   & \n   \\multirow{2}{=}{\\setlength\\parskip{\\baselineskip}%\n   Computed as $\\lambda_{t+1} = \\lambda_{t} + x$, where $x \\sim \\Gamma\\left(\\alpha(b_{t},b_{t+1}),\\beta(b_{t},b_{t+1})\\right)$} \\rule[-10pt]{0pt}{20pt} \\\\\n    \\cline{1-2}\n    Load \\rule[-5pt]{0pt}{30pt} &\\multirow{2}{=}{\\setlength\\parskip{\\baselineskip}%\nCluster selected based on transition probability $p(k_{t+1} | k_t, w_t, w_{t+1})$ \\newline Normalised profile  randomly selected from bank $b_{t+1} = (k_{t+1}, w_{t+1})$} \\rule[-5pt]{0pt}{30pt} &\n \\\\ \n\\cline{1-1}\n\\cline{3-3} \n   EV \\rule[-5pt]{0pt}{30pt} & & Random variable from discrete distribution $p(\\lambda_{t+1}|\\lambda_t, b_t, b_{t+1})$ \\rule[-10pt]{0pt}{20pt} \\\\\n \\bottomrule\n\\end{tabularx}\n  \\caption{Markov chain mechanism for selecting behaviour clusters, profiles and scaling factors for input data in subsequent days}\n\\label{tab:loadnextday}\n\\end{table*}\n\nThis section presents the data that is fed into the model presented in \\Cref{system}. Interaction with this data will shape the policies learned through RL \\citep{Sutton1998} and should reflect resource intermittency and uncertainty to maximise the expectation of rewards in a robust way without over-fitting. EV demand $\\textrm{d}_{\\textrm{EV},i}^t$ and availability $\\upmu_i^t$, PV production $\\textrm{p}_{\\textrm{PV},i}^t$ and electricity consumption $\\textrm{d}_i^{t}$ are drawn from large representative datasets.\n\n\\subsection{Data selection and pre-processing}\nLoad and PV generation profiles are obtained from the Customer Led Network Revolution (CLNR), a UK-based smart grid demonstration project \\citep{TC1a,TC5}, and mobility data from the English National Travel Survey (NTS) \\citep{DepartmentforTransport2019}. The NTS does not focus on EVs only and offers a less biased view into the general population's travel pattern than small-scale EV trials data, both due to the smaller volume of data available compared to for generic cars and because the self-selected EV early trial participants may not be representative of patterns once EVs become widely adopted. It is implicitly assumed that electrification will not affect transport patterns \\citep{Crozier2018}.\n\nNTS data from 82,455 households from 2002 to 2017 results in 1,272,834 full days of travel profiles. Load and PV data from 11,907 customers between 2011 and 2014 yields 620,702 and 22,670 full days of data, respectively.  Profiles are converted to hourly resolution and single missing points replaced with the figure from the same time the day or week before or after which has the lowest sum of squares of differences between the previous and subsequent point. Tested with available data, this yields absolute errors with mean 0.13 and 0.08 kWh and 99th percentile 1.09 and 0.81 kWh for PV and load data.  PV sources have nominal capacities between 1.35 and 2.02 kWp.\n\nThe at home-availability of the vehicles is inferred from the recorded journeys' origin and destination.  EV energy consumption profiles are obtained using representative consumption factors from a tank-to-wheel model proposed in \\citep{Crozier2018}, dependent on travel speed and type (rural, urban, motorway). \n\n\\subsection{Markov chain}\n\\begin{figure*}[!b]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{f_load_EV_omni.pdf}\n\\end{center}\n\\caption{Scaling factors for normalised profiles (i.e. total daily loads in kWh) in subsequent days. Linear correlation can be observed for the load profiles,  while more complex patterns are exhibited for EV consumption. $\\rho$ is the Pearson correlation coefficient.}\n\\label{fig:Corr}\n\\end{figure*}\n\nDuring learning, agents continuously receive experience to learn from. However,  numerous subsequent days of data are not available for single agents. We design a Markov chain mechanism to feed consistent profiles for successive days, using both consistent scaling factors and behaviour clusters.\n\nDaily profiles for load and travel are normalised such that $\\sum_{t=0..24}{x^t}=1$, and clustered using K-means, minimising the within-cluster sum-of-squares \\citep{Lloyd1982} in four clusters for both weekday and weekend data (with one for no travel). The features used for load profiles clustering are normalised peak magnitude and time and normalised values over critical time windows, and those for travel are normalised values between 6 am and 10 pm. PV profiles were grouped per month.\n\nProbabilistic Markov chain transition rules are shown in \\Cref{tab:loadnextday}. Transition probabilities for clusters $k$ and scaling factors $\\lambda$ are obtained from available transitions between subsequent days in the datasets for each week day type $w$ (week day or weekend day). \\Cref{fig:Corr} shows that subsequent PV and load scaling factors follow strong linear correlation, with the residuals of the perfect correlation following gamma distributions with zero mean, whereas EV load scaling factors follow more complex patterns, so transitions probabilities are computed between 50 discrete intervals.\n\n\\section{Case study results and discussion}\\label{results}\nThis section compares the performance of the residential flexibility coordination strategies presented in \\Cref{RLSection} to baseline and upper bound scenarios for increasing numbers of prosumers. The performance of traditionally used MARL strategies drops at scale, while that of the novel optimisation-based methodology using marginal rewards is maintained.\n\n\\subsection{Set-up}\nThe MARL algorithm is trained in off-line simulations using historical data prior to online implementation. This means agents do not trial unsuccessful actions with real-life impacts during learning. Moreover, the computation burden is taken prior to implementation, while prosumers only apply pre-learned policies, avoiding the computational challenges of large-scale real-time control. \n\nThe learning occurs over 50 epochs consisting of an exploration, an update and an evaluation phase. First,  the environment is explored over two training episodes of duration $|\\mathcal{T}| = 24$ hours.  Learning in batches of multiple episodes helps stabilise learning in the stochastic environment.  Then,  Q-tables are updated based on the rules presented in \\Cref{methodologies}. Finally, an evaluation is performed using a deterministic greedy policy on new evaluation data.  Ten repetitions are performed such that the learning may be assessed over different trajectories.\n\nThe Social Cost of Carbon is set at 70 \u00a3\/tCO$_2$, consistent with the UK 2030 target \\citep{Hirst2018}. Weather \\citep{WeatherWunderground2020}, electricity time-of-use prices \\citep{OctopusEnergy2019} and grid carbon intensity \\citep{NationalGridESO2020} are from January 2020, where relevant specified for London, UK. The low solar heat gains in January are neglected \\citep{Brown2020}. Other relevant parameters for the case studies are listed in \\Cref{app:inputs}.\n\nAs performed on a Intel(R) Core(TM) i7-9800X CPU @ 3.80GHz, computation time for a learning trajectory is $2^\\prime45^{\\prime\\prime}$ for one agent and $97^\\prime5^{\\prime\\prime}$ for 30 agents, including evaluation points. The policy can then be directly applied at the household level during operation.\n\nCase study results using different experience sources, reward definitions and MARL structures are presented in \\Cref{fig:results}. Acronyms for each strategy are tabulated in the legend. Positive values denote savings relative to a baseline scenario where all agents are passive, i.e. not using their flexibility with EVs charged immediately and no flexible loads delayed. As the Q-learning policies are first initialised with zero values, in the first epoch of learning completely random action values are chosen, which provides rewards far below the baseline. As agents collect experience and update their policies at each epoch, improved policies are learned, some of which are able to outperform the baseline. An upper bound is provided by results from \\say{omniscient} convex optimisations, which are however not achievable in practice for three main reasons. Firstly, they use perfect knowledge of all the environment variables in the present and future, despite uncertainty in renewable generation, mix of the grid, and customer behaviour. Optimisation with inaccurate data would lead to suboptimal results. Secondly, prosumers may not be willing to yield their data and direct control to an external entity. Finally, central optimisations become computationally expensive for real-time control of large numbers of prosumers.\n\n\\subsection{Results}\nResults presented in \\Cref{fig:results} show that only the algorithms learning from optimisations maintained stable coordination performance at scale, while the performance of traditionally used MARL algorithms would drop in this context of stochasticity and partial observation.  The optimisation-based algorithm which uses marginal rewards (MO) performed best.  We further elaborate on the results in the subsections below.\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{results_vs_nag_20211210.pdf}\n\n\n\\end{center}\n\\caption{The left-hand side plot shows the five-epoch moving average of evaluation rewards relative to baseline rewards for a single prosumer. The right-hand side plot shows the mean of the final 10 evaluations against the number of prosumers.  Lines show median values and shaded areas the 25th and 75th percentiles over the 10 repetitions. The best-performing MARL structure is displayed for each exploration source and reward definition pair.  The performance of the baseline MARL algorithm (TE, orange) drops as the number of concurrently learning agents in the stochastic environment increases; the best-performing alternative algorithm proposed (MO, purple) maintains high performance at scale.}\n\\label{fig:results}\n\\end{figure*}\n\n\\subsubsection{Environment exploration-based learning}\nThe centralised MARL structure is favoured for environment exploration-based learning (continuous lines in \\Cref{fig:results}).  A single policy uses experience collected by all agents, rather than each agent learning from their own experience only.  \n\n\\Cref{fig:results} shows that environment exploration-based MARL using total rewards (TE, orange), the baseline MARL framework, exhibits a high performance for a single agent. However, savings drop as the number of cooperating agents increases, down to around zero from ten agents.  Coordination challenges arise for independent learners to isolate the contribution of their actions to total rewards from the stochasticity of the environment, compounded by other simultaneously learning agents' random explorations, and the non-stationarity of their on-policy behaviour \\citep{Matignon2012}.  \n\nUsing advantage rewards (AE, grey), based on estimates of the long-term value of actions relative to that of the baseline action, yields superior results beyond two agents. However, as AE uses the total reward $Q^0$-table as an intermediary step, results similarly drops for increasing numbers of agents.\n\nUsing marginal rewards (ME, dark green), the value of each agent's action relative to the baseline action is singled out immediately by an additional simulation and used as a reward at each time step.  This improves the performance relative to TE and AE for five agents and more, though still with declining performance as the number of agents increases.\n\n\\subsubsection{Optimisation-based learning}\nOptimisation-based learning generally favours the distributed MARL structure, with agents able to converge to distinct compatible policies (dashed lines in \\Cref{fig:results}).