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+{"text":"\\section{The domain of complex metrics}\n\nA Riemannian metric on a manifold $M$ is a positive-definite symmetric bilinear form $g: T_x \\times T_x \\to \\mathbb R$ on the tangent space $T_x$ at each point $x \\in M$. The metrics we shall consider will be defined by symmetric $\\mathbb R$-bilinear maps $g: T_x \\times T_x \\to \\mathbb C$ at each point, with an appropriate generalization of the positivity condition.\n\nTo see what condition we should require, let us consider the simplest example of a field theory: a free real scalar field of mass $m$. Then the space of `fields' $\\Phi_M$ is the vector space C$^{\\infty}(M;\\mathbb R)$ of smooth functions, and in the exponent of the path-integral we have the quadratic form\n\\begin{eqnarray*}\n{\\rm i} S_g(\\phi)  \\ \\  &=& \\  \\ \\frac{1}{2}\\int_M ( d\\phi\\wedge*d\\phi + m^2 \\phi\\wedge *\\phi) \\\\ &=& \\  \\ \\frac{1}{2}\\int_M \\left\\{\\sum g^{ij}\\frac{\\partial \\phi}{\\partial x^i}\\frac{\\partial \\phi}{\\partial x^j} + m^2\\phi^2 \\right\\} ( \\det g)^{1\/2} |dx^1 \\ldots dx^d|.\n\\end{eqnarray*}\nHere $(g^{ij})$ denotes the inverse of the matrix $g = (g_{ij})$, and $*$ is the Hodge star-operator  defined by the metric, which takes differential forms of degree $p$ to  forms of degree $d-p$ twisted by the orientation bundle.  (We shall not assume the space-time $M$ is orientable.) In particular the star-operator takes the constant function 1 to the volume element \n$$*1 \\ = \\ {\\rm vol}_g \\ = \\ ({\\rm det} g)^{1\/2}|dx^1 \\ldots dx^d| \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (3)$$\nNotice that for a Lorentzian metric $g$ the volume element $*1$ is pure imaginary. This agrees with the fact that the `action' $S_g$ should be real for a Lorentzian manifold. We want the real part of the quadratic form i$S_g$ to be positive-definite for all the complex metrics we allow. This imposes two conditions. First,  we need the real part of the twisted $d$-form vol$_g$ defined by the formula (3) to be a positive volume-form on $M$. We therefore require that $\\det g$, which is invariantly defined up to multiplication by a positive real number, is \\emph{not} real and negative, and  we choose $(\\det g)^{1\/2}$ to have positive real part. \n\nThe second condition we need is that the real part of the matrix (det$g)^{1\/2}g^{-1}$ --- or equivalently of the inverse matrix $({\\rm det}g)^{-1\/2}g$ --- is positive-definite.  The two conditions together would give us a domain whose  Shilov boundary (like that of the Siegel generalized half-plane) contains indefinite real quadratic forms of all signatures, and not only the Lorentzian ones. But we shall impose further conditions. A clue to what more is needed comes from the theory of the electromagnetic field on $M$, with its field-strength given by a real 2-form $F$ on $M$, and with the action-functional\n$${\\rm i} S_g(F) \\ = \\ \\frac{1}{2}\\int_M F \\wedge * F.$$\nThe Hodge $*$-operator makes sense for a complex metric: for a $p$-form $\\alpha$ we define a twisted $(d-p)$-form $*\\alpha$  by taking the inner-product of $\\alpha$ with vol$_g \\ = \\ *1$, using the complex inner-product $g$.  We regard vol$_g$ as an element of the complex line $|\\wedge^d(T^*_x)|_{\\mathbb C}$, where $|\\wedge^d(T^*_x)|$ is the tensor product of $\\wedge^d(T^*_x)$ with the real orientation line of $T_x$, and $|\\wedge^d(T^*_x)|_{\\mathbb C}$ is its complexification, \\emph{but} with the convention that the orientation-reversing automorphisms of $T_x$ act \\emph{antilinearly}. We say that an element of the real part of the line is positive if it is a positive volume-element.\n\nFor the electromagnetic field we need the real part of the quadratic form\n$$\\wedge^2(T_x^*)  \\ \\longrightarrow \\ |\\wedge^d(T_x^*)|_{\\mathbb C}$$ \ngiven by $F \\mapsto F\\wedge *F$ to be positive-definite. \n\nThis makes it natural, if we are going to consider space-time manifolds $M$ of all dimensions, to propose \n\n\\bigskip\n\n\\noindent{\\bf Definition 2.1} \\ \\  {\\it On a $d$-dimensional real vector space $V$ a quadratic form $g:V \\to \\mathbb C$ is called an {\\rm allowable} complex metric if, for all degrees $p \\geq 0$, the real part of the quadratic form \n$$\\wedge^p(V^*) \\ \\longrightarrow \\ |\\wedge^d(V^*)|_{\\mathbb C}$$\ngiven by  $\\alpha \\mapsto \\alpha \\wedge * \\alpha$ is positive-definite.} \n\n\\bigskip\n\nFortunately, this definition has an equivalent formulation which is much more explicit and illuminating.\n\n\\bigskip\n\n\n\\noindent{\\bf Theorem 2.2} \\ \\ {\\it Definition 2.1 is equivalent to: there is a basis of the real vector space $V$ in which the quadratic form $g$ can be written\n$$\\lambda_1 y_1^2 + \\lambda_2 y_2^2 + \\ldots + \\lambda_d y_d^2,$$\nwhere the $y_i$ are coordinates with respect to the basis, and the $\\lambda_i$ are non-zero complex numbers, not on the negative real axis, such that}\n$$|\\arg (\\lambda_1) | + |\\arg (\\lambda_2) | + \\ldots + |\\arg (\\lambda_d)| < \\pi. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (4)$$ \n\n\\bigskip\n\nThe complex-valued quadratic forms  $g:V \\to \\mathbb C$ on a real vector space $V$ which satisfy the conditions of (2.1) or (2.2) form an open subset  $Q_{\\mathbb C}(V)$ of the complex vector space $S^2(V^*_{\\mathbb C})$.   It  follows from Theorem 2.2 that the real inner products with signature $(d-1,1)$ --- but not those with other signatures --- lie on the boundary of the domain $Q_{\\mathbb C}(V)$. For if the metric is real then each $|\\arg(\\lambda_i)|$ is either 0 or $\\pi$, and the inequality (4)  shows that at most \\emph{one} of the $|\\arg(\\lambda_i)|$ can become $\\pi$ on the boundary. \n\nAnother consequence of (4) is that \n$$\\max \\arg \\lambda_i \\ - \\ \\min \\arg \\lambda_i \\ < \\ \\pi,$$\nwhich shows  that when $v$ runs through $V$ the complex numbers $g(v)$ form a closed convex cone in $\\mathbb C$ disjoint from the open negative real axis. In particular, $g(v)$ can never be real and negative.\n  \n  Using the criterion mentioned at the end of Section 1 we see that the real Lorentzian metrics --- and no other nondegenerate metrics ---  belong to the \\emph{Shilov} boundary of $Q_{\\mathbb C}(V)$, when it is regarded as a bounded domain in an affine variety (cf. the proof of 2.7 below).  Indeed if $g = \\sum \\lambda_j y_j^2$ is a complex metric for which the inequality (4) becomes an equality, and at least two of the eigenvalues $\\lambda_j$ and $\\lambda_k$ are not on the negative real axis, then (after rescaling the basis vectors $e_j$ and $e_k$ so that $|\\lambda_j | = |\\lambda_k| = 1$) we get a holomorphic curve through $g$, in the closure of $Q_{\\mathbb C}(V)$, by changing $\\lambda_j$ to $(\\lambda_j)^{1+z}$ and $\\lambda_k$ to $(\\lambda_k)^{1-\\varepsilon z}$, where $\\varepsilon$ is $+1$ or $-1$ according as the arguments of $\\lambda_j$ and $\\lambda_k$ have the same or opposite signs. \n  \n  In fact the Shilov boundary of $Q_{\\mathbb C}(V)$ contains \\emph{two} disjoint copies of the space of Lorentzian metrics on $V$, for an eigenvalue $\\lambda$ can approach the negative real axis either from above or from below. The two copies are interchanged by the complex-conjugation map on $Q_{\\mathbb C}(V)$. Because of our choice to make the orientation-reversing elements of GL$(V)$  act antilinearly on the orientation-line of $V$, we can say that the nondegenerate points of the Shilov boundary of $Q_{\\mathbb C}(V)$ are the \\emph{time-oriented} Lorentzian metrics.\n\n\\bigskip\n\n\nWe define the space Met$_{\\mathbb C}(M)$ of allowable complex metrics on a smooth manifold $M$ as the space of smooth sections of the bundle on $M$ whose fibre at $x$ is $Q_{\\mathbb C}(T_x)$. \n\n\\bigskip\n\nBefore giving the surprisingly simple proof of Theorem 2.2\nlet us say what motivated  the two different-looking  conditions. The desire to make the real parts of natural quadratic action functionals positive-definite hardly needs further comment, but choosing to focus on the  `higher abelian gauge field' actions $\\alpha \\wedge  * \\alpha$ --- the `Ramond-Ramond' fields of superstring theory ---  may well seem arbitrary. Why not allow other kinds of tensor fields? Our conditions do not imply that they will be positive-definite. Witten has kindly suggested to us a justification for our focus, based on properties of the classical energy-mometum tensor explained in [WW]. Including the higher gauge theories does, in any case, impose an \\emph{upper} bound on the class of complex metrics we can allow, for the  partition functions of these theories on a  $d$-dimensional torus $M$ with a flat Riemannian metric $g$ are explicitly known (cf. [Ke], [Sz](4.4)), and we can see to which complex metrics they can be analytically continued. The gauge-equivalence classes of fields form an infinite-dimensional Lie group which is a product of a torus, a lattice, and an infinite-dimensional real vector space, and the partition function is the product of three corresponding factors.  More precisely, an abelian gauge $(p-1)$-field $A$ has a \\emph{field-strength} $F_A$, a closed $p$-form on $M$ with integral periods, which determines $A$ up to the finite-dimensional torus $H^{p-1}(M;\\mathbb T)$ of flat gauge fields with $F_A = 0$. The space of fields is therefore a product\n$$H^{p-1}(M;\\mathbb T) \\ \\times \\ \\Phi_p \\ \\times \\ \\Gamma_p,$$\nwhere $\\Phi_p$ is the vector space of exact $p$-forms on $M$, and $\\Gamma_p \\cong {\\rm Harm}_{\\mathbb Z}^p(M) \\cong H^p(M;\\mathbb Z)$ is the finite-dimensional lattice of harmonic (and hence constant) $p$-forms with integral periods. The partition function is a Gaussian integral on this product: the torus of flat fields contributes its volume (for an appropriate metric determined by the geometry of $M$), the lattice $\\Gamma_p$  of harmonic $p$-forms contributes its theta-function $$\\sum_{\\alpha \\in \\Gamma_p}  \\ \\exp \\left(-\\frac{1}{2} \\int_M\\ \\alpha \\wedge *\\alpha \\right ) \\ ,$$\nwhile the vector space $\\Phi_p$ contributes an `analytic torsion' which is a power of the determinant of the Laplace operator acting on smooth functions on $M$ (with the zero-eigenvalue omitted) --- an analogue of the Dedekind eta-function, but with the lattice of characters of the torus $M$ replacing the lattice $\\mathbb Z + \\tau \\mathbb Z \\subset \\mathbb C$. Of these three  factors, the first clearly extends holomorphically to the space of all flat complex metrics on $M$, and the analytic torsion can be continued to a non-vanishing holomorphic function in the open set of complex metrics $g$ for which (det  $g)^{-1\/2}g$ belongs to the Siegel domain $\\mathbb U(V))$; but the theta-function cannot be continued beyond those metrics for which the real part of the form $\\int \\alpha \\wedge *\\alpha$ is positive. \n\n\n\\bigskip\n\nApproaching from the opposite direction, the inequality (4) is motivated by \nthe traditional analytical continuation of vacuum expection values to an open subset of the $k$-fold product of complexified Minkowski space $\\mathbb M_{\\mathbb C}$. The Wightman axioms imply that the expectation values extend holomorphically to a domain $\\mathcal U_k$ called the `permuted extended tube'\\footnote{A set of points $x_1, \\ldots , x_k$ belongs to $\\mathcal U_k$ if, after ordering them suitably, there is an element $\\gamma$ of the complexified Lorentz group such that the imaginary part of $\\gamma(x_i - x_{i+1})$ belongs to the forward light-cone for each $i$. }, which is functorially associated to $\\mathbb M_{\\mathbb C}$ with its $\\mathbb C$-bilinear metric.  