\n\nComparing trajectories in \\Cref{fig:results},  learning from the total rewards obtained by an optimiser (TO, light blue) yields lower savings than when using environment explorations (TE). The learned policies yield negative savings, i.e. would provide worse outcomes than inflexible agents. The omniscient optimiser takes precise, extreme decisions thanks to its perfect knowledge of all current and future system variables, importing at very high $\\psi$ values when it is optimal to do so.  RL algorithms on the other hand are used under partial observability, aiming for actions that statistically perform well under uncertainty.  Agents independently picking TO-based decisive actions in a stochastic environment do not yield optimal outcomes. Assessing the long-term advantage of actions from optimisations (AO, dark blue) follows a similar trend, whilst providing marginally superior savings relative to TO.\n\nOptimisation-based learning using marginal rewards (MO, purple) offers the highest savings as the additional baseline simulations are best able to isolate the contribution of individual actions from variations caused by both the environment and other agents.  When increasing the number of agents, the strategy is able to learn from optimal, stable, consistently behaving agents. Savings of 6.18p per agent per hour, or \u00a345.11 per agent per month are obtained on average for 30 agents, corresponding to a 33.7\\% reduction from baseline costs.  65.9\\% of savings stem from reduced battery depreciation, 20.32\\% from distribution grid congestion, 11.1\\% from grid energy, and 2.7\\% from greenhouse gas emissions.\n\nThe count-based strategy learning from optimisations (CO, light green) seeks to reproduce the state-action patterns of the omniscient optimiser with perfect knowledge of system variables and perfect control of agents for local decision-making under partial observability.  It provides results lower than the high performances of MO, though with a stable performance at scale. Savings of \u00a321.09 per agent per month on average for 30 agents are obtained. The battery and distribution grid costs increase by an equivalent of 6.0\\% and 7.7\\% of total savings respectively, while grid energy and greenhouse gas emissions costs reductions represent 59.7\\% and 54.0\\% of total savings.\n\nBoth the MO and CO strategies exhibit stable performance at scale, though converging to different types of policy. The MO policy saves more by smoothing out the charging and distribution grid utilisation profiles despite smaller savings in imports and emissions costs, while CO derives a larger advantage from the grid price differentials in grid imports, though with higher battery and distribution grid costs. The weight applied to each of those competing objectives in the objective function directly impacts the policies that are learned. Examples of how the individual home energy management system decision variables (heating, energy consumption, battery charging) vary based on the controller are illustrated in \\Cref{app:example_case_study}.\n\nOverall, the new class of optimisation-based learning performs significantly better across different numbers of prosumers, with higher savings and lower inter-quartile range than environment-based learning at scale. This superior performance requires computations to run optimisations on historical data, and to perform baseline simulations to compute marginal rewards, though computational time for pre-learning is not strictly a limiting factor as it is performed off-line ahead of implementation. \n\nA fundamental challenge in MARL has been the trade-off between fully centralised value functions, which are impractical for more than a handful of agents, or, in a more straightforward approach, independent learning of individual action-value functions by each agent in independent Q-learning (IQL) \\citep{Tan1993}. However, an ongoing issue with this approach has been that of convergence at scale, as agents do not have explicit representations of interactions between agents, and each agent's learning is confounded by the learning and exploration of others \\citep{Rashid2020}. As shown in \\Cref{fig:results}, the Pareto selection, non-stationarity and stochasticity issues presented in \\Cref{sec:gapanalysis} have prevented environment exploration-based learners from achieving successful MARL cooperation at scale for agents under partial observability in a stochastic environment. This case study of coordinated residential energy management shows that the novel combination of marginal rewards, which help agents isolate their marginal contribution to total rewards, and the learning from results of convex optimisations, where agents learn successful policy equilibriums from omniscient, stable, and consistent solutions, offer significant improvements on these scalability and convergence issues.  \n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, a novel class of strategies has addressed the scalability issue of residential energy flexibility coordination in a cost-efficient and privacy-preserving manner.  The combination of off-line optimisations with multi-agent reinforcement learning provides high, stable coordination performance at scale.\n\nWe identified in the literature that the concept of RL-based implicit energy coordination, where energy prosumers cooperate towards global objectives based on local information only, had been under-researched beyond frequency droop control with limited number of agents. The scalability of such methods was identified as a key gap that we have sought to bridge. The novel coordination mechanism proposed in this paper thus satisfies the criteria for successful residential energy coordination set out in the introduction, as tested with large banks of real data in the case studies:\n\n\\begin{itemize}\n\n\\item Computational scalability: The scalability of traditional learning algorithms is significantly improved thanks to fixed-size Q-tables to avoid the curse of dimensionality, so that policies can be learned for larger number of agents. The proposed method does not require expensive communication and control appliances at the prosumer level, as pre-learned policies are directly applied with no further communication and no exponential time real-time optimisations needed. This is a crucial benefit for applications with physical limitations in hardware availability and processing time.  \n\n\\item Performance scalability: The coordination performance remains high for increasing numbers of prosumers despite the challenges of partial observability,  environment stochasticity and concurrently learning of agents, thanks to learning from the results of global omniscient optimisations on historical data, and to rewards signals that isolate individual contributions to global rewards. Significant value of \u00a345.11 per agent per month was obtained in the presented case study for 30 agents,  thanks to savings in energy,  prosumer storage and societal greenhouse gas emissions-related costs. Those savings do not drop with increasing number of agents, as opposed to with standard MARL approaches.\n\n\\item Acceptability: The approach does not rely on sharing of personal data, thermal discomfort, or hindrance\/delay of activities, and the appliances are controlled locally. This cost-efficient and privacy-preserving implicit coordination approach could help integrate distributed energy resources such as residential energy, otherwise excluded from energy systems' flexibility management. \n\n\\end{itemize}\n\nImportant future work is a more detailed assessment of the impacts of the coordination strategies on power flows, as well as an evaluation of the generalisation and adaptability potential of policies when used by other households or if household characteristics change over time.  Moreover, while all agents readily reduce individual costs through participation in the framework, further game-theoretic tools could be used to design a post-operation reward scheme.\n\n\\section*{Acknowledgement}\nThis work was supported by the Saven European Scholarship and by the UK Research and Innovation and the Engineering and Physical Sciences Research Council (award references EP\/S000887\/1, EP\/S031901\/1, and EP\/T028564\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOver the last decade topological phases of matter have attracted many attentions for providing tremendous insights both in fundamental and experimental aspects of condensed matter physics \\cite{Chiu2016}. In addition to the gapped phases which initially ignited the field, recently discovery of gapless topological phases stimulated many works towards the understanding of nontrivial topology of gapless systems \\cite{review1}. Among those, Weyl and Dirac semimetals are of particular interest due to their experimental discovery and many unique physical properties \\cite{review1}. The Dirac\/Weyl smimetals (DSMs\/WSMs) are characterized by isolated point touchings of two degenerate\/nondegenerate bands in momentum space. Weyl nodes can be generated by splitting of degenerate Dirac nodes usually via breaking of either time-reversal ($\\mathcal{T}$) or inversion ($\\mathcal{I}$) symmetries or both. In the standard form, the low energy excitations near the Weyl nodes, disperse linearly along all three momentum directions with each node carries monopole charge of $\\pm 1$. As a result, on the surface, there exists a Fermi arc that connects a pair of Weyl nodes with opposite chiralities. Recently, generalization of WSMs to multi-Weyl nodes have been proposed where each nodes have higher-order dispersion in one or more directions and consequently caries monopole charge of larger than one \\cite{MultiWeyl1,MultiWeyl2}. \\\\\n\\indent On the other hand, DSMs\/WSMs can also be classified into type-I and type-II, based on the tilting of their nodes. In the standard type-I WSMs, Fermi surfaces are point-like while in the type-II WSMs, Weyl nodes are tilted resulting in formation of electron and hole pockets producing finite density of states at the Fermi level \\cite{typeII1,typeII2}. In a conventional type-I and II WSMs, two Weyl nodes with opposite chiralities have same types, however, recently, a theoretical proposal \\cite{hybridWeyloriginal}, introduced a new WSM where a pair of Weyl nodes with different chiralities can have different types, forming the so-called \\emph{hybrid Weyl semimetals}.\\\\\n\\indent Besides the DSMs\/WSMs, another class of three-dimensional nodal semimetals are Luttinger semimetals (LSMs) where possess a quadratic band touching (QBT) point between doubly degenerate valence and conduction bands of $J=3\/2$ (effective) fermions at an isolated point in the Brillouin zone.\nThe LSM provides the low-energy description for a plethora of both strongly and weakly correlated compounds, such as the 227 pyrochlore iridates (Ln2Ir2O7, with Ln being a lanthanide element) \\cite{LSMapp1,LSMapp2, LSMapp3,LSMapp4}, half-Heusler compounds (ternary alloys such as LnPtBi, LnPdBi) \\cite{LSMapp5,LSMapp6}, HgTe \\cite{semiconductor1,semiconductor2,QBToriginal,Ruan2016,mottQBT}, and gray-tin \\cite{LSMapp7,LSMapp8}.\nMoreover, LSMs proved to show many interesting behaviors\\cite{LSMprop1,LSMprop2,LSMprop3,LSMprop4,LSMprop5}, specially in the presence of interaction, for example, investigation of the magnetic and superconducting orders actively have been explored \\cite{LSMapp9, Roy-Ghorashi2019, GhorashiPRB2017, GhorashiPRL2018, Ghorashi2019,SatoSC32,LSMBoettcher2,LSMSCLiu1,LSMBoettcher1,Szabo-Bitan32-2018}.\\\\\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{FlashpickickedLSMv4.png}\n    \\caption{Schematic picture of summary of the results obtained in this work. Starting from a Luttinger semimetal via two different nonuniform ($k$-dependent) periodic kickings, where break inversion ($\\mathcal{I}$) and time-reversal ($\\mathcal{T}$) symmetries while preserving their combinations ($\\mathcal{IT}$), we have obtained a \\emph{hybrid dispersion Dirac semimetals} (e.g., $k_z\\Gamma_5$ where $\\Gamma_i$ are Dirac matrices [see text for details]) and tilted LSM (e.g., $k_z\\Gamma_0$). Then, by applying an external magnetic field, $J_z$, in parallel to the direction of nodes (or kick direction), \\emph{hybrid Weyl phases} can be generated.}\n    \\label{fig:adpic}\n\\end{figure*}\n\\indent The coexistence of various Weyl nodes with different charges and\/or types is an interesting phenomena that could help towards understanding as well as manipulation of the properties of various Weyl nodes in an equal footing setup. Besides a few works reporting the coexistence of type-I and II Dirac\/Weyl nodes \\cite{hybridWSMexp,hybridDSM1,hybridWSM1,WSMexp1&2,weylcoexPRL}, there have been also proposals\\cite{LSMFloq1,Ghorashi2018}, claiming dynamical generation of various Weyl nodes of different types and\/or charges in one system. Application of the light is shown to be a powerful method to change the material properties \\cite{FloquetRev}. In particular, the conversion of a topologically trivial phase into a nontrivial one using periodic driving has attracted enormous attention in the past decade \\cite{FloquetRev,Floquet1, Floquet2, Floquet3}. Specifically many proposals on Floquet WSMs in various systems exist, such as Dirac semimetals \\cite{FloqWeyl1,FloqWeyl2}, band insulators \\cite{FloqWeyl3}, stacked graphene \\cite{FloqWeyl4}, line-nodal semimetals \\cite{FloqWeyl2,FloqWeyl5}, and crossing-line semimetals \\cite{FloqWeyl6,FloqWeyl7}. Also, proposals have been made to create tunable WSMs in pyrochlore iridates with Zeeman fields \\cite{Weylpyro1,Weylpyro2}. Very recently, using circular\/elliptic polarized light on Luttinger Hamiltonian in the high-frequency limit we have shown a very rich phase diagram of various Weyl semimetals, including coexistence of type-I and II as well as single and double Weyl nodes \\cite{Ghorashi2018} . \\\\\n \\indent Despite the several dynamical proposals for the generation of different Weyl phases, a promising setup for the realization of hybrid Dirac and Weyl semimetals is still lacking. In this work, we tackle this issue by an alternative way of periodic driving, in particular the periodic kicking. Using the periodic $\\delta$-function kicks can typically simplify theoretical studies by allowing to perform calculations analytically to a large extent (in contrast to sinusoidal driving or elliptical\/circular light) \\cite{ Floquet3,kicking2}.\nHowever, for the sake of comparison we also briefly discuss the smooth driving case to show that some of the features of our discussion can be hold up in smooth driving setup as well as long as the perturbation breaks inversion (uniaxially) and time-reversal symmetres but preserve their combinations. In this paper, we show two examples of such perturbations which along with an external magnetic field induce various hybrid Dirac and Weyl phases, including a new \\emph{hybrid dispersion Dirac semimetal}. Figure.~\\ref{fig:adpic} summarizes the result of this work.\n\n\\section{Model and Formalism}\n\\subsection{Model}\nWe start with reviewing the main ingredients of Luttinger Hamiltonian in the non-equilibrium limit, which can be represented as,\n\n\\begin{align}\n    H_{L}(\\vec{k})=\\int \\frac{d^3\\vec{k}}{(2\\pi)^3}\\Psi^{\\dagger}_{\\vec{k}} h_L(\\vec{k}) \\Psi_{\\vec{k}},\n\\end{align}\nwhere\n\\begin{align}\n    h_{L}(\\vec{k})=&(\\frac{k^2}{2m_0}-\\mu)\\Gamma_0-\\frac{1}{2m_1}\\sum^3_{a=1} d_a(\\vec{k})\\Gamma_a\\cr\n    -&\\frac{1}{2m_2}\\sum^5_{a=4} d_a(\\vec{k})\\Gamma_a\n\\end{align}\nwhere $k^2=k^2_x+k^2_y+k^2_z$ and,\n\\begin{align}\n    \\Psi^T_{k}= (c_{\\vec{k},3\/2}, c_{\\vec{k},1\/2}, c_{\\vec{k},-1\/2}, c_{\\vec{k},-3\/2}).\n\\end{align}\n$\\mu$ is the chemical potential measured from the band touching point. $\\Gamma_a$ are the well-known gamma matrices which are given by,\n\\begin{align}\n    \\Gamma_1=\\tau_3\\sigma_2,\\,\\, \\Gamma_2=\\tau_3\\sigma_1,\\,\\, \\Gamma_3=\\tau_2,\\,\\,\n    \\Gamma_4=\\tau_1,\\,\\, \\Gamma_5=\\tau_3\\sigma_3,\n\\end{align}\n and satisfy $\\{\\Gamma_a,\\Gamma_b\\}=\\delta_{a,b}$, while $\\Gamma_0$ is four dimensional identity matrix. $\\tau$ and $\\sigma$ denote space of sign and magnitude of spin projection $m_s\\in\\{\\pm 3\/2, \\pm 1\/2\\}$, respectively. $d_a(\\vex{k})$ are given as,\n \\begin{align}\n     d_1=\\sqrt{3}k_y k_z,\\,\\, d_2=\\sqrt{3}k_x k_z,\\,\\, d_3=\\sqrt{3}k_x k_y,\\,\\,\\cr\n     d_4=\\frac{\\sqrt{3}}{2}(k_x^2-k_y^2),\\,\\, d_5=\\frac{1}{2}(2k_z^2-k_x^2-k_y^2)\n \\end{align}\n The five mutually anticommuting $\\Gamma$ matrices can be written\nin terms of spin-$3\/2$ matrices according to,\n\\begin{align}\n    \\Gamma_1=&\\frac{1}{\\sqrt{3}}\\{J_y,J_z\\},\\,\\,\\Gamma_2=\\frac{1}{\\sqrt{3}}\\{J_x,J_z\\},\\,\\,\\Gamma_3=\\frac{1}{\\sqrt{3}}\\{J_x,J_y\\},\\cr\n    \\Gamma_4=&\\frac{1}{\\sqrt{3}}(J^2_x-J^2_y),\\,\\,\\Gamma_5=J^2_z-\\frac{5}{4},\n\\end{align}\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveup.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveopposite.pdf}\n    \\caption{The band structure for LSM along the $k_z$ axis with $\\lambda_1=0.6$ and (a) $\\lambda_2=0.1$, (b) $\\lambda_2=0.6$.}\n    \\label{fig:LSMbare}\n\\end{figure}\n Here we take the isotropic limit of $m_1=m_2 \\equiv m$. Therefore, the Luttinger Hamiltonian can also be written in an alternative way,\n\\begin{align}\n    h_L(\\vec{k})= [(\\lambda_1+5\\lambda_2\/2)k^2-\\mu]\\Gamma_0-2\\lambda_2(\\vec{J}.\\vec{k})^2\n\\end{align}\n with $\\vec{J}=(J_x,J_y,J_z)$ and $\\vec{k}=(k_x,k_y,k_z)$ and we used $\\lambda_1=1\/2m_0$ and $\\lambda_2=1\/4m$. $J_{x,y,z}$ are effective spin-$3\/2$ operators,\n \\begin{eqnarray}\n&\\,J_z=\\begin{bmatrix}\n        \\frac{3}{2} & 0 & 0 & 0 \\\\\n        0 & \\frac{1}{2} & 0 & 0\\\\\n        0 & 0 & -\\frac{1}{2} & 0 \\\\\n        0 & 0 & 0 & -\\frac{3}{2} \\\\\n      \\end{bmatrix},J_x=\\begin{bmatrix}\n        0 & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & 1 & 0\\\\\n        0 & 1 & 0 & \\frac{\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}\\cr\n      &\\,J_y=\\begin{bmatrix}\n        0 & \\frac{-i\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & -i & 0\\\\\n        0 & 1 & 0 & \\frac{-i\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}.\n\\end{eqnarray}\nThe energy dispersions are $E(k)=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}-\\mu$ for the $j=3\/2$ and the $j=1\/2$ bands, respectively. Four bands come in doubly degenerate pairs as a result of time-reversal (with antiunitary operator $\\mathcal{T}=\\Gamma_1\\Gamma_3\\mathcal{K}$ and $\\vec{k}\\rightarrow -\\vec{k}$ where $\\mathcal{K}$ is complex conjugation) and inversion ($\\mathcal{I}=I_{4\\times4}$ and $\\vec{k}\\rightarrow -\\vec{k}$) symmetries. For $\\lambda_{2}<2\\lambda_{1}$\n($\\lambda_{2}>2\\lambda_{1}$), the degenerate bands curve the same (opposite) way as shown in Figure.~(\\ref{fig:LSMbare}). In the case of both bands bending the same way, Eq. (2) is widely used to model heavy- and light-hole bands in zinc-blende semiconductors \\cite{semiconductor1} and many properties of such a dispersion have been studied in the literature, including a recent study on the realization of fully gapped topological superconductivity with \\emph{p}-wave pairing which has states with exotic cubic and linear dispersions coexisting on the surface \\cite{congjunWuPRL16,GhorashiPRB2017}. On the other hand, when bands bend oppositely, the model in Eq. (2) is known as Luttinger semimetal with QBT and is used to describe behavior of certain pyrochlore iridates as well as some doped half-Heusler alloys such as LaPtBi \\cite{Chadov2010,Lin2010,halfheusler3}.\n\n\n\\subsection{Periodic driving}\nA general time-dependent problem with $H(t)=H_0+V(t)$, can be tackled using Floquet theory when $V(t+T)=V(t)$ is periodic. To proceed, we can expand the periodic potential in a Fourier series as\n\\begin{align}\n    V(t)=V_0+\\sum^{\\infty}_{n=1} \\big(V_n e^{i\\omega nt}+V_{-n}e^{-i\\omega nt}\\big).\n\\end{align}\n In the limit of fast driving regime, in which the driving frequency is larger than any natural energy scale in the problem, one can obtain the effective Hamiltonian and Floquet operators perturbatively up to  $\\mathcal{O}(1\/\\omega^2)$ \\cite{Floquet3}. The Floquet operator, $\\mathcal{F}(t)$, is the unitary time-evolution $\\hat{U}(t)$ after one period of drive can be factorized as,\n\\begin{align}\n    \\mathcal{F}(t)=\\exp[-i \\alpha\\Lambda(\\vec{k})]\\exp[-i H_L T]=\\exp[-i H_{eff} T].\n\\end{align}\nSo the dynamics can have three stages: initial kick at $t_i$, the evolution of system with $H(t)$ in the interval $t_f-t_i$ and final kick at $t_f$ which describes the \"micromotion\" \\cite{Floquet3}. Then, the time-evolution operator can be expressed as,\n\\begin{align}\n    \\hat{U}(t_i\\rightarrow t_f)= \\hat{U}(t_f)^{\\dagger}e^{-iH_{eff}(t_f-t_i)}\\hat{U}(t_i),\n\\end{align}\nwhere $\\hat{U}(t)=e^{-i\\mathcal{F}(t)}$. $\\mathcal{F}(t)$ is a time-periodic operator with zero average over one period.  Lets set $t_i=0$, then $H_{eff}$ and $\\mathcal{F}(t)$ can be expanded as,\n\\begin{align}\n    H_{eff}=\\sum_{n=0}^{\\infty}\\frac{1}{\\omega^n}H_{eff}^n,\\,\\,\\mathcal{F}(t)=\\sum_{n=1}^{\\infty}\\frac{1}{\\omega^n}\\mathcal{F}^n.\n\\end{align}\nA generic quantum $\\delta$-kick can be implemented with following perturbation,\n\\begin{align}\n    H_{kick}(t)=\\alpha \\Lambda(\\vec{k}) \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)\n\\end{align}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5p_05_kz.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5m_05_kz.pdf}\n    \\caption{The band structure of LSM along the $k_z$ axis in the presence of uniform strain ($\\alpha \\Gamma_5$) with (a) $\\alpha=0.5$ (tensile) and (b) $\\alpha=-0.5$ (compression). $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used.}\n    \\label{fig:G5}\n\\end{figure}\nwhere $T=2\\pi\/\\omega$, $\\alpha$ is the kicking strength and $\\Lambda(\\vec{k})$ is the matrix representation of a perturbation, which in general can be a function of momentum, $\\vec{k}$, and could be used to mimic a nonuniform kicking. Following a perturbative expansion the effective Hamiltonian for $\\delta$-kick, with $H_{kick,n}=\\alpha \\Lambda(\\vec{k})\/T\\,\\text{for all}\\,\\,n$, can be obtained as \\cite{Floquet3,kicking2},\n\\begin{align}\\label{effkick}\n    H^{kick}_{eff}=&\\,H_L+\\frac{\\alpha\\Lambda(\\vec{k})}{T}+\\frac{1}{24}[[\\alpha\\Lambda(\\vec{k}),H_L],\\alpha\\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n   \n   \n\\end{align}\nWe note that the third term in Eq.~(\\ref{effkick}), can be dropped in the limit of weak kicks, i.e, to the first order of kick strength. Therefore, a dynamical kicking effectively is equivalent to directly adding the kick term as a perturbation to the bare Hamiltonian, something which is not always relevant in the equilibrium. In the rest of this work we only focus on the physics of the effective Hamiltonian and we leave \"micromotion\" physics for future studies. \\\\\nSimilarly, the effective Hamiltonian for the case of smooth driving of the form $H(t)=H_0+\\alpha \\Lambda(\\vex{k})\\cos(\\omega t)$ can be obtained as \\cite{Floquet3},\n\\begin{align}\\label{effsmooth}\n    H^{cos}_{eff}=H_0+\\frac{1}{4\\omega^2}[[\\alpha \\Lambda(\\vec{k}),H_0],\\alpha \\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n\\end{align}\nUnlike Eq.~(\\ref{effkick}) of periodic kicking, the first correction to the $H_0$ in the case of smooth driving is $\\mathcal{O}(\\alpha^2)$.