It is a basic result in the Wightman theory (cf. [SW], or [Ka](2.1)) that $\\mathcal U_k$ contains the configuration space Conf$_k(\\mathbb E)$ of all $k$-tuples of \\emph{distinct} points of the standard Euclidean subspace $\\mathbb E \\subset \\mathbb M_{\\mathbb C}$. For a $d$-dimensional real vector space $V$ with a complex metric the complexification $V_{\\mathbb C}$ is isomorphic to $\\mathbb M_{\\mathbb C}$, uniquely up to a complex Lorentz transformation, and so the domain $\\mathcal U_k(V)$ is well-defined in $(V_{\\mathbb C})^k$. In  the next section we shall give a definition of a quantum field theory on space-times $M$ with complex metrics: it implies that the expectation values are smooth functions on the configuration spaces Conf$_k(M)$ of distinct $k$-tuples in $M$.  That makes it natural to\nask which (constant) complex metrics on $V$  have the property that the configuration space Conf$_k(V)$  is contained in the holomorphic envelope of $\\mathcal U_k(V)$, i.e.  the largest Stein manifold to which all holomorphic functions on $\\mathcal U_k(V)$ automatically extend. The original motivation of condition (4) was\n\n\\bigskip\n\n\\noindent{\\bf Proposition 2.3} \\ {\\it If a complex metric on a $d$-dimensional real vector space $V$ satisfies condition (4)  then ${\\rm Conf}_k(V)$ is contained in the holomorphic envelope of $\\mathcal U_k(V)$.}\n\n\\bigskip\n\n We shall postpone the  proof of this result to an appendix at the end of this section. \n\n\\bigskip\n     \n\\noindent{\\it Proof of Theorem 2.2} \\ \\  The first point is to show that a quadratic form which satisfies the conditions of Definition 2.1 can be written in the diagonal form $\\sum \\lambda_j y_j^2$ with respect to  real coordinates $y_j$ on $V$. To diagonalize a complex form $g = A+ {\\rm i} B$ with respect to a real basis is to diagonalize its real and imaginary parts simultaneously, which is possible if either $A$ or $B$ --- or, more generally, a real linear combination of them such as the real part of $({\\rm det}g)^{-1\/2}g$ --- is positive-definite. But 2.1, applied when  $p=1$, implies that the real part of $({\\rm det}g)^{-1\/2}g$ is positive.\n\nSuppose now that $g$ is diagonalized  with respect to a basis $\\{e_i\\}$ of $V$. Then the form $\\alpha \\mapsto \\alpha \\wedge * \\alpha$ on $\\wedge^p(V^*)$ is diagonal with respect to the basis $\\{e_S^* = e^*_{i_1}\\wedge \\ldots \\wedge e^*_{i_p}\\}$, where $\\{e_i^*\\}$ is the dual basis to $\\{e_i\\}$, and $S$ runs through $p$-tuples $S = (i_1, \\ldots , i_p)$. The value of the form  $\\alpha \\wedge * \\alpha$ on the basis element $e^*_S$ is\n$$(\\lambda_1 \\ldots \\lambda_d)^{1\/2}\\prod_{i \\in S}\\lambda_i^{-1},$$\nwhich has positive real part if its argument\n$$\\frac{1}{2}\\left\\{\\sum_{i \\in S} {\\rm arg}(\\lambda_i) - \\sum_{i \\not \\in S} {\\rm arg}(\\lambda_i)\\right\\}$$\nlies in the open interval $(-\\pi\/2,\\pi\/2)$. But to say that this is true for every subset $S$ of $\\{1, \\ldots , d\\}$ is precisely condition $(4)$.  $\\spadesuit$\n\n\\bigskip\n\n\nThe proof of Theorem 2.2 shows that to give an element $g$ of $Q_{\\mathbb C}(V)$ is the same as to give a finite sequence $\\theta_1 \\geq \\theta_2 \\geq \\ldots \\geq \\theta_m$ in the interval $(-\\pi, \\pi)$ together with a  decomposition\n $$V = V_1 \\oplus \\ldots \\oplus V_m$$\n such that\n $$\\sum_k \\dim V_k\\cdot |\\theta_k|<\\pi. $$ Thus on $V_k$ the bilinear form $g$ is ${\\rm e}^{{\\rm i} \\theta_k}$ times a real positive-definite form.  The only ambiguity in this description is that if, say, $\\theta_k = \\theta_{k+1}$ we can  replace $V_k$ by $V_k \\oplus V_{k+1}$ and omit $\\theta_{k+1}$ and $V_{k+1}$. This means that the subspace $P = \\bigoplus {\\rm e}^{-{\\rm i} \\theta_k\/2}V_k$ of the complexification $V_{\\mathbb C}$ of $V$ is \\emph{canonically} associated to the form $g$. On the real subspace $P$ the complex bilinear form $g$ is real and positive-definite. Our argument  gives us  canonical isomorphisms \n $$V \\ = \\ \\exp({\\rm i}\\pi\\Theta\/2)(P) \\ \\subset \\ P_{\\mathbb C} \\ = \\ V_{\\mathbb C},$$\n where $\\Theta:P \\to P$ is the self-adjoint operator which is multiplication by $\\theta_k$ on $P_k = {\\rm e}^{-{\\rm i}\\theta_k\/2} V_k$. Condition $(4)$ becomes the assertion that $\\Theta$ has trace-norm\\footnote{The trace-norm is the sum of the absolute values of the eigenvalues.} $||\\Theta ||_1 < 1$. \nThis shows that the space $Q_{\\mathbb C}(V)$ is parametrized by the pairs $(g_0, \\Theta)$, where $g_0$ is a positive-definite inner-product on $V$ and $\\Theta$ belongs to the  convex open set $\\Pi(V,g_0)$ of operators in $V$ which are self-adjoint with respect to $g_0$ and satisfy $||\\Theta||_1 < 1$, i.e. the interior of the convex hull of the rank 1 orthogonal projections in $V$.  In fact we have proved\n \n \\bigskip\n \n \\noindent{\\bf Proposition 2.4} \\ \\ $Q_{\\mathbb C}(V)$ {\\it is a fibre-bundle over the space of positive-definite inner products on $V$ whose fibre at a point $g_0$ is $\\Pi(V,g_0)$. Equivalently, choosing a reference inner-product on $V$, we have\n $$Q_{\\mathbb C}(V) \\ \\cong \\ {\\rm GL}(V) \\times_{{\\rm O}(V)} \\Pi(V).$$\n In particular, $Q_{\\mathbb C}(V)$ is contractible.}\n \n \\bigskip\n\nIt is an important fact that an allowable complex metric on $V$ remains allowable when restricted to any subspace $W$ of $V$. This follows from an analogous property of the trace-norm, but we shall give a direct proof, as its point of view on the angles $\\theta_i$ as critical values  helps give a feeling for allowable complex metrics.\n\n\\bigskip \n\n \\noindent{\\bf Proposition 2.5} \\  \\ {\\it If $g \\in Q_{\\mathbb C}(V)$ and $W$ is any vector subspace of $V$ then $g|W$ belongs to $Q_{\\mathbb C}(W)$. }\n \n \\bigskip\n\n\\noindent{\\it Proof} \\ \\ \\  For any $g \\in Q_{\\mathbb C}(V)$ the function $v \\mapsto {\\rm arg}(g(v))$ is a smooth map from the real projective space $\\mathbb P(V)$ to the open interval $(-\\pi, \\pi) \\subset \\mathbb R$.  By rescaling the basis elements $\\{e_k\\}$  we can write $g$ as $\\sum{\\rm e}^{{\\rm i}\\theta_k}y_k^2$. The numbers $\\theta_k$ are precisely the critical values of arg$(g)$.  We shall order the basis elements so that\n$$\\pi \\ > \\ \\theta_1 \\ \\geq \\ \\theta_2 \\ \\geq \\ \\ldots \\ \\geq \\ \\theta_d \\ > \\ -\\pi.$$\n\n\nFor each vector subspace $A$ of $V$  let us write $\\theta^A$ and $\\theta_A$ for the supremum and infimum of arg$(g)$ on $\\mathbb P(A)$.  Then we have\n$$\\theta_k  \\ \\ = \\ \\ {\\rm sup}\\{\\theta_A: {\\rm dim}(A) = k\\} \\ \\ = \\ \\ {\\rm inf}\\{\\theta^A: {\\rm dim}(A) = d-k+1\\}.$$\nIt is enough to prove Proposition 2.5 when $W$ is a subspace of $V$ of codimension 1. In that case the preceding characterization of the critical values shows that if \n$\\theta'_1 \\geq \\ldots \\geq \\theta'_{d-1}$\nare the critical values of arg$(g|W)$  we have $\\theta_k \\geq \\theta'_k \\geq \\theta_{k+1}$. The critical values for $g|W$ therefore \\emph{interleave} those for $g$:\n$$\\theta_1 \\geq \\theta'_1 \\geq \\theta_2 \\geq \\theta'_2 \\geq \\ldots \\geq \\theta_{d-1} \\geq \\theta'_{d-1} \\geq \\theta_d.$$\nThis implies that $\\sum |\\theta'_k| \\leq \\sum |\\theta_k| < \\pi$, as we want. $ \\ \\spadesuit$\n\n\n  \n\\bigskip\n\nIn Section 5 we shall need the following variant of the preceding formulation. \nSuppose that $Z$ is a $d$-dimensional complex vector space with a nondegenerate quadratic form $g$. (Any such pair $(Z,g)$ is isomorphic to $\\mathbb C^d$ with the standard form $\\sum z^2_k$.) Let $\\mathcal R(Z)$ denote the space of all $d$-dimensional real subspaces $A$ of $Z$ such that $g|A$ belongs to $Q_{\\mathbb C}(A)$. This is an open subset of the Grassmannian of all real subspaces of $Z$. If $Z_{\\mathbb R}$ is any $d$-dimensional real vector subspace of $Z$ for which $g|z_{\\mathbb R}$ is real and positive-definite then  the projection $A \\subset Z \\to Z_{\\mathbb R}$ is an isomorphism, for any non-zero element of its kernel would have the form i$v$ with $v \\in Z_{\\mathbb R}$, and so $g({\\rm i}v)$ would be real and negative, which cannot happen if $g|A$ is allowable.\n\n\\bigskip\n\n\\noindent{\\bf Proposition 2.6}  \\ \\ {\\it The space $\\mathcal R(Z)$ is contractible, and is  isomorphic to}\n$$ {\\rm O_{\\mathbb C}}(Z) \\times_{{\\rm O}(Z_{\\mathbb R})} \\Pi(Z_{\\mathbb R}).$$\n\n\\bigskip\n\n\\noindent{\\it Proof} \\ \\ This is essentially a reformulation of what has been said, but it may be helpful to relate the spaces $Q_{\\mathbb C}(V)$ and $\\mathcal R(Z)$ by considering, for a complex quadratic vector space  $(Z,g)$ as above, the intermediate space $\\mathcal R(V;Z)$ of $\\mathbb R$-linear embeddings $f:V \\to Z$ of the real vector space $V$  such that $f^*(g)$ is allowable. This space has two connected components, corresponding to the orientation of the projection $V \\to Z_{\\mathbb R}$.\n\nThe groups GL$(V)$ and O$_{\\mathbb C}(Z)$ act by right- and left-composition on $\\mathcal R(V;Z)$, and each action is free. Thus $\\mathcal R(V;Z)$ is at the same time a principal GL$(V)$-bundle with base $\\mathcal R(Z)$ and a principal O$_{\\mathbb C}(Z)$-bundle with base $Q_{\\mathbb C}(V)$. But the Lie groups GL$(V)$ and O$_{\\mathbb C}(Z)$ are homotopy equivalent to their maximal compact subgroups, i.e. in both cases to the compact orthogonal group O$_d$. More precisely, the contractibility of $Q_{\\mathbb C}(V)$ implies that $\\mathcal R(V;Z)$ is homotopy-equivalent to the fibre O$_{\\mathbb C}(Z)f$ for any  $f\\in \\mathcal R(V;Z)$. If we choose $f$ so that $f^*(g)$ is a positive-definite real form on $V$ this gives us a homotopy-equivalence $ {\\rm O}(V) \\to {\\rm O}_{\\mathbb C}(Z)f \\to \\mathcal R(V;Z)$. But O$(V)$ is also contained in and equivalent to the fibre $f{\\rm GL}(V)$ of the other fibration $\\mathcal R(V;Z) \\to \\mathcal R(Z)$, which implies the contractibility of its base $\\mathcal R(Z)$. \\ $\\spadesuit$\n\n\\bigskip\n\nThe last property of $Q_{\\mathbb C}(V)$ which we shall record briefly, for the sake of experts, is\n\n\\bigskip\n \n \n\\noindent{\\bf Proposition 2.7} \\  \\ {\\it The domain $Q_{\\mathbb C}(V)$ is holomorphically convex, i.e.\\ a `domain of holomorphy'.}\n\n\\bigskip\n\n\\noindent{\\it Proof} \\ \\ The Siegel domain $\\mathbb U(V)$ of  complex-valued inner products with positive-definite real part on a real vector space $V$ is known to be a domain of holomorphy in $S^2(V^*_{\\mathbb C})$. So therefore is the product\n$$\\prod_{0 \\leq p \\leq d\/2} \\mathbb U(\\wedge^p(V))$$\ninside its ambient complex vector space. The space $Q_{\\mathbb C}(V)$ is the intersection of this product domain with the affine variety which is the natural embedding of $S^2(V^*_{\\mathbb C})$ in this ambient vector space, and so it too is a domain of holomorphy. \\ $\\spadesuit$\n\n\\bigskip\n\n\n\\bigskip\n \n\\noindent {\\bf The two-dimensional case}\n\n\\bigskip\n\nThe case $d = 2$ is especially simple because then the matrix $(\\det g)^{- 1\/2}g$ depends only on the conformal structure, and decouples from  the volume element.\n\nA non-degenerate complex inner product $g$ on a 2-dimensional real vector space $V$ is determined up to a scalar multiple by its two distinct null-directions in the complexified space $V_{\\mathbb C}$. We can think of these as two points of the Riemann sphere $\\mathbb P(V_{\\mathbb C})$. Then  $(\\det g)^{-1\/2}g$ has positive real part precisely when the two points lie one in each of the open hemispheres of the sphere $\\mathbb P(V_{\\mathbb C})$ separated by the real equatorial circle $\\mathbb P(V)$. When the two points move to distinct points of the equator we get a Lorentzian inner product, with its two light-directions in $\\mathbb P(V)$.\n\nA point of the sphere $\\mathbb P(V_{\\mathbb C})$ not on the equator can be regarded as a complex structure on the real vector space $V$, and the two hemispheres correspond to the two possibilities for the orientation which a complex structure defines. On a smooth surface $\\Sigma$ any almost-complex structure is integrable, so a point of Met$_{\\mathbb C}(\\Sigma)$ is a pair of complex structures on $\\Sigma$ of opposite orientations, together with a complex volume element. The Riemannian metrics are those for which the two complex structures are complex-conjugate to each other, and the volume element is real.\n\nWhen $d=2$ the domain $Q_{\\mathbb C}(V)$ is thus a 3-dimensional polydisc, one disc for each of the complex structures, and the third for the volume-element.\n\n\\bigskip\n\n\\newpage\n\n\\noindent{\\bf The one-dimensional case: electric circuits}\n\n\\bigskip\n\nOur concept of an allowable complex metric does not at first look interesting in the one-dimensional case, but if we allow \\emph{singular} 1-manifolds --- identified with finite graphs $M$ --- we find that complex metrics arise naturally in electrical circuit theory. A Riemannian metric on $M$ is determined (up to isometry) by the assignment of a positive real number to each edge of the graph, and can be interpreted as its \\emph{resistance} when the edge is regarded as a wire in an electrical circuit. A state of the system (perhaps with current entering or leaving at each node) is determined by a continuous potential function $\\phi:M \\to \\mathbb R$ which is smooth on each closed edge, and whose gradient is the current flowing in the circuit. Because $\\phi$ is determined only up to adding a constant we shall normalize it by $\\int_M \\phi = 0.$  The energy of a state is \n$$\\frac{1}{2}\\int_M ||\\nabla \\phi ||^2 {\\rm d}s,$$\nand so the system can be regarded as a free massless field theory on the graph: in particular the vacuum expectation value $\\langle \\phi(x)\\phi(y)\\rangle$, when $x$ and $y$ are two nodes of the graph, is the ratio of the potential-difference $\\phi(x) - \\phi(y)$ to the current flowing in at $x$ and out at $y$ when no current is allowed to enter or leave at other nodes.\n\nWe encounter complex metrics when we consider a circuit in which  an alternating current with frequency $\\omega$ is flowing, and in which each branch has not only a resistance $R$ but also a positive inductance $L$ and a positive capacitance $C$. In that situation the volume element $\\sqrt g = R$ is replaced by the \\emph{impedance} $$\\sqrt g = R + {\\rm i}\\omega L + 1\/{\\rm i} \\omega C,$$ \na complex number which defines an allowable metric because Re$\\sqrt g > 0$.\n\nQuite apart from electric circuitry, however, singular one-dimensional manifolds with allowable complex metrics can arise in quantum field theory as the Gromov-Hausdorff limits of non-singular space-times of higher dimension. For example, if we embed a smooth graph $M$ in $\\mathbb R^3$, then for almost all sufficiently small $\\varepsilon >0$ the boundary of the $\\varepsilon$-neighbourhood of $M$ is a smooth surface $M_{\\varepsilon}$ whose limit is $M$ as $\\varepsilon \\to 0$: this is one way of viewing the passage from closed  string theory to quantum field theory.\n\n\\bigskip\n\n\\newpage\n\n\\noindent{\\bf Appendix to Section 2: proof of 2.3}\n\n\\bigskip\n\nIf $V$ is a real vector space with an allowable complex metric then  the preceding discussion shows that it can be identified with the subspace\n$$V \\ = \\ \\exp({\\rm i} \\Theta \/2)(\\mathbb E)$$\nof $\\mathbb M_{\\mathbb C}$. Here $\\mathbb E = \\mathbb R^d$ is the standard Euclidean subspace of $\\mathbb M_{\\mathbb C}$, and  $\\Theta$ is a real diagonal matrix whose entries $\\theta_1, \\ldots , \\theta_d$ belong to the `generalized octahedron' $\\Pi_0 \\subset \\mathbb R^d$ consisting of those $\\Theta$ whose diagonal entries $\\theta_1, \\ldots , \\theta_d$ satisfy  the inequality (4).  We want to prove  that $\\exp({\\rm i} \\Theta \/2)$ maps each $k$-tuple ${\\bf x} = \\{x_1,\\ldots , x_k\\}$ of distinct points of $\\mathbb E$ to a point of the holomorphic envelope $\\hat \\mathcal U_k$ of the Wightman permuted extended tube $\\mathcal U_k$. In fact we shall prove the stronger statement that $\\exp(\\rm i\\Theta\/2)({\\bf x}) \\in \\hat \\mathcal U_k$ when $\\Theta$ is a \\emph{complex} diagonal matrix with Re$(\\Theta) \\in \\Pi_0$.\n\n\n\nThe crucial fact is that $\\Pi_0$ is the convex hull of its intersection $\\Pi_{00}$ with the coordinate axes in $\\mathbb R^d$,  (i.e.\\ $\\Pi_{00}$ consists of the diagonal matrices with only one entry  $\\theta_r$  non-zero, and  $-\\pi < \\theta_r < \\pi$). Our strategy is to show that  $\\exp(\\rm i \\Theta \/2)({\\bf x}) \\in \\mathcal U_k$ when Re$(\\Theta) \\in \\Pi_{00}$, and to deduce that the same is true when Re$(\\Theta)$ belongs to the convex hull $\\Pi_0$.  The essential tool  is Bochner's `tube theorem' ([H\\\"or] Thm 2.5.10), which asserts that if $P$ is a connected open subset of $\\mathbb R^d$ then a holomorphic function defined in the tube domain $P\\times \\rm i\\mathbb R^d$ extends holomorphically to the tube domain $P' \\times \\rm i\\mathbb R^d$, where $P'$ is the convex hull of $P$.\n\nHaving fixed a $k$-tuple ${\\bf x}$ in $\\mathbb M_{\\mathbb C}$, let us first show that if Re$(\\Theta) \\in \\Pi_{00}$ then $\\exp({\\rm i} \\Theta \/2)({\\bf x})$ is contained in $\\mathcal U_k$. Suppose that the non-zero diagonal element of $\\Theta$ is in the $r^{\\rm th}$ place. Because $\\mathcal U_k$ in invariant under the orthogonal group O$(\\mathbb E)$ we can assume that the $r^{\\rm th}$ basis vector $e_r$ of $\\mathbb E$ is the Wick-rotated time-axis of $\\mathbb M$, so that $e_r$ belongs to $\\rm i C$, where $C$ is the forward light-cone in $\\mathbb M$. With respect to the real structure $\\mathbb M_{\\mathbb C} = \\mathbb M \\oplus \\rm i \\mathbb M$ the imaginary part of the $k$-tuple $${\\bf y} \\ =  \\ \\exp(\\rm i\\Theta\/2)({\\bf x})$$  lies on the line $\\mathbb R e_r$, and so, after ordering the points appropriately, ${\\bf y}$ will belong to the forward tube in $\\mathbb M_{\\mathbb C}$ providing the points of ${\\bf x}$ have distinct $r^{\\rm th}$ coordinates. But if the $r^{\\rm th}$ coordinates of Im$({\\bf y})$ are not distinct, we can make them so by choosing a unit vector $e \\in \\mathbb E$ perpendicular to $e_r$ such that the coordinates $\\langle {\\bf x}, e \\rangle$ are distinct, and rotating the $k$-tuple ${\\bf y}$ by a small amount in the $\\{e,e_r\\}$-plane, again using the O$(\\mathbb E)$-invariance of $\\mathcal U_k$.\n\nWe now know that $\\mathcal U_k$ contains an open neighbourhood of $\\Pi_{00} \\times \\rm i\\mathbb R^d$ in $\\mathbb C^d$. To apply Bochner's  theorem we need to know that the envelope $\\hat \\mathcal U_k$ contains a \\emph{tube}  $P \\times \\rm i\\mathbb R^d$,  where $P$ is an open neighbourhood of $\\Pi_{00}$ in $\\mathbb R^d$.  In fact it is enough, by induction, to treat the case  $d = 2$, for that case, together with Bochner's theorem, implies that a function holomorphic in a neighbourhood of $(\\Pi_0(\\mathbb R^r) \\cup \\Pi_{00}(\\mathbb R^{d-r}) \\times \\rm i\\mathbb R^d$ is holomorphic in a neighbourhood of $(\\Pi_0(\\mathbb R^{r+1}) \\cup \\Pi_{00}(\\mathbb R^{d-r-1})) \\times \\rm i\\mathbb R^d$.\n\nTo reduce the  $d=2$ case to the standard Bochner theorem it is enough to prove the following\n\n\\bigskip\n\n\\noindent{\\bf Lemma 2.8} \\ {\\it Let $L$ be the L-shaped subset $(\\{0\\} \\times [0,1)) \\cup ([0,1) \\times \\{0\\})$ of the quadrant $(\\mathbb R_+)^2$. Then any holomorphic function $F$ defined in a neighbourhood of $L \\times \\rm i\\mathbb R^2 \\subset \\mathbb C^2$ can be extended  holomorphically to  $P \\times \\rm i\\mathbb R^2$, where $P$ is the intersection of $(\\mathbb R_+)^2$ with a neighbourhood of $L$ in $\\mathbb R^2$.}\n\n\\bigskip\n\n\\noindent{\\it Proof} \\ For any $t \\in (0,1\/2)$ we define $D_t$ as the intersection of the two  unit discs $\\{z \\in \\mathbb C: |z - (1-t)| \\leq 1\\}$ and $\\{z \\in \\mathbb C: |z + (1-t)| \\leq 1 \\}$. Then we define $f:D_t \\to \\mathbb C^2$ by\n$$f(z) \\ = \\ (-\\log ((1-t)-z), -\\log ((1-t) +z).$$\nThe map $f$ is a holomorphic embedding in a neighbourhood of $D_t$ in $\\mathbb C$, and Re $f(\\partial D_t)$ is contained in the coordinate axes of $\\mathbb R^2$. If we choose \n $T = (1 - {\\rm e}^{-1})\/2$ then Re $f(\\partial D_T)$ is precisely the closure of $L$.\n  \nFor any $\\eta \\in \\mathbb R^2$, define $f_{\\eta}:D_T \\to \\mathbb C^2$ by $ f_{\\eta}(z) = f(z)+ \\rm i\\eta$. Then the holomorphic map $F$ is defined in a neighbourhood of the  curve $f_{\\eta}(\\partial D_T)$, and if we can show that $F \\circ f_{\\eta}$ extends holomorphically over $D_T$ then we shall have continued $F$ analytically to the tube domain $f(\\mathring D_T) + \\rm i\\mathbb R^2$, and the proof will be complete.\n\nWhen  a function $F$ is holomorphic in an open domain containing the boundary of a holomorphically-embedded disc  --- in this case $f_{\\eta}(D_T)$ --- then to show that $F$ can be extended over the whole disc  the standard method is to show that the disc can be moved holomorphically, keeping its boundary within the domain of $F$, until the whole disc is contained in the domain of $F$; the Cauchy integral formula then defines the desired extension. In our case we can deform  $f_{\\eta}(D_T)$ through the family $f_{\\eta}(D_t)$ as $t$ decreases from $T$ towards 0. As $t \\downarrow 0$ the domain $D_t$ shrinks to the origin in $\\mathbb C$, and $f_{\\eta}(D_t) \\to \\rm i\\eta$, which is contained in the domain of $F$. \\ \\ $\\spadesuit$ \n\n\n\n\\newpage\n\n\\section{Quantum field theories as functors}\n\nThe traditional Wightman approach to quantum field theory is not  well-adapted to important examples such as gauge theories, especially when the space-time is not flat. Another formulation --- potentially more general --- views a $d$-dimensional field theory as something more like a group representation, except that the group is replaced by a {\\it category} $\\mathcal C_d^{\\mathbb C}$ of space-time manifolds. The guiding principle of this approach is to preserve as much as possible of the path-integral intuition. We shall present it very briefly here, with minimal motivation.