\\\\\n\\indent In the following we first investigate the effect of uniform kicking ($k$-independent) and show that some of the known results can be retrieved. Then we turn to nonuniform periodic kicking and show that by introducing $k$-dependent $\\mathcal{IT}$ symmetric kicks, interestingly,  different hybrid Dirac\/Weyl semimetals can be obtained. Finally, we briefly compare the case of smooth driving of Eq.~(\\ref{effsmooth}) with the results obtained by periodic kicking.\n\n\n \\section{Uniform kicking}\nThere are many possibilities of uniform kicking, including: five $\\Gamma_j$, and ten commutators $\\Gamma_{ij}=[\\Gamma_i,\\Gamma_j]\/(2i)$ with $i > j$ , which can be expressed in terms of the products of odd number of spin-$3\/2$ matrices. Here we focus on uniform kicks which are proportional to $\\Lambda=\\Gamma_j$. The effective Hamiltonian for such kicks, in a sufficiently weak kick limit, is given by,\n\\begin{align}\\label{effG5}\n    H^j_{eff}=h_L(\\vec{k})+\\frac{\\alpha}{T}\\Gamma_j,\n\\end{align}\nwith spectrum,\n\\begin{align}\n    E^j_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2\\sum_i d^2_i T^2 - 4\\lambda_2\\alpha T d_j+\\alpha^2}}{T},\n\\end{align}\n\nOf particular interest is the $\\Gamma_5$ kicking, that breaks the rotational invariance and can be thought of as the effect of an external strain in $z$ direction \\cite{Ruan2016,QBToriginal}. It is known that a uniaxial strain, can realize two different phases in LSMs depending on the sign of $\\alpha$ \\cite{Ruan2016,QBToriginal,Weylpyro1,Weylpyro2,LSMmagnetic}. Figure.~\\ref{fig:G5}a(b) shows the bandstructure corresponding to effective Hamiltonian of Eq.~(\\ref{effG5}) with $\\Gamma_j=\\Gamma_5$ and for $\\alpha > 0$ ($\\alpha<0$) which represents a Dirac semimetal (trivial\/topological insulator).\n\nIn the case of a tensile strain, there are two Dirac nodes at $k_z=\\pm\\frac{\\alpha}{2\\lambda_2 T}$ ($\\alpha > 0 $). Now if we add an external magnetic field, we expect that each Dirac nodes split to two Weyl points and form Weyl semimetal phases due to breaking of $\\mathcal{T}$ symmetry. Interestingly, even in the topological insulator limit of $\\alpha<0$, one can realize Weyl semimetals depending on the strength of magnetic field \\cite{LSMmagnetic}. We also note that magnetic field alone can drive a Luttinger semimetal to a Weyl semimetal phase \\cite{LSMmagnetic,Weylpyro2}. Moreover, in the experimentally relevant system of pyrochlore irridates, it is shown that magnetic field can give rise to rich phase diagram of topological semimetals \\cite{Weylpyro2}. It is important to mention that an external magnetic field can have orbital effects for sufficiently large strength. However, here and in the rest of this paper, we neglect the orbital effects of magnetic field for sake of simplicity following many works in the literature \\cite{Weylpyro1, Weylpyro2,LSMFloq1} and we refer readers to some of the works which explored the effects of magnetic field in Landau levels and quantum oscillations \\cite{LSMmagnetic,LSMquantumoscillation}. The detailed analysis of the effect of magnetic field (also possible pseudomagnetic fields due to other perturbations) will be left for future studies.\n\n\\section{Nonuniform kicking}\nNow lets turn to the main part of this work, where we investigate the effect of nonuniform kicks. Similar to the uniform kicking discussed in the previous section, we can consider numerous possibilities of nonuniform driving. However, here, we restrict ourselves to a \"linear in momentum\" kicks of the form $k_i\\Gamma_i$ where $\\Gamma_i$ are time-reversal even. The very first consequence of such choice of the kicks, is the breaking of both $\\mathcal{T}$ and $\\mathcal{I}$ symmetries, while preserving their combinations $\\mathcal{IT}$, which is a necessary symmetry for keeping double degeneracy of the bands intact. Specifically, we focus on two types of nonuniform kicking: (a) $k_z\\Gamma_5$ and (b) a $k_z\\Gamma_0$.\n\n\\begin{figure*}[htb]\n \\centering\n    \\includegraphics[width=0.42\\textwidth]{Fig4a.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz05_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz07_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz11_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz15_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz20_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz30_kz-R.pdf}\\includegraphics[width=0.42\\textwidth]{kzG5_PH_kxkz.pdf}\n    \\caption{The band structure of LSM in the presence of $k_z\\Gamma_5$ with kick strength of $\\alpha=0.8$  and (a) $h=0$, $k_y-k_z$ plane showing quadratic dispersion along the $k_y$ for the node at $\\Gamma$ point, (b) $h=0.5$, (c) $h=0.7$, (d) $h=1.1$, (e) $h=1.5$, (f) $h=2$, (g) $h=3$ and (h) same as (a) but with $\\lambda_1=0$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots (except (h) where $\\lambda_1=0$).}\n    \\label{fig:kzG5}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_5$}\n\nAs it is mentioned above the $\\Gamma_5$ can be interpreted as the effect of strain in $k_z$ directions \\cite{Ruan2016}, similarly, $k_z\\Gamma_5$ also breaks the rotational invariance, then in principle we might still think of $k_z\\Gamma_5$ term as a type of nonuniform strain or strain gradient, even though full microscopic derivation of such strain could shed more light. The presence of $k_z$ suggests that microscopically on a lattice, $\\Gamma_5$ now acts non-local and directly modifies hopping terms. On the hand, one can convert the $k_z\\Gamma_5$ to the lattice model and explicitly write it down as, $\\sum_{\\sigma\\in\\{1\/2,3\/2\\}}(C^{\\dagger}_{k_{\\perp},i,\\sigma,\\uparrow}C_{k_{\\perp},j,\\sigma,\\uparrow}-C^{\\dagger}_{k_{\\perp},i,\\sigma,\\downarrow}C_{k_{\\perp},j,\\sigma,\\downarrow})\/2i+h.c$. This is similar to the spin current operator along $z$-axis. Therefore, we can also think of the $k_z\\Gamma_5$ kick as an applied spin current (or related to it) along $z$ direction. However, it is essential to note that besides the rotational invariance the $k_z\\Gamma_5$ kick also breaks inversion and time-reversal symmetries while preserving their combinations. Then the bands of driven system are still degenerate. Therefore, here, without restricting ourselves to any particular microscopic interpretation, we emphasize that to achieve the results obtained in this section one need perturbations which break both $\\mathcal{I}$ and $\\mathcal{T}$ but preserve $\\mathcal{IT}$.\\\\\nThe effective Hamiltonian for $k_z\\Gamma_5$ kick can be written as, $H_{eff}=h_{L}(\\vec{k})+\\frac{\\alpha}{T}k_z\\Gamma_5$, with following spectrum,\n\\begin{align}\n    E_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2T^2\\sum_i d^2_i - 4\\lambda_2\\alpha T k_z d_5+\\alpha^2 k^2_z}}{T}.\n\\end{align}\nFigure.~\\ref{fig:kzG5}, shows the spectrum in $k_x-k_z$ plane. There are two Dirac nodes at $k_z=0$ and $k_z=\\frac{\\alpha}{2\\lambda_2 T}$. While one of the nodes is always pinned to $\\Gamma$ point the other node can be tuned by kick parameters. Therefore, for fixed set of parameters the distance between two nodes for $k_z\\Gamma_5$ is half of the case with uniform strain $\\Gamma_5$. Moreover, we observe three main differences in compare to the case with uniform strain as is shown in Figure.~\\ref{fig:G5}. (i) There is no major difference between the tensile ($\\alpha >0 $) and  compressive ($\\alpha <0 $) strain. In both cases system drives into a Dirac semimetal phase. (ii) Unlike the uniform tensile strain, due to broken inversion symmetry two Dirac nodes reside at different energies and (iii) the most important of all, remarkably, one of the nodes is quadratic while the other is linear, realizing an unique \\emph{hybrid Dirac semimetal}. Interestingly, the \"quadratic\" Dirac node (or a QBT which is linearly dispersed in $k_z$ direction) is pinned at the $\\Gamma$ point and the \"energy difference\" as well as distance between the nodes can be controlled with kick strength. Moreover, while the linear Dirac node is tilted, the node centered around the $\\Gamma$ point no matter how strong is the kick, shows no tilt. It should be emphasized, here, we define hybrid Dirac semimetal mainly based on the different dispersion of two Dirac nodes instead of their tilts, even though as we discussed above, they show different tilting (and types) as well. Moreover, We note that such hybrid phase is unique to the Dirac semimetals because it is impossible to realize a Weyl semimetal with pair of nodes with different dispersion or (magnitude of monopole charges).\\\\\n\\indent Next, lets apply an external magnetic field $h J_z$ where $h$ denotes the strength of magnetic field. The magnetic field splits Dirac nodes to Weyl nodes in $k_z$ direction. In the presence of a magnetic field an analytical expression of eigenvalues for the general $\\lambda_1,\\lambda_2$ can not be achieved, so in the following we proceed by solving the model numerically. Figure.~\\ref{fig:kzG5} depicts the evolution of the Weyl nodes by strength of external magnetic field. Starting from $h \\ll \\alpha $, there are 8 nodes on $k_z$ axis, four single and four double Weyl nodes, which are indicated in Figure.~\\ref{fig:kzG5} by the red and blue dots, respectively. Interestingly, we observe that one or more of the Weyl pairs realize a \\emph{hybrid Weyl phases} and they can survive up to a decently strong magnetic field (Figure.~\\ref{fig:kzG5}(b-f)). While the other Weyl nodes do not possess hybrid types, they show different tilts for each nodes of the same pair. Only at a very strong magnetic field (Figure.~\\ref{fig:kzG5}(g)) all of the Weyl points show a type-I structure.\nMoreover, we see that by increasing the magnetic field, from initial eight nodes, four of them merge and gap out and we left off with only four nodes (two single and two double nodes) at $h>>\\alpha$ limit. \\\\\n\\indent Finally, we comment on the particle-hole symmetric limit of $\\lambda_1=0$. In this limit we can get energies analytically even in presence of an external magnetic field as,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nwith 8 nodes located at,\n\\begin{align}\n    k^{\\pm,\\pm}_{z,I}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 4h\\lambda_2T^2}}{4\\lambda_2 T},\\cr\n    k^{\\pm,\\pm}_{z,II}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 8h\\lambda_2T^2}}{4\\lambda_2 T}.\n\\end{align}\nAs is clear from the above equations, four out of the eight nodes, $k^{\\pm,-}_{z,I,II}$, gap out by increasing $h$. However, it is noteworthy that even though a \\emph{hybrid dispersion Dirac semimetal phase} can still be generated, but now in the limit of $\\lambda_1=0$, the two nodes have the same energies Figure.~\\ref{fig:kzG5}(h), therefore, no longer a hybrid Weyl phase can be achieved in the presence of an external magnetic field.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_kz.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz02.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz04.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz06.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz08.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz15.pdf}\n\\caption{The band structure of LSM with tilted QBT with kick strength of $\\alpha=0.6$  and (a) $h=0$, (b) $h=0.2$, (c) $h=0.4$, (d) $h=0.6$, (e) $h=0.8$ and (f) $h=1.5$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots.}\n\\label{fig:kzG0}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_0$: Tilted QBT}\n\nThe second class of nonuniform kicks which we investigate in this work, is $\\Lambda=k_z\\Gamma_0$. The effective spectrum for such driving can be obtained as,\n\\begin{align}\\label{tQBT}\n    E_{\\pm}(\\vec{k})=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}+\\frac{\\alpha k_z}{T}.