\n\nRoughly, the objects of the category $\\mathcal C_d^{\\mathbb C}$ are compact smooth $(d-1)$-dimensional manifolds $\\Sigma$ equipped with complex metrics $g \\in {\\rm Met}_{\\mathbb C}(\\Sigma)$. A morphism from $\\Sigma_0$ to $\\Sigma_1$ is a cobordism $M$ from $\\Sigma_0$ to $\\Sigma_1$, also with a complex metric. We shall write $M:\\Sigma_0 \\leadsto \\Sigma_1$ to indicate a cobordism. Composition of morphisms is by concatenation of the cobordisms.  The reason for the word `roughly' is that, because there is no canonical way to give a smooth structure to the concatenation of two smooth cobordisms, we must modify the definition slightly so that an object of $\\mathcal C_d^{\\mathbb C}$ is not a $(d-1)$-manifold but rather is a {\\it germ} of a $d$-manifold along a given $(d-1)$-manifold $\\Sigma$ --- i.e. $\\Sigma$ is given as a closed submanifold of a $d$-manifold $U$, but any two open neighbourhoods of $\\Sigma$ in $U$ define the same object of $\\mathcal C_d^{\\mathbb C}$. We require $\\Sigma$ to be \\emph{two-sided} in $U$, and equipped with a \\emph{co-orientation} which tells us which side is incoming and which is outgoing. (Nevertheless, we shall usually suppress  the thickening $U$, the co-orientation, and the complex metric $g$ from the notation.) Furthermore, two morphisms $M$ and $M'$ from $\\Sigma_0$ to $\\Sigma_1$ are identified if there is an isometry $M \\to M'$ which is the identity on the germs $\\Sigma_0$ and $\\Sigma_1$. (We shall return below to the question of the existence of identity morphisms in the cobordism category.)\n\n\n\\bigskip\n\nIn terms of the category $\\mathcal C_d^{\\mathbb C}$ we make the \n\n\\bigskip\n\n\\noindent{\\bf Definition} \\ {\\it A $d$-dimensional field theory is a \\emph{holomorphic} functor from $\\mathcal C_d^{\\mathbb C}$ to the category of Fr\\'echet topological vector spaces and \\emph{nuclear} (i.e. trace-class) linear maps which takes disjoint unions to tensor products.}\n\n\\bigskip\n\nUnfortunately, almost every word in this definition requires further explication.\n\n We shall write $E_{\\Sigma}$ for the vector space associated to an object $\\Sigma$, and $Z_M: E_{\\Sigma_0} \\to E_{\\Sigma_1}$ for the linear map associated to a cobordism $M: \\Sigma_0 \\leadsto \\Sigma_1$. To say that the functor is `holomorphic' means that, for a given smooth manifold-germ $\\Sigma \\subset U$, the topological vector spaces $E_{\\Sigma}$ form a locally trivial holomorphic vector bundle on the complex manifold Met$_{\\mathbb C}(U)$ of complex metrics on $U$, and that the maps $Z_M:E_{\\Sigma_0} \\to E_{\\Sigma_1}$ define a morphism of holomorphic vector bundles on the manifold Met$_{\\mathbb C}(M)$ (to which the bundles $\\{E_{\\Sigma_0}\\}$ and $\\{E_{\\Sigma_1}\\}$ are pulled back).\n\nIn practice, theories are usually defined on cobordism categories where the manifolds are required to have additional structure such as an orientation or a spin-structure. These can easily be included, but are not relevant to  our account. For the same reason we do not mention that, for a theory including fermions, the vector spaces $E_{\\Sigma}$ will have a mod 2 grading, and the usual sign-conventions must be applied when we speak of their tensor products.\n\n\\bigskip\n\nBecause our objects $\\Sigma \\subset U$ are really germs of $d$-manifolds, we automatically have a family of cobordisms $\\Sigma' \\leadsto \\Sigma$ embedded in $U$, each diffeomorphic to the trivial cobordism $\\Sigma \\times [0,1]$ with the outgoing boundary $\\Sigma \\times \\{1\\}$ corresponding to $\\Sigma \\subset U$. These cobordisms can be ordered by inclusion, giving us a direct system of objects $\\Sigma'$ with cobordisms to $\\Sigma$. Similarly, looking downstream rather than upstream, we have a family of cobordisms $\\Sigma \\leadsto \\Sigma''$ contained in $U$, giving us an inverse system of objects $\\Sigma''$ to which $\\Sigma$ maps.  For any field theory, therefore, there are natural maps\n$$\\lim_{\\rightarrow} E_{\\Sigma'} \\ \\  \\to \\ \\ E_{\\Sigma} \\ \\ \\to \\ \\ \\lim_{\\leftarrow} E_{\\Sigma''},$$\nwhich we shall write\n$$\\check E_{\\Sigma} \\ \\to \\ E_{\\Sigma} \\ \\to \\ \\hat E_{\\Sigma},$$\nintroducing the notations $\\check E_{\\Sigma} = \\varinjlim E_{\\Sigma'}$ and $\\hat E_{\\Sigma} = \\varprojlim E_{\\Sigma''}$ for the upstream and downstream limits. \n\nWe shall assume the functor has the \\emph{continuity} property that each of these maps is injective with dense image.  The space $\\hat E_{\\Sigma}$, being the inverse-limit of a countable sequence of nuclear maps of Fr\\'echet spaces, is a \\emph{nuclear} Fr\\'echet space\\footnote {A very useful concise account of nuclear spaces can be found in [C].}. The other space $\\check E_{\\Sigma}$ is also nuclear, but  usually not metrizable: it is the dual of the nuclear Fr\\'echet space $\\hat E_{\\Sigma^*}$, where $\\Sigma^*$ denotes the germ $\\Sigma$ with its co-orientation reversed. As this is such a basic point, we have included a proof as an Appendix at the end of this section. \n\nWhen we have a cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ we automatically get maps $\\check E_{\\Sigma_0} \\to \\check E_{\\Sigma_1}$ and $\\hat E_{\\Sigma_0} \\to \\hat E_{\\Sigma_1}$.\nThe space $E_{\\Sigma}$ with which we began plays only a provisional role in the theory, serving to construct the fundamental nuclear spaces between which it is sandwiched. \n\n \n\n\\bigskip\n\nThe essential requirement we place on the functor is that it takes disjoint unions to tensor products, i.e., we are given an isomorphism of functors\n$$\\check E_{\\Sigma} \\otimes \\check E_{\\Sigma'} \\ \\to \\ \\check E_{\\Sigma \\sqcup \\Sigma'} ,$$\nwhich is associative and commutative in terms of the usual isomorphisms for the disjoint union and tensor product. There is a unique natural concept of tensor product here, because all the vector spaces are nuclear, and $\\check E_{\\Sigma} \\otimes \\check E_{\\Sigma'} \\cong \\check E_{\\Sigma \\sqcup \\Sigma'}$ is equivalent to $\\hat E_{\\Sigma} \\otimes \\hat E_{\\Sigma'} \\cong \\hat E_{\\Sigma \\sqcup \\Sigma'}$.  The functoriality means that we are assuming \n$$Z_M \\otimes Z_{M'} \\ = \\ Z_{M \\sqcup M'}$$\nfor two cobordisms $M$ and $M'$. \n\n\\bigskip\n\nThe tensoring assumption implies that $E_{\\emptyset} = \\mathbb C$, where $\\emptyset$ denotes the empty $(d-1)$-manifold.  Thus for a \\emph{closed} $d$-manifold $M$ we have a \\emph{partition function} $Z_M \\in {\\rm End}(E_{\\emptyset}) = \\mathbb C$. The whole structure of the theory is a way of expressing the sense in which the number $Z_M$ depends \\emph{locally} on $M$.\n\n\n\\bigskip \n\nIn this discussion we have still committed an abuse of language: the ``category\" $\\mathcal C_d^{\\mathbb C}$ is not really a category because it does not have identity maps. We could deal with this by agreeing that an isomorphism $\\Sigma_0 \\to \\Sigma_1$ is a cobordism of zero length, but then these degenerate cobordisms are represented by operators which are not nuclear. The true replacement for the missing identity operators is our assumption that the maps $\\check E_{\\Sigma} \\to \\hat E_{\\Sigma}$ are injective with dense image. To avoid the abuse of language we can say that  a field theory is a functor $\\Sigma \\mapsto E_{\\Sigma}$ from $(d-1)$-manifolds and isomorphisms to vector spaces, together with a transformation $Z_M: E_{\\Sigma_0} \\to E_{\\Sigma_1}$ for each cobordism. Whichever line we take, we must  assume that an isomorphism $f:\\Sigma_0 \\to \\Sigma_1$ of germs of $d$-manifolds  induces an isomorphism $f_*: E_{\\Sigma_0} \\to E_{\\Sigma_1}$ which depends smoothly on $f$, in the sense that for any family $P \\times \\Sigma_0 \\to \\Sigma_1$ parametrized by a finite-dimensional manifold $P$ the induced map $P \\times E_{\\Sigma_0} \\to E_{\\Sigma_1}$ is smooth.\n \n\n \n \\bigskip\n\n\n\\bigskip\n\nLet us explain briefly \nhow to get from this functorial picture to the traditional language of local observables and vacuum expectation values. For a point $x$ of a $d$-manifold $M$ we define the vector space $\\mathcal O_x$ of observables at $x$ as follows. We consider the family of all closed discs $D$ smoothly embedded in $M$ which contain $x$ in the interior $\\mathring D$. If $D' \\subset \\mathring D$ then $D \\setminus \\mathring D'$ is a cobordism  $\\partial D' \\leadsto \\partial D$ and gives us a trace-class map $E_{\\partial D'} \\to E_{\\partial D}$. We therefore have an inverse system $\\{E_{\\partial D}\\}$ indexed by the discs $D$, and we define $\\mathcal O_x$ as its inverse-limit.\n\nNow suppose that $M$ is closed, and that $x_1, \\ \\ldots \\  x_k$ are distinct points of $M$. Let $D_1, \\ \\ldots \\ D_k$ be disjoint discs in $M$ with $x_i \\in \\mathring D_i$. Then $M' = M \\setminus \\bigcup \\mathring D_i$ is a cobordism from $\\bigsqcup \\partial D_i$ to the empty $(d-1)$-manifold $\\emptyset$, and defines $Z_{M'}:E_{\\sqcup \\partial D_i} \\to E_{\\emptyset} =\\mathbb C$. Using the tensoring property  we can write this\n$$Z_{M'} : \\bigotimes E_{\\partial D_i} \\  \\longrightarrow \\ \\mathbb C,$$\nand then we can pass to the inverse-limits to get the expectation-value map\n$$\\bigotimes \\mathcal O_{x_i} \\ \\longrightarrow \\ \\mathbb C.$$\n\n\n\nWe might prefer the language of ``field operators\" to that of vacuum expectation values. If the space-time $M$ is a cobordism $\\Sigma_0 \\leadsto \\Sigma_1$, then for any $x$ in the interior of $M$ --- say $x \\in \\mathring D \\subset M$ --- the cobordisms $M \\setminus \\mathring D$ from $\\partial D \\sqcup \\Sigma_0$ to $\\Sigma_1$ define maps \n$$\\mathcal O_x \\  \\to \\ {\\rm Hom}_{nucl}(E_{\\Sigma_0}; E_{\\Sigma_1}),$$\nwhile if $x$ lies on a hypersurface $\\Sigma$ an observable at $x$ defines a map $\\check E_{\\Sigma} \\to \\hat E_{\\Sigma}$, i.e. it acts on $E_{\\Sigma}$ as an unbounded operator. But on a Lorentzian space-time $M$ we sometimes want to make the observables at all points $x \\in M$ act on a \\emph{single} vector space, and to ask whether they commute when space-like separated. We shall postpone that discussion to Section 5.\n\n\\bigskip\n\nOne observable which we should mention is the \\emph{energy-momentum tensor}. If we think of a field theory as analogous to a group representation then the energy-momentum tensor is the analogue of the induced representation of Lie algebras: for every cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ it is the derivative of the operator $Z_M$ with respect to the metric of $M$. This makes it a distributional symmetric tensor-density $T^{ij}$ on $\\mathring M$ with values in Hom$_{nucl}(E_{\\Sigma_0};E_{\\Sigma_1})$. If we cover $M$ with small balls $D_i$, then by using a partition of unity we can write an infinitesimal change in the metric as the sum of contributions supported in the interiors of the $D_i$, and so the change in $Z_M$ is the sum of contributions coming from the spaces $E_{\\partial D_i}$, and hence from a field operators placed at the centres of the balls $D_i$.  