\n\\end{align}\nThe mere effect of such driving is to effectively tilt the QBT in $k_z$ direction as shown in Figure.~\\ref{fig:kzG0}(a). However, the tilted QBT behaves very differently in the presence of an external perturbation. To clarify further, we rewrite the Eq.~(\\ref{tQBT}) at $k_x=k_y=0$ as,\n\\begin{align}\n     E_{\\pm}(\\vec{k})=&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^{2}+\\frac{\\alpha k_z}{T}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})(k_z+\\mathcal{A}_z)^{2}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z+(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}^2_z\\cr\n     \\simeq&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z,\n\\end{align}\nwhere in the last line we assumed $\\mathcal{A}_z<1$. Now by comparing the first and the last line of the above equation, we obtain $\\mathcal{A}_z=\\frac{\\alpha}{2(\\lambda_{1}\\mp2\\lambda_{2})T}$. Therefore, we can think of a tilted QBT as a QBT with a pseudo-electromagnetic potential $\\mathcal{A}= (0, 0, \\mathcal{A}_z)$ which is proportional to kick strength. In particular, interesting nontrivial topological phases can emerge by applying an external magnetic field in the presence of tilt, which otherwise are absent in a conventional QBT. As we mentioned previously, it is known that in the presence of a magnetic field, the QBT splits to multiple Weyl points. However, when the QBT is tilted, due to the competition between the external magnetic field and pseudo-magnetic field due to the $\\mathcal{A}$ (proportional to kick strength) there are various regimes which different types of WSM phases can be generated. Starting with the weak field limit ($\\alpha >> h$), there are two pairs of nodes, two double and two single nodes. In this regime, all Weyl nodes have a type-II nature (Figure.~\\ref{fig:kzG0}(b)). By further increasing of the field, for intermediate fields ($h \\lesssim \\alpha$), hybrid Weyl pairs are realized where one of nodes in each of single and double pairs are type-II and the others are type-I. Interestingly, the realized hybrid WSM can survive up to a very strong magnetic field. This could be experimentally beneficent as it demonstrates that hybrid WSMs generated here, are accessible in a broad range of external fields. Similar to the $k_z\\Gamma_5$ kick, we can get the analytical expression of energies and nodes position in the particle-hole limit,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z+\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nand,\n\\begin{align}\n    k^{\\pm}_{z,I}=\\pm\\sqrt{\\frac{h}{2\\lambda_2}},\\,\\,k^{\\pm}_{z,II}=\\pm\\frac{\\sqrt{h}}{2\\sqrt{\\lambda_2}}=\\frac{k^{\\pm}_{z,I}}{\\sqrt{2}}.\n\\end{align}\nUnlike the $k_z\\Gamma_5$ kick, the presence of particle-hole symmetry does not prevent the generation of hybrid Weyl phases, because does not generate any other Dirac nodes and as we mentioned before only tilts the QBT. Moreover, the position of Weyl nodes are independent of kick strength and can be tuned solely using magnetic field.\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kxkz.pdf}\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kykz.pdf}\n    \\caption{The bandstructure of LSM at $k_x-k_z$ plane ($k_y=0$) in presence of $k_z\\Gamma_5$ smooth driving with $\\alpha=5$. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:kzG5smooth}\n\\end{figure}\n\n\\subsection{Smooth driving}\nHere, we briefly discuss the case of smooth driving using Eq.~(\\ref{effsmooth}) and make comparison with the results obtained by periodic kicking in previous sections. First of all, any type of perturbation which is proportional to the identity will not modify the system in the case of smooth driving, as it obviously commutes with Hamiltonian. However, the $\\Lambda(\\vec{k})=\\alpha k_z\\Gamma_5$ can modify the system when is applied via smooth driving. We obtain the effective Hamiltonian as,\n\\begin{align}\n    H^{cos}_{eff}(\\vec{k})=&H_0+\\frac{\\lambda_2\\alpha^2k^2_z}{\\omega}\\bigg(2k_xk_z\\{J_x,J_z\\}+2k_yk_z\\{J_y,J_z\\}\\cr\n    +&(k^2_x-k^2_y)(J^2_x-J^2_y)+2k_xk_y\\{J_x,J_y\\}\\bigg).\n\\end{align}\nwith energies,\n\\begin{align}\n    E^{\\pm}(\\vec{k})=\\lambda_1 k^2\\pm \\frac{\\lambda_2\\sqrt{f(\\vec{k})\\left[k^2_z\\alpha^2-2\\omega^2\\right]+4k^4\\omega^4}}{\\omega^2},\n\\end{align}\nwhere each $f(\\vec{k})=3 (k_x^2+k_y^2)k_z^2 (k_x^2+k_y^2 +4k_z^2)\\alpha^2$ and $E^{\\pm}$ are doubly degenerate. Figure.~\\ref{fig:kzG5smooth} shows the spectrum for $k_y=0$ plane, where four linear Dirac nodes coexist with a QBT at which is located at $\\Gamma$ point. A similar plot can be obtained for $k_x=0$ plane, then the system also possesses line-nodes in $k_x-k_y$ plane at $k_z\\neq 0$. Therefore, in addition to the coexistence of Dirac nodes and QBT at $k_x-k_z$ and $k_y-k_z$ planes, a smooth driving can lead to a richer phase diagram. This analysis was only for the purpose of comparison with kicked driving results, so a detailed analysis of such models will be left for future works.\n\n\\subsection{Effect of Lattice Regularization}\n\nLets now look at the effect of lattice regularization. One of the typical effect of lattice versus continuum model is the appearance of more nodes, usually at the boundaries of the Brillioun Zone \\cite{Ghorashi2018}. However, we show that the main features discussed in the previous subsections survives in lattice models. For the sake of brevity we restrict ourselves to $h=0$ limit. We consider a cubic lattice, which captures the physics of continuum model for $\\alpha,\\omega,\\vec{k}<<1$. In the $k_z$ cut (Figure.~\\ref{fig:latt}), we find: (i) two nodes at BZ boundaries, $k_z=\\pm \\pi$, as we expected, (ii) one at the $\\Gamma$ point with quadratic dispersion in $k_i \\perp k_z$, in agreement with the result of continuum model, (iii) however, due to lattice regularization there can be two nodes (instead of the node at $k_z=\\frac{\\alpha}{2\\lambda_2 T}$ in the continuum model) with linear dispersion away from the boundaries and the $\\Gamma$ point, which are located at $k_z=\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T}), k_z=-\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T})+\\pi$. Figure.~\\ref{fig:latt}, shows the lattice bandstructure along $k_z$ for the case of $k_z\\Gamma_5$ driving. Therefore, we confirm that the generation of the hybrid dispersion Dirac semimetal (and consequently Weyl semimetal) persists in the lattice model.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_03_lattice.pdf}\n    \\caption{The $k_x=k_y=0$ cut of the bandstructure of LSM on a cubic lattice in presence of $\\alpha\\sin(k_z)\\Gamma_5$ ($\\alpha=0.3$) along the $k_z$ axis. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:latt}\n\\end{figure}\n\\section{DISCUSSION AND CONCLUDING REMARKS}\n\nWe have proposed a dynamical way to realize hybrid Dirac and Weyl semimetals in a three-dimensional Luttinger semimetal via applying a nonuniform (momentum-dependent) periodic $\\delta$-kick. We explicitly demonstrated this through two examples of nonuniform kicking which break both inversion and time-reversal symmetries while preserving their combinations. We have identified the first example of an unusual hybrid Dirac semimetal phase where two nodes not only have different types but also have different dispersions ( a linear Dirac and QBT coexist). Then by applying an external magnetic field we demonstrated the emergence of hybrid Weyl semimetals. Next, we found that the combination of a tilted QBT with an external magnetic field provides a promising setup for generation of hybrid Weyl semimetals. Moreover, by interpreting the tilted QBT as a QBT with emergent psudomagnetic field proportional to kick strength, we discussed the interplay between the kick strength and the external magnetic field. \\\\\n\\indent We note that the experimental realization of models discussed in this work can be difficult, specifically for the case of uniform strain which requires very fast time scales for periodic driving. Despite all the difficulties there are some proposals for the fast dynamical generation of strain \\cite{straindyn1,straindyn2,kickgraphene}. However, the possibility of interpreting the $k_z\\Gamma_5$ kick as an applied spin current (or proportional to it) along the $z$-direction is potentially a promising route, considering the recent developments in the field of ultrafast spintronics \\cite{ultrafastspin}. Moreover, the 2D QBT has been already proposed to be realized in optical lattices \\cite{QBToptical}, therefore, in practice a 3D QBT could be realized in optical lattices as a potential experimental setup where parameters can be tuned at will.\\\\\n\\indent The LSMs can describe the low-energy physics of many experimental candidates. Therefore, this work opens up a promising way for the realization of various hybrid Dirac and Weyl semimetals. Therefore, it could motivate further studies on the lacking investigation of physical properties of the hybrid Weyl semimetals. \\\\\nSome of the possible future directions are: the investigation of bulk and surface transport properties in the hybrid dispersion Dirac semimetals introduced here, in particular, how some of the most interesting properties of a typical Dirac semimetal, such as transport anomalies \\cite{review1}, would be different. Moreover, it would be interesting to see whether other multi-band systems with higher-spin can show similar physics.\n\n\\section*{Acknowledgement}\nWe thank Matthew Foster and Bitan Roy for useful comments. We also acknowledge useful comments and suggestions from the anonymous referees which helped to improve this manuscript. This work was supported by the U.S. Army Research Office Grant No.W911NF-18-1-0290. we also acknowledge partial support from NSF CAREER Grant No. DMR-1455233 and ONR Grant No. ONR-N00014-16-1-3158.\n\n\n\n\n\n\n\n\\section{Introduction}\nOver the last decade topological phases of matter have attracted many attentions for providing tremendous insights both in fundamental and experimental aspects of condensed matter physics \\cite{Chiu2016}. In addition to the gapped phases which initially ignited the field, recently discovery of gapless topological phases stimulated many works towards the understanding of nontrivial topology of gapless systems \\cite{review1}. Among those, Weyl and Dirac semimetals are of particular interest due to their experimental discovery and many unique physical properties \\cite{review1}. The Dirac\/Weyl smimetals (DSMs\/WSMs) are characterized by isolated point touchings of two degenerate\/nondegenerate bands in momentum space. Weyl nodes can be generated by splitting of degenerate Dirac nodes usually via breaking of either time-reversal ($\\mathcal{T}$) or inversion ($\\mathcal{I}$) symmetries or both. In the standard form, the low energy excitations near the Weyl nodes, disperse linearly along all three momentum directions with each node carries monopole charge of $\\pm 1$. As a result, on the surface, there exists a Fermi arc that connects a pair of Weyl nodes with opposite chiralities. Recently, generalization of WSMs to multi-Weyl nodes have been proposed where each nodes have higher-order dispersion in one or more directions and consequently caries monopole charge of larger than one \\cite{MultiWeyl1,MultiWeyl2}. \\\\\n\\indent On the other hand, DSMs\/WSMs can also be classified into type-I and type-II, based on the tilting of their nodes. In the standard type-I WSMs, Fermi surfaces are point-like while in the type-II WSMs, Weyl nodes are tilted resulting in formation of electron and hole pockets producing finite density of states at the Fermi level \\cite{typeII1,typeII2}. In a conventional type-I and II WSMs, two Weyl nodes with opposite chiralities have same types, however, recently, a theoretical proposal \\cite{hybridWeyloriginal}, introduced a new WSM where a pair of Weyl nodes with different chiralities can have different types, forming the so-called \\emph{hybrid Weyl semimetals}.