But to develop this picture properly needs much more discussion, which we shall not embark on here;  it probably requires the assumption that the theory is asymptotically conformal at short distances.  The case of a 2-dimensional conformal theory is treated fully in Section 9 of [Se2].\n\n\\bigskip\n \n\n\n\n\\noindent {\\bf Lorentzian manifolds}\n\n\\bigskip\n\n\nThere is a category  $\\mathcal C^{\\rm Lor}_d$ which at first sight looks more relevant to quantum field theory than $\\mathcal C_d^{\\mathbb C}$.  Its objects are compact Riemannian manifolds of dimension $(d-1)$ and its morphisms are $d$-dimensional cobordisms equipped with real Lorentzian metrics. Fredenhagen and his coworkers (cf. [BF]) have developed the theory of quantum fields in curved space-time using a version of this category.  The category $\\mathcal C^{\\rm Lor}_d$ lies ``on the boundary\" of the category $\\mathcal C_d^{\\mathbb C}$. In section 5 we shall discuss the sense in which a representation of $\\mathcal C_d^{\\mathbb C}$ has a ``boundary value\" on $\\mathcal C_d^{\\rm Lor}$, at least if it is \\emph{unitary}. \n\n\\bigskip\n\n\n\n\\noindent {\\bf Unitarity}\n\n\n\n\\nopagebreak\n\n\n\\bigskip\n\nSo far we have not asked for an inner product on the topological vector space $E_{\\Sigma}$ associated to a $(d-1)$-manifold $\\Sigma$. Our main concern in this work is with {\\it unitary} theories, even though not  all interesting quantum field theories are unitary.\n\nTo define unitarity in our context, recall that,  if $\\Sigma^*$ denotes the manifold germ $\\Sigma$ with its co-orientation reversed, then $\\check E_{\\Sigma^*}$ is the dual topological vector space to $\\hat E_{\\Sigma}$. \nFurthermore, a cobordism $M: \\Sigma_0 \\leadsto \\Sigma_1$  can also be regarded as a cobordism from $\\Sigma^*_1$ to $\\Sigma^*_0$, and the two maps $E_{\\Sigma_0} \\to E_{\\Sigma_1}$ and $E_{\\Sigma_1^*} \\to E_{\\Sigma_0^*}$ are automatically algebraic transposes of each other. Thus $\\Sigma \\mapsto \\Sigma^*$ is a {\\it contravariant} functor.\n\n\nIn a unitary theory we shall not expect the vector space $E_{\\Sigma}$ to have an inner product for every $(d-1)$-manifold $\\Sigma$. A complex metric $g \\in {\\rm Met}_{\\mathbb C}(\\Sigma)$ has a complex conjugate $\\bar g$.\nIf we write $\\bar \\Sigma$ for $\\Sigma$ with the metric $\\bar g$ but with its co-orientation unchanged\\footnote{If our theory is defined on a category of \\emph{oriented} space-time manifolds, we must give $\\bar \\Sigma$  the opposite orientation to $\\Sigma$, although the same co-orientation in $U$.} then $\\Sigma \\mapsto \\bar \\Sigma$ is a \\emph{covariant} functor. It is natural to require that there is an antilinear involution\n$$E_{ \\bar\\Sigma} \\cong \\bar E_{\\Sigma}. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (5)$$\n\n\n\n\\bigskip\n\nFor a theory satisfying condition (5) the conjugate dual of the vector space $\\check E_{\\Sigma}$ is $\\hat E_{\\bar\\Sigma^*}$.  We expect $\\check E_{\\Sigma}$ to have an inner product only when $\\Sigma \\cong \\bar\\Sigma^*$, i.e. when the $d$-manifold germ $\\Sigma \\subset U$ admits a reflection with fixed-point set $\\Sigma$ which reverses the co-orientation and changes the metric to its complex conjugate.\nSuch a hypersurface-germ $\\Sigma$ will be called {\\it time-symmetric}. Its metric is real and Riemannian when restricted to the $(d-1)$-dimensional hypersurface $\\Sigma$ itself.\n\n\\bigskip\n\nWe can now define a {\\it unitary} theory as one which satisfies two conditions:\n\n\\bigskip\n\n(i) \\ \\ the reality condition (5), and\n\n\\bigskip\n\n(ii) \\ \\ {\\it reflection-positivity}, in the sense that when we have a time-symmetric hypersurface $\\Sigma \\cong \\bar \\Sigma^*$ the hermitian duality between $\\check E_{\\Sigma}$ and $\\check E_{\\bar \\Sigma}$ is positive-definite. \n\n\\bigskip\n\nFor a unitary theory, when we have a time-symmetric germ $\\Sigma$ we can complete the pre-Hilbert space $\\check E_{\\Sigma}$ to obtain a Hilbert space $E^{Hilb}_{\\Sigma}$ with\n$$\\check E_{\\Sigma}  \\ \\to \\ E_{\\Sigma}^{Hilb} \\ \\to \\ \\hat E_{\\Sigma}.$$ \n\n\n\n\n\n\n\n\n\n\n\\noindent{\\bf The theory on flat tori}\n\n\\bigskip\n\nThe partition function of a theory on oriented flat Riemannian tori already gives us a lot of information about the theory.  The moduli space of such tori is the double-coset space$${\\rm O}_d \\backslash {\\rm GL}_d(\\mathbb R)\/{\\rm SL}_d(\\mathbb Z) \\ \\cong \\ Q(\\mathbb R^d)\/{\\rm SL}_d(\\mathbb Z),$$\nwhere $Q(\\mathbb R^d) = {\\rm O}_d\\backslash {\\rm GL}_d(\\mathbb R)$ is the space of positive-definite real $d \\times d$ matrices. This space is an orbifold, so the partition function is best described as a smooth function $Z: Q(\\mathbb R^d) \\to \\mathbb C$ which is invariant under SL$_d(\\mathbb Z)$. Our axioms imply that $Z$ extends to a \\emph{holomorphic} function\n$$Q_{\\mathbb C}(\\mathbb R^d) \\ \\to \\ \\mathbb C,$$\nbut they also imply very strong constraints beyond that. Notably, each choice of a surjection $\\mathbb Z^d = \\pi_1(M) \\to \\mathbb Z$ gives us a way of writing the torus $M$ as a cobordism $\\tilde M : \\Sigma \\leadsto \\Sigma$ from a $(d-1)$-dimensional torus $\\Sigma$ to itself, and then we have $Z(M) = {\\rm trace} (Z_{\\tilde M})$, where \\mbox{$Z_{\\tilde M}:E_{\\Sigma} \\to E_{\\Sigma}$} is a nuclear operator in the vector space $E_{\\Sigma}$, which is graded by the characters $\\chi$ of the translation-group $T_{\\Sigma}$ of $\\Sigma$. More explicitly, $M$ is constructed from the product manifold $\\tilde M \\times [0,t]$ by attaching the ends to each other after translating by a vector $\\xi \\in T_{\\Sigma}$, and  we  have\n$$Z(A,t,\\xi) \\ = \\ \\sum_{i,\\chi} n_{i,\\chi }\\chi(\\xi) \\  {\\rm e}^{-\\lambda_i t},$$\nwhere $\\{\\lambda_i = \\lambda_i(A) \\}$ is the sequence (tending to $+\\infty$) of eigenvalues of the Hamiltonian operator on  $E_{\\Sigma}$, and the $n_{i,\\chi}$ are positive integers which, for each $i$, vanish for all but finitely many characters $\\chi$. \n\n\\bigskip\n\n\n\\noindent{\\bf Appendix  to Section 3: \\ The duality $(\\check E_{\\Sigma})^* \\ \\cong \\ \\hat E_{\\Sigma^*}$}\n\n\\bigskip\n\nTo keep things as general as possible, we suppose that $\\Sigma \\mapsto E_{\\Sigma}$ is a functor from the $d$-dimensional cobordism category to a category of metrizable topological vector spaces and nuclear maps. We suppose also that the category of vector spaces is equipped with a  tensor product functor\\footnote{For example, we could work with the category of Hilbert spaces with the natural Hilbert space tensor product.} which is coherently associative and commutative, and that we are given natural isomorphisms $E_{\\Sigma_1} \\otimes E_{\\Sigma_2} \\ \\to \\ E_{\\Sigma_1 \\sqcup \\Sigma_2}$.\n\n\\bigskip\n\nComposable cobordisms\n$\\Sigma_1 \\leadsto \\Sigma_2 \\leadsto \\Sigma_3$ \n  give us  maps\n$$E_{\\Sigma_1} \\to E_{\\Sigma_2} \\to E_{\\Sigma_3}. \\ \\qquad  \\ \\qquad \\ (6)$$\nBy reinterpreting  $\\Sigma_1 \\leadsto \\Sigma_2$ as a cobordism $\\Sigma_1 \\sqcup \\Sigma_2^* \\leadsto \\emptyset$ we get a map $E_{\\Sigma_1} \\otimes E_{\\Sigma_2^*} \\to \\mathbb C$, and hence $E_{\\Sigma_1} \\to (E_{\\Sigma_2^*})^*$. Similarly, we can reinterpret $\\Sigma_2 \\leadsto \\Sigma_3$ as $\\emptyset \\leadsto \\Sigma_2^* \\sqcup \\Sigma_3$, which gives $(E_{\\Sigma_2^*})^* \\to E_{\\Sigma_3}$. It is easy to see that the composite $E_{\\Sigma_1} \\to (E_{\\Sigma_2^*})^* \\to E_{\\Sigma_3}$ coincides with $E_{\\Sigma_1} \\to E_{\\Sigma_2} \\to E_{\\Sigma_3}$. \n\nYet again, performing the reinterpretations in the reverse order, we get maps\n$$(E_{\\Sigma^*_1})^* \\to E_{\\Sigma_2} \\to (E_{\\Sigma^*_3})^*$$\nwhose composite  is the transpose of the map induced by the composite cobordism $\\Sigma_3^* \\leadsto \\Sigma_1^*$.\n\n\\bigskip\n\nNow suppose that we have an infinite sequence of cobordisms \n$$\\ldots \\leadsto \\Sigma_{i+1} \\leadsto \\Sigma_{i} \\leadsto \\Sigma_{i-1} \\leadsto \\ldots \\ , \\qquad \\qquad (7)$$ indexed by $i \\geq 0$, which form the downstream tail of a manifold-germ $\\Sigma$, i.e. the sequence which we used above to define the space $\\hat E_{\\Sigma} = \\lim_{\\leftarrow} E_{\\Sigma_i}$. \nLet us perform the two manipulations that we performed on (6) alternately on the sequence (7), thereby obtaining a sequence whose even terms are $E_{\\Sigma_{2i}}$ and whose odd terms are $(E_{\\Sigma^*_{2i+1}})^*$.  The inverse-limit of the whole sequence is the same as that of any cofinal subsequence. Considering the cofinal subsequence of even terms shows that the inverse-limit is $\\hat E_{\\Sigma}$. But the inverse-limit of the cofinal sequence of odd terms is\n$$\\lim_{\\leftarrow} \\ (E_{\\Sigma_{21+1}^*})^* \\ = \\ (\\lim_{\\rightarrow} E_{\\Sigma^*_{21+1}})^*.$$ This  shows that $\\hat E_{\\Sigma} \\cong (\\check E_{\\Sigma^*})^*$. But, because $\\hat E_{\\Sigma}$ is automatically a nuclear Fr\\'echet space, we can dualize again and conclude that $(\\hat E_{\\Sigma})^* \\cong \\check E_{\\Sigma^*}$ also.\n\n\\bigskip \n\n\n\\section{Some analogies from representation theory}\n\n\\nopagebreak\n\nThe relation between representations of the category $\\mathcal C_d^{\\mathbb C}$ and of the Lorentzian category $\\mathcal C_d^{\\rm Lor}$ which lies ``on its boundary\"  follows a pattern familiar in the representation theory of many Lie groups which have a special class of unitary representations characterized as  the boundary values of holomorphic representations of a complex semigroup by contraction operators. The essential features can all be seen in the simplest example.\n\n\\bigskip\n\nThe group $G = {\\rm PSL}_2(\\mathbb R)$ is the group of M\\\"obius transformations of the Riemann sphere $\\Sigma = \\mathbb C \\cup \\infty$ which map the open upper half-plane $\\mathbb U$ to itself. It lies on the boundary of the complex sub-semigroup of $G_{\\mathbb C} = {\\rm PSL}_2(\\mathbb C)$ consisting of M\\\"obius transformations which map the closure of $\\mathbb U$ into its own interior. It is natural, however, to consider a slightly larger semigroup $G^<_{\\mathbb C}$ by including the degenerate M\\\"obius transformations which collapse $\\mathbb U$ to a single point in $\\mathbb U$ --- these correspond to complex $2 \\times 2$ matrices of rank one. The resulting semigroup is then a contractible open subset of the 3-dimensional complex projective space formed from the $2 \\times 2$ matrices.  The topological boundary of $G^<_{\\mathbb C}$ consists of the M\\\"obius transformations which take $\\mathbb U$ to a disc or point in the upper  half-plane which touches the real axis, and the Shilov boundary consists of the group $G$ of real M\\\"obius transformations ---  an open solid torus --- compactified by its 2-torus boundary, which is the hyperboloid  det$(A) = 0$ in $\\mathbb P^3_{\\mathbb R}$ consisting of the degenerate real M\\\"obius transformations. (Thus the complete Shilov boundary is the part of $\\mathbb P^3_{\\mathbb R}$ where det$(A) \\geq 0$.)\n\n\n\n\\bigskip\n\nThe irreducible unitary representations of the group $G = {\\rm PSL}_2(\\mathbb R)$ are essentially\\footnote{We shall ignore the  ``supplementary\" series, which is of measure zero in the space of representations.} of two kinds, the {\\it principal series} and the {\\it discrete series}. The best-known principal series representation is the action of $G$ on the Hilbert space of $1\/2$-densities on the circle $\\mathbb P^1_{\\mathbb R}$ which is the boundary of $\\mathbb U$ --- the general member of the series is the action on densities of complex degree $s$ with Re$(s) = 1\/2$. The best-known discrete series representation is the action of $G$ on the square-summable holomorphic 1-forms on $\\mathbb U$,  with the natural norm \n$$\\parallel \\alpha \\parallel ^2 = {\\rm i}\\int_{\\mathbb U} \\alpha \\wedge \\bar \\alpha$$\n --- more generally, for each positive integer $p$ we have the action on holomorphic $p$-forms $\\alpha = f(z)(dz)^{\\otimes p}$, when one must divide $\\alpha \\wedge \\bar \\alpha$ by the $(p-1)^{\\rm st}$ power of the $G$-invariant area form on the Poincar\\'e plane $\\mathbb U$ to define the norm. \n \n The discrete series representations obviously extend to bounded holomorphic representations of the semigroup $G^<_{\\mathbb C}$ by contraction operators. They are singled out by this `positive energy' property: the principal series representations cannot extend to $G^<_{\\mathbb C}$, because when $|a| < 1$ the element $w \\mapsto aw$ (here $w = (z-{\\rm i})\/(z+{\\rm i})$ is the coordinate in the unit-disc model $|w|<1$ of $\\mathbb U$) of the semigroup $G^<_{\\mathbb C}$ would be represented by an operator whose eigenvalues are $a^n$ for \\emph{all} $n \\in \\mathbb Z$.  But let us notice that, though  the discrete series representations are unitary on the boundary group $G = {\\rm PSL}_2(\\mathbb R)$, the degenerate elements of $G^<_{\\mathbb C}$, which collapse $\\mathbb U$ to a point $p \\in \\mathbb U$, are represented by bounded operators of rank 1. So these unitary representations of PSL$_2(\\mathbb R)$ do not extend unitarily to the whole Shilov boundary: the degenerate elements correspond to unbounded rank 1 operators $\\xi \\mapsto \\langle \\zeta , \\xi \\rangle \\eta$, where $\\eta$ and $\\zeta$ are ``non-normalizable elements\" of the Hilbert space --- i.e. they belong to an appropriate completion of it.\n  \n\\bigskip \n\nThe group $G$ is a subgroup of the group Diff$^+(S^1)$ of orientation-preserving diffeomorphisms of the circle. This infinite-dimensional Lie group does not possess a complexification, though its Lie algebra, the space of smooth vector fields on the circle, can of course be complexified. The beginning of the present work was the observation, made in the 1980s quite independently by the two authors and also by Yu. Neretin ([N], [Se1]), that there is an infinite-dimensional complex semigroup $\\mathcal A$ which has exactly the same relation to Diff$^+(S^1)$ as $G_{\\mathbb C}^<$ has to $G = {\\rm PSL}_2(\\mathbb R)$.  Its elements are complex annuli with parametrized boundary circles: one can think of them as `` exponentiations\" of outward-pointing complex vector fields defined on a circle in the the complex plane. The annuli form a complex semigroup when concatenated as cobordisms, and the lowest-weight or ``positive-energy\" representations of Diff$^+(S^1)$ --- and of loop groups --- which arise in 2-dimensional conformal field theory are precisely those which are boundary values of holomorphic representations of the semigroup $\\mathcal A$ by trace-class operators.\n\n\\bigskip\n\nThe discussion of PSL$_2(\\mathbb R)$ generalizes to the symplectic group $G = {\\rm Sp}(V) \\cong {\\rm Sp}_{2n}(\\mathbb R)$ of a real symplectic vector space $V$ of dimension $2n$. The role of the upper half-plane $\\mathbb U$ is played by the Siegel `generalized  upper half-plane' --- the domain $\\mathbb U(V)$ of positive  Lagrangian subspaces of the complexification $V_{\\mathbb C}$  described in Section 1. The group $G$ lies on the boundary of a semigroup $G^<_{\\mathbb C}$  which is the Siegel domain $\\mathbb U(\\tilde V \\oplus V)$, where $\\tilde V$ denotes $V$ with sign of its symplectic form reversed. A generic element of this domain is the graph of a complex symplectic transformation of $V_{\\mathbb C}$ which maps the closure of $\\mathbb U(V)$ into its own interior, but, just as was the case with PSL$_2(\\mathbb C)$, there are degenerate elements which map $\\mathbb U(V)$ non-injectively into itself.\nThe complex semigroup $G_{\\mathbb C}^<$ has been carefully studied by Roger Howe [H], who called it the {\\it oscillator semigroup}.  \n\nThe Shilov boundary of $G^<_{\\mathbb C}$ is the Grassmannian of real Lagrangian subspaces of $\\tilde V \\oplus V$: generically, these are the graphs of elements of the real  group $G = {\\rm Sp}(V)$, but this group is compactified by the addition of Lagrangian subspaces which intersect the axes of $\\tilde V \\oplus V$ nontrivially, and thus correspond to Lagrangian correspondences from $V$ to $V$ which are not actual maps $V \\to V$. Once again, whereas Sp$^<(V_{\\mathbb C})$ is a genuine semigroup, the composition-law of the real group Sp$(V)$ does not extend to the compactification.\n\n\\bigskip\n\n\n\\bigskip\n\n\n\nThe group $G = {\\rm Sp}(V)$ has a discrete series of unitary representations generalizing those of PSL$_2(\\mathbb R)$. The most important   is the \\emph{metaplectic representation} --- actually a representation of a double covering $\\tilde G$ of Sp$(V)$ --- which is the action on the {\\it quantization} $\\mathcal H_V$ of the symplectic space $V$.  The Hilbert space $\\mathcal H_V$ is characterized by the property that it contains a copy of the ray $(\\bigwedge^n (W))^{\\otimes (1\/2)}$ for each point $W$ of the domain $\\mathbb U(V)$ ---  the square-root of the natural hermitian holomorphic line bundle $\\{\\bigwedge^n(W)\\}$ on $\\mathbb U(V)$ is canonical up to multiplication by $\\pm 1$, and is holomorphically embedded in $\\mathcal H_V$. It is acted on by $\\tilde G$ rather than $G$.\n\nThe action of $\\tilde G$ on $\\mathcal H_V$ is the boundary-value of a holomorphic projective representation of the oscillator semigroup $G_{\\mathbb C}^<$. For $G_{\\mathbb C}^<$ is just the domain $\\mathbb U(\\tilde V \\oplus V)$, each point of which defines a ray in\n$$\\mathcal H_{\\tilde V \\oplus V} \\ \\cong \\  \\mathcal H_V^* \\otimes \\mathcal H_V \\ \\cong \\  {\\rm End}_{HS}(\\mathcal H_V),$$\nwhere End$_{HS}$ denotes the Hilbert-Schmidt endomorphisms.  (A more careful discussion shows that $G^<_{\\mathbb C}$ is represented by operators of trace class.)\n\n\\bigskip\n\nWhen $n = 1$ the group Sp$(V)$ is SL$_2(\\mathbb R)$, a double covering of the group PSL$_2(\\mathbb R)$ of M\\\"obius transformations we considered before. To relate the  cases of PSL$_2(\\mathbb R)$ and Sp$(V)$, recall that PSL$_2(\\mathbb C)$ is an open subspace of the complex projective space $\\mathbb P^3_{\\mathbb C}$ formed from the vector space of $2 \\times 2$ matrices: in fact it is the complement of the quadric $Q^2_{\\mathbb C} \\cong \\mathbb P^1_{\\mathbb C} \\times \\mathbb P^1_{\\mathbb C}$  defined by the vanishing of the determinant, i.e. by the matrices of rank 1.   The double covering group SL$_2(\\mathbb C)$ sits inside the Grassmannian of complex Lagrangian subspaces of $\\mathbb C^4$, which is a quadric 3-fold $Q^3_{\\mathbb C}$ in $\\mathbb P^4_{\\mathbb C}$: it is a non-singular hyperplane section (corresponding to the Lagrangian condition) of the Klein quadric formed by all the lines in $\\mathbb P^3(\\mathbb C)$. The quadric $Q^3_{\\mathbb C}$ is the branched double-covering of the projective space $\\mathbb P^3_{\\mathbb C}$ of $2 \\times 2$ matrices, branched along the quadric $Q^2_{\\mathbb C}$ of rank 1 matrices. The contractible semigroup SL$^<_2(\\mathbb C)$ is the open subset of the Lagrangian Grassmannian of $\\mathbb C^4$ consisting of the positive Lagrangian subspaces, and it is a double covering of PSL$^<_2(\\mathbb C)$.\n\n\\bigskip\n \n\n\\section{Unitarity and global hyperbolicity}\n\nIn the previous section we saw how a holomorphic representation of a complex semigroup by contraction operators on a Hilbert space can give rise --- on passing to the boundary --- to a unitary representation of a group which is a dense open subset of the Shilov boundary of the semigroup. The remaining points of the Shilov boundary are not represented by unitary operators; the representation extends to them only in  some ``weak\" sense. We now come to the analogue of this phenomenon in quantum field theory, where the Lorentzian cobordism category $\\mathcal C_d^{\\rm Lor}$ lies on the boundary of $\\mathcal C_d^{\\mathbb C}$, and the role of the open dense subgroup of the Shilov boundary is played by the subcategory of \\emph{globally hyperbolic} cobordisms which we shall define below. We should  mention, however, that although the category of globally hyperbolic cobordisms is very natural, the category $\\mathcal C_d^{{\\rm Lor}}$ may be smaller than the optimal category we could put on the boundary of $\\mathcal C_d^{\\mathbb C}$. For example, the Lorentzian cobordisms could possibly be allowed to contain `black holes' surrounded by horizons, rather analogous  to the `cobordisms-with-boundaries' used to describe two-dimensional theories with both open and closed strings. We shall not pursue such speculations here.\n\n\\bigskip\n\nWhen we have a theory defined on $\\mathcal C_d^{\\mathbb C}$ let us first consider how to extend the assignment $\\Sigma \\mapsto E_{\\Sigma}$ to  a \\emph{Lorentzian} germ\n $\\Sigma \\subset U$,  with $\\Sigma$ co-oriented in $U$.  We can identify $U$ with $\\Sigma \\times (-\\varepsilon, \\varepsilon)$ by exponentiating the geodesic curves emanating perpendicularly from $\\Sigma$.  The metric then takes the form $h_t - {\\rm d}t^2$, where $t \\mapsto h_t$ is a smooth map from $(-\\varepsilon, \\varepsilon)$ to the manifold of Riemannian metrics on $\\Sigma$. If the germ is time-symmetric then we can define $E_{\\Sigma}$ by replacing the Lorentzian metric by the `Wick rotated' Riemannian metric  $h_{{\\rm i}t} + {\\rm d}t^2$, which makes sense because if $h_t = h_{-t}$ then $h_t$ is a function of $t^2$, so that $h_{{\\rm i}t}$ is defined and real. But this does not help for a general hypersurface, and in any case seems rather arbitrary: we shall return to this point in Remark 5.3 below.