\\\\\n\\indent Besides the DSMs\/WSMs, another class of three-dimensional nodal semimetals are Luttinger semimetals (LSMs) where possess a quadratic band touching (QBT) point between doubly degenerate valence and conduction bands of $J=3\/2$ (effective) fermions at an isolated point in the Brillouin zone.\nThe LSM provides the low-energy description for a plethora of both strongly and weakly correlated compounds, such as the 227 pyrochlore iridates (Ln2Ir2O7, with Ln being a lanthanide element) \\cite{LSMapp1,LSMapp2, LSMapp3,LSMapp4}, half-Heusler compounds (ternary alloys such as LnPtBi, LnPdBi) \\cite{LSMapp5,LSMapp6}, HgTe \\cite{semiconductor1,semiconductor2,QBToriginal,Ruan2016,mottQBT}, and gray-tin \\cite{LSMapp7,LSMapp8}.\nMoreover, LSMs proved to show many interesting behaviors\\cite{LSMprop1,LSMprop2,LSMprop3,LSMprop4,LSMprop5}, specially in the presence of interaction, for example, investigation of the magnetic and superconducting orders actively have been explored \\cite{LSMapp9, Roy-Ghorashi2019, GhorashiPRB2017, GhorashiPRL2018, Ghorashi2019,SatoSC32,LSMBoettcher2,LSMSCLiu1,LSMBoettcher1,Szabo-Bitan32-2018}.\\\\\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{FlashpickickedLSMv4.png}\n    \\caption{Schematic picture of summary of the results obtained in this work. Starting from a Luttinger semimetal via two different nonuniform ($k$-dependent) periodic kickings, where break inversion ($\\mathcal{I}$) and time-reversal ($\\mathcal{T}$) symmetries while preserving their combinations ($\\mathcal{IT}$), we have obtained a \\emph{hybrid dispersion Dirac semimetals} (e.g., $k_z\\Gamma_5$ where $\\Gamma_i$ are Dirac matrices [see text for details]) and tilted LSM (e.g., $k_z\\Gamma_0$). Then, by applying an external magnetic field, $J_z$, in parallel to the direction of nodes (or kick direction), \\emph{hybrid Weyl phases} can be generated.}\n    \\label{fig:adpic}\n\\end{figure*}\n\\indent The coexistence of various Weyl nodes with different charges and\/or types is an interesting phenomena that could help towards understanding as well as manipulation of the properties of various Weyl nodes in an equal footing setup. Besides a few works reporting the coexistence of type-I and II Dirac\/Weyl nodes \\cite{hybridWSMexp,hybridDSM1,hybridWSM1,WSMexp1&2,weylcoexPRL}, there have been also proposals\\cite{LSMFloq1,Ghorashi2018}, claiming dynamical generation of various Weyl nodes of different types and\/or charges in one system. Application of the light is shown to be a powerful method to change the material properties \\cite{FloquetRev}. In particular, the conversion of a topologically trivial phase into a nontrivial one using periodic driving has attracted enormous attention in the past decade \\cite{FloquetRev,Floquet1, Floquet2, Floquet3}. Specifically many proposals on Floquet WSMs in various systems exist, such as Dirac semimetals \\cite{FloqWeyl1,FloqWeyl2}, band insulators \\cite{FloqWeyl3}, stacked graphene \\cite{FloqWeyl4}, line-nodal semimetals \\cite{FloqWeyl2,FloqWeyl5}, and crossing-line semimetals \\cite{FloqWeyl6,FloqWeyl7}. Also, proposals have been made to create tunable WSMs in pyrochlore iridates with Zeeman fields \\cite{Weylpyro1,Weylpyro2}. Very recently, using circular\/elliptic polarized light on Luttinger Hamiltonian in the high-frequency limit we have shown a very rich phase diagram of various Weyl semimetals, including coexistence of type-I and II as well as single and double Weyl nodes \\cite{Ghorashi2018} . \\\\\n \\indent Despite the several dynamical proposals for the generation of different Weyl phases, a promising setup for the realization of hybrid Dirac and Weyl semimetals is still lacking. In this work, we tackle this issue by an alternative way of periodic driving, in particular the periodic kicking. Using the periodic $\\delta$-function kicks can typically simplify theoretical studies by allowing to perform calculations analytically to a large extent (in contrast to sinusoidal driving or elliptical\/circular light) \\cite{ Floquet3,kicking2}.\nHowever, for the sake of comparison we also briefly discuss the smooth driving case to show that some of the features of our discussion can be hold up in smooth driving setup as well as long as the perturbation breaks inversion (uniaxially) and time-reversal symmetres but preserve their combinations. In this paper, we show two examples of such perturbations which along with an external magnetic field induce various hybrid Dirac and Weyl phases, including a new \\emph{hybrid dispersion Dirac semimetal}. Figure.~\\ref{fig:adpic} summarizes the result of this work.\n\n\\section{Model and Formalism}\n\\subsection{Model}\nWe start with reviewing the main ingredients of Luttinger Hamiltonian in the non-equilibrium limit, which can be represented as,\n\n\\begin{align}\n    H_{L}(\\vec{k})=\\int \\frac{d^3\\vec{k}}{(2\\pi)^3}\\Psi^{\\dagger}_{\\vec{k}} h_L(\\vec{k}) \\Psi_{\\vec{k}},\n\\end{align}\nwhere\n\\begin{align}\n    h_{L}(\\vec{k})=&(\\frac{k^2}{2m_0}-\\mu)\\Gamma_0-\\frac{1}{2m_1}\\sum^3_{a=1} d_a(\\vec{k})\\Gamma_a\\cr\n    -&\\frac{1}{2m_2}\\sum^5_{a=4} d_a(\\vec{k})\\Gamma_a\n\\end{align}\nwhere $k^2=k^2_x+k^2_y+k^2_z$ and,\n\\begin{align}\n    \\Psi^T_{k}= (c_{\\vec{k},3\/2}, c_{\\vec{k},1\/2}, c_{\\vec{k},-1\/2}, c_{\\vec{k},-3\/2}).\n\\end{align}\n$\\mu$ is the chemical potential measured from the band touching point. $\\Gamma_a$ are the well-known gamma matrices which are given by,\n\\begin{align}\n    \\Gamma_1=\\tau_3\\sigma_2,\\,\\, \\Gamma_2=\\tau_3\\sigma_1,\\,\\, \\Gamma_3=\\tau_2,\\,\\,\n    \\Gamma_4=\\tau_1,\\,\\, \\Gamma_5=\\tau_3\\sigma_3,\n\\end{align}\n and satisfy $\\{\\Gamma_a,\\Gamma_b\\}=\\delta_{a,b}$, while $\\Gamma_0$ is four dimensional identity matrix. $\\tau$ and $\\sigma$ denote space of sign and magnitude of spin projection $m_s\\in\\{\\pm 3\/2, \\pm 1\/2\\}$, respectively. $d_a(\\vex{k})$ are given as,\n \\begin{align}\n     d_1=\\sqrt{3}k_y k_z,\\,\\, d_2=\\sqrt{3}k_x k_z,\\,\\, d_3=\\sqrt{3}k_x k_y,\\,\\,\\cr\n     d_4=\\frac{\\sqrt{3}}{2}(k_x^2-k_y^2),\\,\\, d_5=\\frac{1}{2}(2k_z^2-k_x^2-k_y^2)\n \\end{align}\n The five mutually anticommuting $\\Gamma$ matrices can be written\nin terms of spin-$3\/2$ matrices according to,\n\\begin{align}\n    \\Gamma_1=&\\frac{1}{\\sqrt{3}}\\{J_y,J_z\\},\\,\\,\\Gamma_2=\\frac{1}{\\sqrt{3}}\\{J_x,J_z\\},\\,\\,\\Gamma_3=\\frac{1}{\\sqrt{3}}\\{J_x,J_y\\},\\cr\n    \\Gamma_4=&\\frac{1}{\\sqrt{3}}(J^2_x-J^2_y),\\,\\,\\Gamma_5=J^2_z-\\frac{5}{4},\n\\end{align}\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveup.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_curveopposite.pdf}\n    \\caption{The band structure for LSM along the $k_z$ axis with $\\lambda_1=0.6$ and (a) $\\lambda_2=0.1$, (b) $\\lambda_2=0.6$.}\n    \\label{fig:LSMbare}\n\\end{figure}\n Here we take the isotropic limit of $m_1=m_2 \\equiv m$. Therefore, the Luttinger Hamiltonian can also be written in an alternative way,\n\\begin{align}\n    h_L(\\vec{k})= [(\\lambda_1+5\\lambda_2\/2)k^2-\\mu]\\Gamma_0-2\\lambda_2(\\vec{J}.\\vec{k})^2\n\\end{align}\n with $\\vec{J}=(J_x,J_y,J_z)$ and $\\vec{k}=(k_x,k_y,k_z)$ and we used $\\lambda_1=1\/2m_0$ and $\\lambda_2=1\/4m$. $J_{x,y,z}$ are effective spin-$3\/2$ operators,\n \\begin{eqnarray}\n&\\,J_z=\\begin{bmatrix}\n        \\frac{3}{2} & 0 & 0 & 0 \\\\\n        0 & \\frac{1}{2} & 0 & 0\\\\\n        0 & 0 & -\\frac{1}{2} & 0 \\\\\n        0 & 0 & 0 & -\\frac{3}{2} \\\\\n      \\end{bmatrix},J_x=\\begin{bmatrix}\n        0 & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & 1 & 0\\\\\n        0 & 1 & 0 & \\frac{\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}\\cr\n      &\\,J_y=\\begin{bmatrix}\n        0 & \\frac{-i\\sqrt{3}}{2} & 0 & 0 \\\\\n        \\frac{\\sqrt{3}}{2} & 0 & -i & 0\\\\\n        0 & 1 & 0 & \\frac{-i\\sqrt{3}}{2} \\\\\n        0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\\n      \\end{bmatrix}.\n\\end{eqnarray}\nThe energy dispersions are $E(k)=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}-\\mu$ for the $j=3\/2$ and the $j=1\/2$ bands, respectively. Four bands come in doubly degenerate pairs as a result of time-reversal (with antiunitary operator $\\mathcal{T}=\\Gamma_1\\Gamma_3\\mathcal{K}$ and $\\vec{k}\\rightarrow -\\vec{k}$ where $\\mathcal{K}$ is complex conjugation) and inversion ($\\mathcal{I}=I_{4\\times4}$ and $\\vec{k}\\rightarrow -\\vec{k}$) symmetries. For $\\lambda_{2}<2\\lambda_{1}$\n($\\lambda_{2}>2\\lambda_{1}$), the degenerate bands curve the same (opposite) way as shown in Figure.~(\\ref{fig:LSMbare}). In the case of both bands bending the same way, Eq. (2) is widely used to model heavy- and light-hole bands in zinc-blende semiconductors \\cite{semiconductor1} and many properties of such a dispersion have been studied in the literature, including a recent study on the realization of fully gapped topological superconductivity with \\emph{p}-wave pairing which has states with exotic cubic and linear dispersions coexisting on the surface \\cite{congjunWuPRL16,GhorashiPRB2017}. On the other hand, when bands bend oppositely, the model in Eq. (2) is known as Luttinger semimetal with QBT and is used to describe behavior of certain pyrochlore iridates as well as some doped half-Heusler alloys such as LaPtBi \\cite{Chadov2010,Lin2010,halfheusler3}.\n\n\n\\subsection{Periodic driving}\nA general time-dependent problem with $H(t)=H_0+V(t)$, can be tackled using Floquet theory when $V(t+T)=V(t)$ is periodic. To proceed, we can expand the periodic potential in a Fourier series as\n\\begin{align}\n    V(t)=V_0+\\sum^{\\infty}_{n=1} \\big(V_n e^{i\\omega nt}+V_{-n}e^{-i\\omega nt}\\big).\n\\end{align}\n In the limit of fast driving regime, in which the driving frequency is larger than any natural energy scale in the problem, one can obtain the effective Hamiltonian and Floquet operators perturbatively up to  $\\mathcal{O}(1\/\\omega^2)$ \\cite{Floquet3}. The Floquet operator, $\\mathcal{F}(t)$, is the unitary time-evolution $\\hat{U}(t)$ after one period of drive can be factorized as,\n\\begin{align}\n    \\mathcal{F}(t)=\\exp[-i \\alpha\\Lambda(\\vec{k})]\\exp[-i H_L T]=\\exp[-i H_{eff} T].\n\\end{align}\nSo the dynamics can have three stages: initial kick at $t_i$, the evolution of system with $H(t)$ in the interval $t_f-t_i$ and final kick at $t_f$ which describes the \"micromotion\" \\cite{Floquet3}. Then, the time-evolution operator can be expressed as,\n\\begin{align}\n    \\hat{U}(t_i\\rightarrow t_f)= \\hat{U}(t_f)^{\\dagger}e^{-iH_{eff}(t_f-t_i)}\\hat{U}(t_i),\n\\end{align}\nwhere $\\hat{U}(t)=e^{-i\\mathcal{F}(t)}$. $\\mathcal{F}(t)$ is a time-periodic operator with zero average over one period.  Lets set $t_i=0$, then $H_{eff}$ and $\\mathcal{F}(t)$ can be expanded as,\n\\begin{align}\n    H_{eff}=\\sum_{n=0}^{\\infty}\\frac{1}{\\omega^n}H_{eff}^n,\\,\\,\\mathcal{F}(t)=\\sum_{n=1}^{\\infty}\\frac{1}{\\omega^n}\\mathcal{F}^n.\n\\end{align}\nA generic quantum $\\delta$-kick can be implemented with following perturbation,\n\\begin{align}\n    H_{kick}(t)=\\alpha \\Lambda(\\vec{k}) \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)\n\\end{align}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5p_05_kz.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_G5m_05_kz.pdf}\n    \\caption{The band structure of LSM along the $k_z$ axis in the presence of uniform strain ($\\alpha \\Gamma_5$) with (a) $\\alpha=0.5$ (tensile) and (b) $\\alpha=-0.5$ (compression). $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used.