\n\n\\bigskip\n\nIt is  less easy to assign an operator $Z_M:E_{\\Sigma_0} \\to E_{\\Sigma_1}$ to a Lorentzian cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$. Even if $M$ is a cylinder topologically, it can be complicated in its ``causal\" structure. Consider, for example, a 2-dimensional cylindrical space-time. We saw in Section 2 that, up to a conformal multiplier, a complex metric on a surface is a pair of complex structures with opposite orientations. At the Shilov boundary the complex structures degenerate to the foliations by the left- and right-moving light-lines of a Lorentzian surface. If each light-line which sets out from the incoming boundary circle of the cylinder eventually reaches the outgoing boundary circle then each family of light-lines gives us a diffeomorphism from the incoming to the outgoing boundary. In fact (cf. [Se2] p.8 and p.16) the isomorphism classes of Lorentzian cylinders of this kind are determined up to conformal equivalence by the pair of diffeomorphisms together with a positive integer which counts the number of times that the left- and right-moving lines emanating from a given point of the incoming circle cross before hitting the outgoing circle. This agrees with the well-known fact that the Hilbert space associated to a circle in 2-dimensional conformal field theory comes with a projective unitary representation of the group Diff$^+(S^1) \\times {\\rm Diff}^+(S^1)$.\n\nBut the light-lines from the incoming circle can behave in a more complicated way. For example, one set of light-lines may spiral closer and closer to a closed limit-cycle of the foliation, a light-line which is a circle parallel to the incoming boundary circle of the annulus. That set of lines will then never reach the outgoing circle. One might think of this phenomenon as  akin to a black hole in the space-time, though, unlike a black hole, the Lorentzian metric here has no singularity. The ``blocked'' foliation is conformally the same as the ``degenerate annulus'' obtained by collapsing the closed light-line to a point, i.e.   a pair of discs with their centre-points identified. This is usually regarded as an ``annulus of infinite length\", and it acts on an irreducible positive-energy representation of Diff$^+(S^1)$ by a projection operator of rank one, like the action of a degenerate complex M\\\"obius transformation  in a discrete-series representation of PSL$_2(\\mathbb R)$.\n\n\n\n\\bigskip\n\n\n\nIn works on general relativity a Lorentzian cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ between Riemannian manifolds is called {\\it globally hyperbolic} if every maximally-extended time-like geodesic in $M$ travels from $\\Sigma_0$ to $\\Sigma_1$. Such an $M$ must be diffeomorphic to $\\Sigma_0 \\times [0,1]$.  It is only for globally hyperbolic manifolds that, for example, the  Cauchy problem for the wave-equation on $M$ is soluble.\n\nOf course here we are only considering {\\it compact} cobordisms, which are not the usual focus in relativity theory. In the compact situation we can take the definition of global hyperbolicity to be the existence of a smooth time-function $t: M \\to [0,1]$ whose gradient is everywhere in the positive light-cone,  and which is therefore a fibration with Riemannian fibres. From $t$ we obtain a diffeomorphism $M \\to \\Sigma_0 \\times [0,1]$  by following the orthogonal trajectories to the time-slices.\n\nThe existence of a time-function on a compact Lorentzian cobordism is clearly an open condition, and so the globally hyperbolic cobordisms form an open subcategory $\\mathcal C_d^{\\rm gh}$ of $\\mathcal C_d^{\\rm Lor}$ which should play the role of the real Lie group to which the holomorphic contraction representations of Section 4 can be extended (though the result (5.2) we prove below is unfortunately weaker).\n\n\\bigskip\n\nFor a globally hyperbolic cobordism equipped with a time-function, the metric, in terms of the  diffeomorphism $M \\to \\Sigma_0 \\times [0,1]$, takes  the form $h_t  + c^2{\\rm d}t^2$ for some function $c: \\Sigma_0 \\times [0,1]\\to {\\rm i}\\mathbb R$. A small deformation $\\delta c$ of $c$ into the right half-plane  changes the Lorentzian metric into an allowable complex metric, and we could hope to define $Z_M$ in the Lorentzian case as the limit of the operators associated to such deformations. That, however, encounters the problem that the deformed metric depends not only on the choice of the deformation $\\delta c$, but, more importantly, on the choice of the time-function, which should be irrelevant to the operator $U_M$. Happily, there is a better point of view, which also shows why the boundary-value of a semigroup of contraction operators is a \\emph{unitary} representation. There is, after all, no obvious reason why the concatenation of a Lorentzian cobordism with its reverse should be represented by the identity operator --- quite unlike what happens with Riemannian cobordisms.  (A possible analogy is the process of making a based loop-space  into a topological group by collapsing paths which retrace their steps.) \n\n\\bigskip\n\nThe passage from $\\mathcal C_d^{\\mathbb C}$ to $\\mathcal C_d^{\\rm Lor}$ is already interesting when $d=1$, i.e. for quantum mechanics rather than quantum field theory --- the case when the Euclidean path-integral can be treated by traditional measure-theory. It is worthwhile to spell out the argument in this case, before passing to higher dimensions. \n\nWe began this work with the relation of positive energy to 1-parameter contraction semigroups. Our first task now is to understand why a holomorphic representation of the category $\\mathcal C_1^{\\mathbb C}$ is just such a 1-parameter semigroup, where the parameter runs through the open half-plane $\\mathbb C_+ = \\{z \\in \\mathbb C: {\\rm Re}(z)>0\\}$. Whereas a Riemannian structure on a closed interval is completely determined by its length, the allowable complex metrics on the interval have an infinite-dimensional moduli-space.\n\nAny complex metric on  $I = [0,1]$ can be pulled back from the holomorphic quadratic differential ${\\rm d}z^2$ on $\\mathbb C$ by means of a smooth embedding $f: I \\to \\mathbb C$ such that $f(0) = 0$ and Re $f'(t) > 0$ for all $t \\in I$. In fact the space Emb$(I;\\mathbb C)$ of such embeddings is isomorphic to Met$_{\\mathbb C}(I)$ {\\it as a complex manifold}. If $f'(t) = 1$ when $t$ is sufficiently close to the ends of the  interval $I$ then the pulled-back metric defines a morphism $I_f: P \\to P$ in the category $\\mathcal C_1^{\\mathbb C}$, where $P$ denotes the object defined by the germ of the standard metric on the line $\\mathbb R$ at the origin. \n\nThe crucial observation is that the operator $Z_f: E_P \\to E_P$ defined by $I_f$ depends only on the point $f(1) \\in \\mathbb C_+$. It is as if $Z_f$ were the `contour integral' of a holomorphic differential on $\\mathbb C$ along the path $f$. The argument is as follows. First, $Z_f$ does not change if $f$ is replaced by $\\tilde f = f \\circ \\phi$ where $\\phi$ is any diffeomorphism $I \\to I$ which is the identity near the ends of the interval. This means that $Z_f$ does not change if $f$ moves along a curve in Emb$(I;\\mathbb C)$ whose tangent vector at each point is the action of an element of the Lie algebra Vect$(\\mathring I)$ of compactly supported vector fields on the interior of $I$. But then --- because $Z_f$ depends holomorphically on $f$ ---  it does not change if each tangent vector is the action of an element of the \\emph{complexified} Lie algebra Vect$_{\\mathbb C}(\\mathring I)$. Finally, if $f,\\tilde f \\in {\\rm Emb}(I;\\mathbb C)$ define two morphisms $P \\to P$ and have $f(1) = \\tilde f(1)$, the tangent vectors to the obvious linear path from $f$ to $\\tilde f$  are given by the action of elements of Vect$_{\\mathbb C}(\\mathring I)$. \n\nWe can therefore write $Z_f = u(z)$, where $z = f(1)$. Obviously we have $u(z_1)u(z_2) = u(z_1 + z_2)$ for any $z_1,z_2 \\in \\mathbb C_+$. Furthermore, because the object $P$ of $\\mathcal C_1^{\\rm gh}$ is time-symmetric, the vector space $\\check E_P$ is a pre-Hilbert space, and the unitarity condition tells us that $u(\\bar z)$ is the hermitian transpose of $u(z)$.\n\nThe desired unitary semigroup $\\{u({\\rm i} T)\\}_{T \\in \\mathbb R}$, which will act on the triple $\\check E_P \\to E_P^{Hilb} \\to \\hat E_P$, can now be defined as follows. As explained in Section 3, any vector $\\xi \\in \\check E_P$ can be written $\\xi = u(\\varepsilon )\\eta$ for some $\\varepsilon > 0$ and some $\\eta \\in E_P$. We define $u({\\rm i} T)\\xi = u(\\varepsilon + {\\rm i}T)\\eta$, which is plainly independent of $\\varepsilon$. Finally, $u({\\rm i} T)$ is unitary because\n\\begin{eqnarray*}\nu(-{\\rm i}T)u({\\rm i}T) \\xi \\ &=&  \\ u(-{\\rm i} T)u(\\varepsilon + {\\rm i} T)\\eta\\\\\n&=& \\ u(-{\\rm i} T)u(\\varepsilon\/2)u(\\varepsilon\/2 + {\\rm i} T)\\eta\\\\\n&=& \\ u(\\varepsilon\/2 - {\\rm i} T)u(\\varepsilon\/2 + {\\rm i} T)\\eta\\\\\n&=& \\ u(\\varepsilon)\\eta \\ = \\ \\xi.\n\\end{eqnarray*}\n\n\\bigskip\n\nTo pass from $d=1$ to higher-dimensional cobordisms we observe that the essential step in our argument was the first case of the following \n\n\\bigskip\n\n\\noindent{\\bf Principle 5.1} \\ \\ \\ {\\it If a  $d$-dimensional cobordism $M$  is a real submanifold of a complex $d$-manifold $M_{\\mathbb C}$, and $M$ has an allowable complex metric induced from a  holomorphic symmetric form $g$ on the tangent bundle $TM_{\\mathbb C}$,   then the linear map $Z_M$ does not change when $M$ is moved around smoothly inside $M_{\\mathbb C}$ (leaving its ends fixed), providing the restriction of $g$ to $M$ remains an allowable complex metric.}\n\n\\bigskip\n\nAs in the $d=1$ case, this principle holds because any infinitesimal movement of $M$ inside $M_{\\mathbb C}$ is given by a complex vector field on $M$, while $Z_M$ depends holomorphically on $M$ and, being invariant under the action of \\mbox{Diff$(M$ rel $\\partial M)$}, does not change when $M$ moves in a direction given by the action of a complexified tangent vector to this group. \n\n\\bigskip\n\nUnfortunately, to  use  the principle we need the cobordism $M$ to be embedded in a complexification $M_{\\mathbb C}$, and the only natural way to ensure this  is to pass from the smooth Lorentzian category $\\mathcal C_d^{\\rm Lor}$ to the corresponding \\emph{real-analytic} cobordism category $\\mathcal C_d^{\\rm Lor,\\omega}$, where both the manifolds and their metrics are assumed real-analytic. Inside this category there is the subcategory $\\mathcal C_d^{\\rm gh,\\omega}$ of globally hyperbolic cobordisms: we shall also assume that the time-function $\\tau:M \\to {\\rm i}[0,1]$ is real-analytic, though that could be avoided, because any smooth function can be approximated real-analytically. \n\n\\bigskip\n\nThere are two ways of thinking about restricting to real-analytic cobordisms. One might think that the smooth cobordism category is the natural object, and try to eliminate the analyticity hypothesis. But one could also think that that the natural allowable space-times really do come surrounded by a thin holomorphic thickening, within which the choice of a smooth totally-real representative is essentially arbitrary. In any case, we can prove the following theorem.