}\n    \\label{fig:G5}\n\\end{figure}\nwhere $T=2\\pi\/\\omega$, $\\alpha$ is the kicking strength and $\\Lambda(\\vec{k})$ is the matrix representation of a perturbation, which in general can be a function of momentum, $\\vec{k}$, and could be used to mimic a nonuniform kicking. Following a perturbative expansion the effective Hamiltonian for $\\delta$-kick, with $H_{kick,n}=\\alpha \\Lambda(\\vec{k})\/T\\,\\text{for all}\\,\\,n$, can be obtained as \\cite{Floquet3,kicking2},\n\\begin{align}\\label{effkick}\n    H^{kick}_{eff}=&\\,H_L+\\frac{\\alpha\\Lambda(\\vec{k})}{T}+\\frac{1}{24}[[\\alpha\\Lambda(\\vec{k}),H_L],\\alpha\\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n   \n   \n\\end{align}\nWe note that the third term in Eq.~(\\ref{effkick}), can be dropped in the limit of weak kicks, i.e, to the first order of kick strength. Therefore, a dynamical kicking effectively is equivalent to directly adding the kick term as a perturbation to the bare Hamiltonian, something which is not always relevant in the equilibrium. In the rest of this work we only focus on the physics of the effective Hamiltonian and we leave \"micromotion\" physics for future studies. \\\\\nSimilarly, the effective Hamiltonian for the case of smooth driving of the form $H(t)=H_0+\\alpha \\Lambda(\\vex{k})\\cos(\\omega t)$ can be obtained as \\cite{Floquet3},\n\\begin{align}\\label{effsmooth}\n    H^{cos}_{eff}=H_0+\\frac{1}{4\\omega^2}[[\\alpha \\Lambda(\\vec{k}),H_0],\\alpha \\Lambda(\\vec{k})]+\\mathcal{O}(1\/\\omega^3).\n\\end{align}\nUnlike Eq.~(\\ref{effkick}) of periodic kicking, the first correction to the $H_0$ in the case of smooth driving is $\\mathcal{O}(\\alpha^2)$.\\\\\n\\indent In the following we first investigate the effect of uniform kicking ($k$-independent) and show that some of the known results can be retrieved. Then we turn to nonuniform periodic kicking and show that by introducing $k$-dependent $\\mathcal{IT}$ symmetric kicks, interestingly,  different hybrid Dirac\/Weyl semimetals can be obtained. Finally, we briefly compare the case of smooth driving of Eq.~(\\ref{effsmooth}) with the results obtained by periodic kicking.\n\n\n \\section{Uniform kicking}\nThere are many possibilities of uniform kicking, including: five $\\Gamma_j$, and ten commutators $\\Gamma_{ij}=[\\Gamma_i,\\Gamma_j]\/(2i)$ with $i > j$ , which can be expressed in terms of the products of odd number of spin-$3\/2$ matrices. Here we focus on uniform kicks which are proportional to $\\Lambda=\\Gamma_j$. The effective Hamiltonian for such kicks, in a sufficiently weak kick limit, is given by,\n\\begin{align}\\label{effG5}\n    H^j_{eff}=h_L(\\vec{k})+\\frac{\\alpha}{T}\\Gamma_j,\n\\end{align}\nwith spectrum,\n\\begin{align}\n    E^j_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2\\sum_i d^2_i T^2 - 4\\lambda_2\\alpha T d_j+\\alpha^2}}{T},\n\\end{align}\n\nOf particular interest is the $\\Gamma_5$ kicking, that breaks the rotational invariance and can be thought of as the effect of an external strain in $z$ direction \\cite{Ruan2016,QBToriginal}. It is known that a uniaxial strain, can realize two different phases in LSMs depending on the sign of $\\alpha$ \\cite{Ruan2016,QBToriginal,Weylpyro1,Weylpyro2,LSMmagnetic}. Figure.~\\ref{fig:G5}a(b) shows the bandstructure corresponding to effective Hamiltonian of Eq.~(\\ref{effG5}) with $\\Gamma_j=\\Gamma_5$ and for $\\alpha > 0$ ($\\alpha<0$) which represents a Dirac semimetal (trivial\/topological insulator).\n\nIn the case of a tensile strain, there are two Dirac nodes at $k_z=\\pm\\frac{\\alpha}{2\\lambda_2 T}$ ($\\alpha > 0 $). Now if we add an external magnetic field, we expect that each Dirac nodes split to two Weyl points and form Weyl semimetal phases due to breaking of $\\mathcal{T}$ symmetry. Interestingly, even in the topological insulator limit of $\\alpha<0$, one can realize Weyl semimetals depending on the strength of magnetic field \\cite{LSMmagnetic}. We also note that magnetic field alone can drive a Luttinger semimetal to a Weyl semimetal phase \\cite{LSMmagnetic,Weylpyro2}. Moreover, in the experimentally relevant system of pyrochlore irridates, it is shown that magnetic field can give rise to rich phase diagram of topological semimetals \\cite{Weylpyro2}. It is important to mention that an external magnetic field can have orbital effects for sufficiently large strength. However, here and in the rest of this paper, we neglect the orbital effects of magnetic field for sake of simplicity following many works in the literature \\cite{Weylpyro1, Weylpyro2,LSMFloq1} and we refer readers to some of the works which explored the effects of magnetic field in Landau levels and quantum oscillations \\cite{LSMmagnetic,LSMquantumoscillation}. The detailed analysis of the effect of magnetic field (also possible pseudomagnetic fields due to other perturbations) will be left for future studies.\n\n\\section{Nonuniform kicking}\nNow lets turn to the main part of this work, where we investigate the effect of nonuniform kicks. Similar to the uniform kicking discussed in the previous section, we can consider numerous possibilities of nonuniform driving. However, here, we restrict ourselves to a \"linear in momentum\" kicks of the form $k_i\\Gamma_i$ where $\\Gamma_i$ are time-reversal even. The very first consequence of such choice of the kicks, is the breaking of both $\\mathcal{T}$ and $\\mathcal{I}$ symmetries, while preserving their combinations $\\mathcal{IT}$, which is a necessary symmetry for keeping double degeneracy of the bands intact. Specifically, we focus on two types of nonuniform kicking: (a) $k_z\\Gamma_5$ and (b) a $k_z\\Gamma_0$.\n\n\\begin{figure*}[htb]\n \\centering\n    \\includegraphics[width=0.42\\textwidth]{Fig4a.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz05_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz07_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz11_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz15_kz-R.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz20_kz-R.pdf}\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_08_Jz30_kz-R.pdf}\\includegraphics[width=0.42\\textwidth]{kzG5_PH_kxkz.pdf}\n    \\caption{The band structure of LSM in the presence of $k_z\\Gamma_5$ with kick strength of $\\alpha=0.8$  and (a) $h=0$, $k_y-k_z$ plane showing quadratic dispersion along the $k_y$ for the node at $\\Gamma$ point, (b) $h=0.5$, (c) $h=0.7$, (d) $h=1.1$, (e) $h=1.5$, (f) $h=2$, (g) $h=3$ and (h) same as (a) but with $\\lambda_1=0$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots (except (h) where $\\lambda_1=0$).}\n    \\label{fig:kzG5}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_5$}\n\nAs it is mentioned above the $\\Gamma_5$ can be interpreted as the effect of strain in $k_z$ directions \\cite{Ruan2016}, similarly, $k_z\\Gamma_5$ also breaks the rotational invariance, then in principle we might still think of $k_z\\Gamma_5$ term as a type of nonuniform strain or strain gradient, even though full microscopic derivation of such strain could shed more light. The presence of $k_z$ suggests that microscopically on a lattice, $\\Gamma_5$ now acts non-local and directly modifies hopping terms. On the hand, one can convert the $k_z\\Gamma_5$ to the lattice model and explicitly write it down as, $\\sum_{\\sigma\\in\\{1\/2,3\/2\\}}(C^{\\dagger}_{k_{\\perp},i,\\sigma,\\uparrow}C_{k_{\\perp},j,\\sigma,\\uparrow}-C^{\\dagger}_{k_{\\perp},i,\\sigma,\\downarrow}C_{k_{\\perp},j,\\sigma,\\downarrow})\/2i+h.c$. This is similar to the spin current operator along $z$-axis. Therefore, we can also think of the $k_z\\Gamma_5$ kick as an applied spin current (or related to it) along $z$ direction. However, it is essential to note that besides the rotational invariance the $k_z\\Gamma_5$ kick also breaks inversion and time-reversal symmetries while preserving their combinations. Then the bands of driven system are still degenerate. Therefore, here, without restricting ourselves to any particular microscopic interpretation, we emphasize that to achieve the results obtained in this section one need perturbations which break both $\\mathcal{I}$ and $\\mathcal{T}$ but preserve $\\mathcal{IT}$.\\\\\nThe effective Hamiltonian for $k_z\\Gamma_5$ kick can be written as, $H_{eff}=h_{L}(\\vec{k})+\\frac{\\alpha}{T}k_z\\Gamma_5$, with following spectrum,\n\\begin{align}\n    E_{\\pm}(\\vec{k})=\\lambda_1k^2\\pm \\frac{\\sqrt{4\\lambda_2^2T^2\\sum_i d^2_i - 4\\lambda_2\\alpha T k_z d_5+\\alpha^2 k^2_z}}{T}.\n\\end{align}\nFigure.~\\ref{fig:kzG5}, shows the spectrum in $k_x-k_z$ plane. There are two Dirac nodes at $k_z=0$ and $k_z=\\frac{\\alpha}{2\\lambda_2 T}$. While one of the nodes is always pinned to $\\Gamma$ point the other node can be tuned by kick parameters. Therefore, for fixed set of parameters the distance between two nodes for $k_z\\Gamma_5$ is half of the case with uniform strain $\\Gamma_5$. Moreover, we observe three main differences in compare to the case with uniform strain as is shown in Figure.~\\ref{fig:G5}. (i) There is no major difference between the tensile ($\\alpha >0 $) and  compressive ($\\alpha <0 $) strain. In both cases system drives into a Dirac semimetal phase. (ii) Unlike the uniform tensile strain, due to broken inversion symmetry two Dirac nodes reside at different energies and (iii) the most important of all, remarkably, one of the nodes is quadratic while the other is linear, realizing an unique \\emph{hybrid Dirac semimetal}. Interestingly, the \"quadratic\" Dirac node (or a QBT which is linearly dispersed in $k_z$ direction) is pinned at the $\\Gamma$ point and the \"energy difference\" as well as distance between the nodes can be controlled with kick strength. Moreover, while the linear Dirac node is tilted, the node centered around the $\\Gamma$ point no matter how strong is the kick, shows no tilt. It should be emphasized, here, we define hybrid Dirac semimetal mainly based on the different dispersion of two Dirac nodes instead of their tilts, even though as we discussed above, they show different tilting (and types) as well. Moreover, We note that such hybrid phase is unique to the Dirac semimetals because it is impossible to realize a Weyl semimetal with pair of nodes with different dispersion or (magnitude of monopole charges).\\\\\n\\indent Next, lets apply an external magnetic field $h J_z$ where $h$ denotes the strength of magnetic field. The magnetic field splits Dirac nodes to Weyl nodes in $k_z$ direction. In the presence of a magnetic field an analytical expression of eigenvalues for the general $\\lambda_1,\\lambda_2$ can not be achieved, so in the following we proceed by solving the model numerically. Figure.~\\ref{fig:kzG5} depicts the evolution of the Weyl nodes by strength of external magnetic field. Starting from $h \\ll \\alpha $, there are 8 nodes on $k_z$ axis, four single and four double Weyl nodes, which are indicated in Figure.~\\ref{fig:kzG5} by the red and blue dots, respectively. Interestingly, we observe that one or more of the Weyl pairs realize a \\emph{hybrid Weyl phases} and they can survive up to a decently strong magnetic field (Figure.~\\ref{fig:kzG5}(b-f)). While the other Weyl nodes do not possess hybrid types, they show different tilts for each nodes of the same pair. Only at a very strong magnetic field (Figure.~\\ref{fig:kzG5}(g)) all of the Weyl points show a type-I structure.\nMoreover, we see that by increasing the magnetic field, from initial eight nodes, four of them merge and gap out and we left off with only four nodes (two single and two double nodes) at $h>>\\alpha$ limit. \\\\\n\\indent Finally, we comment on the particle-hole symmetric limit of $\\lambda_1=0$. In this limit we can get energies analytically even in presence of an external magnetic field as,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nwith 8 nodes located at,\n\\begin{align}\n    k^{\\pm,\\pm}_{z,I}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 4h\\lambda_2T^2}}{4\\lambda_2 T},\\cr\n    k^{\\pm,\\pm}_{z,II}=&\\,\\frac{\\alpha\\pm \\sqrt{\\alpha^2\\pm 8h\\lambda_2T^2}}{4\\lambda_2 T}.\n\\end{align}\nAs is clear from the above equations, four out of the eight nodes, $k^{\\pm,-}_{z,I,II}$, gap out by increasing $h$. However, it is noteworthy that even though a \\emph{hybrid dispersion Dirac semimetal phase} can still be generated, but now in the limit of $\\lambda_1=0$, the two nodes have the same energies Figure.~\\ref{fig:kzG5}(h), therefore, no longer a hybrid Weyl phase can be achieved in the presence of an external magnetic field.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_kz.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz02.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz04.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz06.pdf}\n\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz08.pdf}\\includegraphics[width=0.3\\textwidth]{LSM_kzG0_06_Jz15.pdf}\n\\caption{The band structure of LSM with tilted QBT with kick strength of $\\alpha=0.6$  and (a) $h=0$, (b) $h=0.2$, (c) $h=0.4$, (d) $h=0.6$, (e) $h=0.8$ and (f) $h=1.5$. $\\lambda_1=\\lambda_2=0.6$ and $\\omega=20$ are used in all plots.}\n\\label{fig:kzG0}\n\\end{figure*}\n\\subsection{$\\Lambda=k_z\\Gamma_0$: Tilted QBT}\n\nThe second class of nonuniform kicks which we investigate in this work, is $\\Lambda=k_z\\Gamma_0$. The effective spectrum for such driving can be obtained as,\n\\begin{align}\\label{tQBT}\n    E_{\\pm}(\\vec{k})=(\\lambda_{1}\\mp2\\lambda_{2})k^{2}+\\frac{\\alpha k_z}{T}.\n\\end{align}\nThe mere effect of such driving is to effectively tilt the QBT in $k_z$ direction as shown in Figure.~\\ref{fig:kzG0}(a). However, the tilted QBT behaves very differently in the presence of an external perturbation. To clarify further, we rewrite the Eq.~(\\ref{tQBT}) at $k_x=k_y=0$ as,\n\\begin{align}\n     E_{\\pm}(\\vec{k})=&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^{2}+\\frac{\\alpha k_z}{T}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})(k_z+\\mathcal{A}_z)^{2}\\cr\n     =&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z+(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}^2_z\\cr\n     \\simeq&\\,(\\lambda_{1}\\mp2\\lambda_{2})k_z^2+2(\\lambda_{1}\\mp2\\lambda_{2})\\mathcal{A}_z k_z,\n\\end{align}\nwhere in the last line we assumed $\\mathcal{A}_z<1$. Now by comparing the first and the last line of the above equation, we obtain $\\mathcal{A}_z=\\frac{\\alpha}{2(\\lambda_{1}\\mp2\\lambda_{2})T}$. Therefore, we can think of a tilted QBT as a QBT with a pseudo-electromagnetic potential $\\mathcal{A}= (0, 0, \\mathcal{A}_z)$ which is proportional to kick strength. In particular, interesting nontrivial topological phases can emerge by applying an external magnetic field in the presence of tilt, which otherwise are absent in a conventional QBT. As we mentioned previously, it is known that in the presence of a magnetic field, the QBT splits to multiple Weyl points. However, when the QBT is tilted, due to the competition between the external magnetic field and pseudo-magnetic field due to the $\\mathcal{A}$ (proportional to kick strength) there are various regimes which different types of WSM phases can be generated. Starting with the weak field limit ($\\alpha >> h$), there are two pairs of nodes, two double and two single nodes. In this regime, all Weyl nodes have a type-II nature (Figure.~\\ref{fig:kzG0}(b)). By further increasing of the field, for intermediate fields ($h \\lesssim \\alpha$), hybrid Weyl pairs are realized where one of nodes in each of single and double pairs are type-II and the others are type-I. Interestingly, the realized hybrid WSM can survive up to a very strong magnetic field. This could be experimentally beneficent as it demonstrates that hybrid WSMs generated here, are accessible in a broad range of external fields. Similar to the $k_z\\Gamma_5$ kick, we can get the analytical expression of energies and nodes position in the particle-hole limit,\n\\begin{align}\n    E^{(1)}_{\\pm}(\\vec{k})=&\\pm\\frac{h}{2}+(2\\lambda_2 k^2_z+\\frac{\\alpha k_z}{T}),\\cr E^{(2)}_{\\pm}(\\vec{k})=&\\pm\\frac{3h}{2}-(2\\lambda_2 k^2_z-\\frac{\\alpha k_z}{T}),\n\\end{align}\nand,\n\\begin{align}\n    k^{\\pm}_{z,I}=\\pm\\sqrt{\\frac{h}{2\\lambda_2}},\\,\\,k^{\\pm}_{z,II}=\\pm\\frac{\\sqrt{h}}{2\\sqrt{\\lambda_2}}=\\frac{k^{\\pm}_{z,I}}{\\sqrt{2}}.\n\\end{align}\nUnlike the $k_z\\Gamma_5$ kick, the presence of particle-hole symmetry does not prevent the generation of hybrid Weyl phases, because does not generate any other Dirac nodes and as we mentioned before only tilts the QBT. Moreover, the position of Weyl nodes are independent of kick strength and can be tuned solely using magnetic field.\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kxkz.pdf}\n    \\includegraphics[width=0.4\\textwidth]{kzG5smooth_5_kykz.pdf}\n    \\caption{The bandstructure of LSM at $k_x-k_z$ plane ($k_y=0$) in presence of $k_z\\Gamma_5$ smooth driving with $\\alpha=5$. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:kzG5smooth}\n\\end{figure}\n\n\\subsection{Smooth driving}\nHere, we briefly discuss the case of smooth driving using Eq.~(\\ref{effsmooth}) and make comparison with the results obtained by periodic kicking in previous sections. First of all, any type of perturbation which is proportional to the identity will not modify the system in the case of smooth driving, as it obviously commutes with Hamiltonian. However, the $\\Lambda(\\vec{k})=\\alpha k_z\\Gamma_5$ can modify the system when is applied via smooth driving. We obtain the effective Hamiltonian as,\n\\begin{align}\n    H^{cos}_{eff}(\\vec{k})=&H_0+\\frac{\\lambda_2\\alpha^2k^2_z}{\\omega}\\bigg(2k_xk_z\\{J_x,J_z\\}+2k_yk_z\\{J_y,J_z\\}\\cr\n    +&(k^2_x-k^2_y)(J^2_x-J^2_y)+2k_xk_y\\{J_x,J_y\\}\\bigg).\n\\end{align}\nwith energies,\n\\begin{align}\n    E^{\\pm}(\\vec{k})=\\lambda_1 k^2\\pm \\frac{\\lambda_2\\sqrt{f(\\vec{k})\\left[k^2_z\\alpha^2-2\\omega^2\\right]+4k^4\\omega^4}}{\\omega^2},\n\\end{align}\nwhere each $f(\\vec{k})=3 (k_x^2+k_y^2)k_z^2 (k_x^2+k_y^2 +4k_z^2)\\alpha^2$ and $E^{\\pm}$ are doubly degenerate. Figure.~\\ref{fig:kzG5smooth} shows the spectrum for $k_y=0$ plane, where four linear Dirac nodes coexist with a QBT at which is located at $\\Gamma$ point. A similar plot can be obtained for $k_x=0$ plane, then the system also possesses line-nodes in $k_x-k_y$ plane at $k_z\\neq 0$. Therefore, in addition to the coexistence of Dirac nodes and QBT at $k_x-k_z$ and $k_y-k_z$ planes, a smooth driving can lead to a richer phase diagram. This analysis was only for the purpose of comparison with kicked driving results, so a detailed analysis of such models will be left for future works.\n\n\\subsection{Effect of Lattice Regularization}\n\nLets now look at the effect of lattice regularization. One of the typical effect of lattice versus continuum model is the appearance of more nodes, usually at the boundaries of the Brillioun Zone \\cite{Ghorashi2018}. However, we show that the main features discussed in the previous subsections survives in lattice models. For the sake of brevity we restrict ourselves to $h=0$ limit. We consider a cubic lattice, which captures the physics of continuum model for $\\alpha,\\omega,\\vec{k}<<1$. In the $k_z$ cut (Figure.~\\ref{fig:latt}), we find: (i) two nodes at BZ boundaries, $k_z=\\pm \\pi$, as we expected, (ii) one at the $\\Gamma$ point with quadratic dispersion in $k_i \\perp k_z$, in agreement with the result of continuum model, (iii) however, due to lattice regularization there can be two nodes (instead of the node at $k_z=\\frac{\\alpha}{2\\lambda_2 T}$ in the continuum model) with linear dispersion away from the boundaries and the $\\Gamma$ point, which are located at $k_z=\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T}), k_z=-\\sin^{-1}(\\frac{\\alpha}{2\\lambda_2 T})+\\pi$. Figure.~\\ref{fig:latt}, shows the lattice bandstructure along $k_z$ for the case of $k_z\\Gamma_5$ driving. Therefore, we confirm that the generation of the hybrid dispersion Dirac semimetal (and consequently Weyl semimetal) persists in the lattice model.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{LSM_kzG5_03_lattice.pdf}\n    \\caption{The $k_x=k_y=0$ cut of the bandstructure of LSM on a cubic lattice in presence of $\\alpha\\sin(k_z)\\Gamma_5$ ($\\alpha=0.3$) along the $k_z$ axis. $\\lambda_1=\\lambda_2=0.6$  and $\\omega=20$ are used.}\n    \\label{fig:latt}\n\\end{figure}\n\\section{DISCUSSION AND CONCLUDING REMARKS}\n\nWe have proposed a dynamical way to realize hybrid Dirac and Weyl semimetals in a three-dimensional Luttinger semimetal via applying a nonuniform (momentum-dependent) periodic $\\delta$-kick. We explicitly demonstrated this through two examples of nonuniform kicking which break both inversion and time-reversal symmetries while preserving their combinations. We have identified the first example of an unusual hybrid Dirac semimetal phase where two nodes not only have different types but also have different dispersions ( a linear Dirac and QBT coexist). Then by applying an external magnetic field we demonstrated the emergence of hybrid Weyl semimetals. Next, we found that the combination of a tilted QBT with an external magnetic field provides a promising setup for generation of hybrid Weyl semimetals. Moreover, by interpreting the tilted QBT as a QBT with emergent psudomagnetic field proportional to kick strength, we discussed the interplay between the kick strength and the external magnetic field. \\\\\n\\indent We note that the experimental realization of models discussed in this work can be difficult, specifically for the case of uniform strain which requires very fast time scales for periodic driving. Despite all the difficulties there are some proposals for the fast dynamical generation of strain \\cite{straindyn1,straindyn2,kickgraphene}. However, the possibility of interpreting the $k_z\\Gamma_5$ kick as an applied spin current (or proportional to it) along the $z$-direction is potentially a promising route, considering the recent developments in the field of ultrafast spintronics \\cite{ultrafastspin}. Moreover, the 2D QBT has been already proposed to be realized in optical lattices \\cite{QBToptical}, therefore, in practice a 3D QBT could be realized in optical lattices as a potential experimental setup where parameters can be tuned at will.\\\\\n\\indent The LSMs can describe the low-energy physics of many experimental candidates. Therefore, this work opens up a promising way for the realization of various hybrid Dirac and Weyl semimetals. Therefore, it could motivate further studies on the lacking investigation of physical properties of the hybrid Weyl semimetals. \\\\\nSome of the possible future directions are: the investigation of bulk and surface transport properties in the hybrid dispersion Dirac semimetals introduced here, in particular, how some of the most interesting properties of a typical Dirac semimetal, such as transport anomalies \\cite{review1}, would be different. Moreover, it would be interesting to see whether other multi-band systems with higher-spin can show similar physics.\n\n\\section*{Acknowledgement}\nWe thank Matthew Foster and Bitan Roy for useful comments. We also acknowledge useful comments and suggestions from the anonymous referees which helped to improve this manuscript. This work was supported by the U.S. Army Research Office Grant No.W911NF-18-1-0290. we also acknowledge partial support from NSF CAREER Grant No. DMR-1455233 and ONR Grant No. ONR-N00014-16-1-3158.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}