\n\n\\bigskip\n\n\\noindent{\\bf Theorem 5.2} \\ \\  {\\it A unitary quantum field theory as defined in Section 3 on the category $\\mathcal C_d^{\\mathbb C}$ induces a functor from $\\mathcal C_d^{\\rm gh, \\omega}$ to topological vector spaces. The functor takes time-symmetric objects to Hilbert spaces, and takes cobordisms between them to unitary operators.} \n\n\\bigskip\n\nTo be quite precise: the theorem asserts that if $\\Sigma$ is a time-symmetric $(d-1)$-manifold germ then there is a Hilbert space $E_{\\Sigma}^{Hilb}$ with\n$$\\check E_{\\Sigma} \\ \\ \\subset \\ \\ E_{\\Sigma}^{Hilb} \\ \\ \\subset \\hat E_{\\Sigma},$$\nand a real-analytic globally hyperbolic cobordism $\\Sigma_0 \\leadsto \\Sigma_1$ between time-symmetric hypersurfaces induces a unitary isomorphism $E_{\\Sigma_0}^{Hilb} \\to E_{\\Sigma_1}^{Hilb}$ which also maps $\\check E_{\\Sigma_0}$ to $\\check E_{\\Sigma_1}$ and $\\hat E_{\\Sigma_0} $ to $\\hat E_{\\Sigma_1}$.\n\n\\bigskip\n\n\n\n\n\n\n\\bigskip\n\n \n  \n\\noindent {\\it Proof of 5.2} \\ \\   Given  a real-analytic globally hyperbolic  cobordism  \\mbox{$M:\\Sigma_0 \\leadsto \\Sigma_1$} we choose a time function $t:M \\to [0,1]$ whose level surfaces foliate $M$ by Riemannian manifolds, and, following the orthogonal trajectories to the foliation, we  identify $M$ with $\\Sigma_0 \\times [0,1]$ as before. \n  \n  Using the real-analyticity assumptions, we can find a complexification $M_{\\mathbb C}$ of $M$ to which both $t$ and $g$ can be extended holomorphically, and we can assume that $\\tau = {\\rm i}t:M_{\\mathbb C} \\to U \\subset \\mathbb C$ is a holomorphic fibre bundle over a neighbourhood $U$ of the interval ${\\rm i} [0,1]$. Furthermore, the isomorphism $\\Sigma_0 \\times [0,1] \\to M$ extends to a holomorphic trivialization of the bundle $M_{\\mathbb C} \\to U$.  For any smooth curve $f:[0,1] \\to U$ such that $f(0) = 0$ and ${\\rm Re} \\    f'(s) > 0$ for $s \\in [0,1]$ this gives us a totally real submanifold $M_f$ of $M_{\\mathbb C}$ sitting over the curve. We can use the morphism associated to the cobordism $M_f$ in exactly the way we used $Z_f$ in  discussing the 1-dimensional case, to obtain a unitary operator $Z_M$ associated to the Lorentzian cobordism.\n  \n  It is important  that $Z_M$ does not depend on the choice of the time-function $t$ defining the foliation. For two choices of $t$ are linearly homotopic, and changing from one to the other amounts to deforming the totally-real embedding $\\Sigma_0 \\times [0,1] \\to M_{\\mathbb C}$ by a real-analytic diffeomorphism of $\\Sigma_0 \\times [0,1]$.\n  \n  \\bigskip\n \n\\noindent{\\bf Remark 5.3} \\ \\ \\   We can apply the principle 5.1  to understand better how a theory defined on $\\mathcal C_d^{\\mathbb C}$ assigns a vector space $E_{\\Sigma}$ to a Lorentzian germ $\\Sigma \\subset U$. \n \n If the Lorentzian metric on $U$ is real-analytic then the complex theory gives us a holomorphic bundle $\\{\\hat E_f\\}$ on the space \n\n$\\mathcal J$   of germs of embeddings $f:(-\\varepsilon,\\varepsilon) \\to \\mathbb C$ such that $f(0)=0$ and Re $f'(t) >0$ for all $t$. In particular, for $\\lambda \\in \\mathbb C_+$ we have the radial paths $f_{\\lambda} \\in \\mathcal J$ for which $f_{\\lambda}(t) = \\lambda t$. But recall that $\\hat E_f$ is the inverse-limit of a sequence of spaces associated to the germs of $f$ at the points $f(t_k)$, for any sequence $\\{t_k \\downarrow 0\\}$. \n\nNow consider two neighbouring rays $f_{\\lambda},f_{\\lambda'}$ with $|\\lambda| = |\\lambda'|$, and choose a sequence $\\{t'_k \\downarrow 0 \\}$ which interleaves $\\{t_k \\}$, i.e. $t_k > t'_k >t_{k+1}$. We can choose a path $f \\in \\mathcal J$ which lies in the sector bounded by the rays $f_{\\lambda}$  and $f_{\\lambda '}$  and coincides with them alternately in the neighbourhoods of the points $\\lambda t_k$ and $\\lambda't'_k$. This $f$ gives us a family of cobordisms from the germ at $\\lambda' t'_k$ to the germ at $\\lambda t_k$, and from the germ at $\\lambda t_{k+1}$ to the germ at $\\lambda 't'_k$. Putting these together, we obtain  inverse canonical isomorphisms between $\\hat E_{f_{\\lambda}}$ and $\\hat E_{f_{\\lambda '}}$. The coherence of these isomorphisms when we consider three nearby rays also follows from the principle 5.1.\n\nBy this means we see that we could have chosen \\emph{any} smooth path $f$ to define $\\hat E_{\\Sigma}$. However the family $\\hat E_f$ has the property that $\\hat E_{\\bar f}$ is the complex-conjugate space to $\\hat E_f$, so that reversing the complex time-direction conjugates the identification of $\\hat E_{\\Sigma}$ with the Euclidean choice $\\hat E_{f_1}$. If the Lorentzian germ $\\Sigma \\subset U$ is time-symmetric --- but not otherwise --- the arguments we have already used will give us a hermitian inner product on $\\check E_{\\Sigma}$.\n \n  \n \n\\bigskip\n\n\\bigskip\n\n\\noindent{\\bf Field operators}\n\n\\nopagebreak\n\n\\bigskip\n\nFinally, we come to the Wick rotation of field operators, though our account will be sketchy. The first step is to understand how the vector space $\\mathcal O_x$ of observables at a point $x$ of a space-time $M$ behaves as the metric of $M$ passes from complex to Lorentzian. We shall continue to assume that $M$ and its Lorentzian metric are real-analytic.\n\nIn Section 3 we associated a space  $\\mathcal O_x$ to a germ at $x$ of a complex metric on a manifold containing $x$: it is the fibre of a bundle on the space Met$_{\\mathbb C}(\\hat x)$ of such germs. If we embed a Lorentzian $M$ in a complexification $M_{\\mathbb C}$ there will be a holomorphic exponential map from a neighbourhood of $0$ in the complexified tangent space $T_x^{\\mathbb C} = T_xM \\otimes \\mathbb C$ to $M_{\\mathbb C}$. Inside $T_x^{\\mathbb C}$ we can consider the $d$-dimensional real vector subspaces $V$ on which the metric induced  from the complex bilinear form of $T_x^{\\mathbb C}$ is allowable. We saw in  (2.6) that these $V$ form a contractible open subset $\\mathcal U$ of the real Grassmannian Gr$_d(T_x^{\\mathbb C})$.  Exponentiating $V$ will give us a germ of a $d$-manifold with a complex metric, and hence a map $\\mathcal U \\to {\\rm Met}_{\\mathbb C}(\\hat x)$. Pulling back the bundle of observables by this map gives us a bundle on $\\mathcal U$, which, using  the principle (5.1) as we did in (5.3), we see to be trivial. Identifying its fibres gives us our definition of $\\mathcal O_x$ for Lorentzian $M$.\n\n\\bigskip\n\nWe need no new ideas to see that for any Lorentzian cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ and any $x \\in \\mathring M$ an element $\\psi \\in \\mathcal O_x$ acts as an operator $E_{\\Sigma_0} \\to E_{\\Sigma_1}$. Furthermore, if $x$ lies on a time-slice $\\Sigma$ we get an operator $\\psi \\in {\\rm Hom}( \\check E_{\\Sigma}; \\hat E_{\\Sigma})$, i.e. an unbounded operator in $E_{\\Sigma}$, simply by considering the cobordisms corresponding to a sequence of successively thinner collars of $\\Sigma$. Indeed the same argument shows that if $x_1, \\ldots , x_k$ are distinct points on $\\Sigma$,  we have a map\n$$\\mathcal O_{x_1} \\otimes \\ldots \\otimes \\mathcal O_{x_k} \\ \\ \\to \\ \\  {\\rm Hom}( \\check E_{\\Sigma}; \\hat E_{\\Sigma})$$\nwhich does not depend on choosing an ordering of the points. \n\n\\bigskip\n\nIn the introduction we  mentioned the Wightman axiom  that field operators at space-like separated points must commute. We can now see how this follows from our framework, at least in a globally hyperbolic space-time. For the spaces $\\check E_{\\Sigma_t} \\subset E_{\\Sigma_t}^{Hilb} \\subset E_{\\Sigma_t}$ for all  times $t_0 \\leq t \\leq t_1$ can be identified with those at time $t_0$ by the unitary propagation $Z_{t,t'}$ from time $t$ to a later time $t'$ to get a single rigged Hilbert space $\\check E \\subset E^{Hilb} \\subset \\hat E$, and we can define an unbounded operator\n$$\\tilde \\psi \\ \\ = \\ \\ Z_{t_0,t}^{-1}\\circ \\psi \\circ Z_{t_0,t}:\\check E \\to \\hat E$$\nfor any $\\psi \\in \\mathcal O_x$ with $x \\in \\Sigma_t$. Furthermore, if we change the choice of time-function on the cobordism, so that $x$ lies on a different time-slice, then $\\tilde \\psi$ will not change.\n\nThe fact that two observables $\\psi,\\psi'$ situated at space-like separated points $x,x'$ give rise to operators $\\tilde \\psi, \\tilde \\psi'$ which are composable, and commute, is now clear. For if $x$ and $x'$ are space-like separated we can choose a single time-slice $\\Sigma_t$ which contains them both, and we see that the composed operator, in either order, is  $Z_{t_0,t}^{-1}\\circ ( \\psi \\otimes \\psi') \\circ Z_{t_0,t}$.\n\n\\bigskip\n\n\n\n\\noindent{\\bf The domain of holomorphicity of the vacuum expectation values}\n\n\\bigskip\n\n\nWe end with a conjecture about a question arising in the traditional treatment of field theories defined in the standard Minkowski space $\\mathbb M = \\mathbb R^{d-1,1}$. There, the Wightman axioms imply that the vacuum expectation values, initially  defined as distributions or other generalized functions on the $k$-fold products $\\mathbb M \\times \\ldots \\times \\mathbb M$, are  boundary values of holomorphic functions defined in an open domain $\\mathcal U_k$ in the complexified space $\\mathbb M_{\\mathbb C}\\times \\ldots \\times \\mathbb M_{\\mathbb C}$. The definition of $\\mathcal U_k$,  known as the `permuted extended tube',  was given in Section 2. Recall that $\\mathcal U_2$ consits of all pairs of points $x,y$ such that $||x-y||^2$ is not real and $\\leq 0$.\n\nIf $k > 2$, however,  $\\mathcal U_k$ is known not to be holomorphically convex, so it cannot be the largest complex manifold to which the expectation values can be analytically continued. It is an old problem to describe this largest manifold $\\mathcal V_k$, or even the  holomorphic envelope of $ \\mathcal U_k$.\n\nThe ideas of this paper suggest  a candidate for $\\mathcal V_k$. It sits over the open subset $\\check \\mathcal V_k$ of all $k$-tuples ${\\bf x} = \\{x_1, \\ldots ,x_k\\}$ of distinct points in $\\mathbb M_{\\mathbb C}$ which lie on some totally-real submanifold $M$ with two properties:\n\n(i) \\ the metric on $M$ induced from $\\mathbb M_{\\mathbb C}$ is allowable, and\n\n(ii) \\ $M$ projects surjectively onto the usual real Euclidean subspace $\\mathbb E = \\mathbb R^d$ of $\\mathbb M_{\\mathbb C} = \\mathbb E \\oplus \\rm i\\mathbb E$.\n\nNotice that, by the remark before Prop.\\ 2.6, the projection $M \\to \\mathbb E$ is a local diffeomorphism if the metric of $M$ is allowable, so (ii) implies that $M$ is the graph of a smooth map $\\mathbb E \\to \\rm i\\mathbb E$.\n\nLet $\\mathcal F_k$ denote the space of all pairs $(M,{\\bf x})$ satisfying the above conditions. It is an infinite-dimensional complex manifold projecting to the open subset $\\check \\mathcal V_k$ of Conf$_k(\\mathbb M_{\\mathbb C})$, and it is easy to see that the map $\\pi: \\mathcal F_k \\to \\check \\mathcal V_k$ is open.  We define $\\mathcal V_k$  as the largest Hausdorff quotient manifold of $\\mathcal F_k$ through which $\\pi$ factorizes and which maps to $\\check V_k$ by a local diffeomorphism. Thus two points $(M,{\\bf x}),\\, ( M',\\bf x)$ of the fibre $\\mathcal F_{k,\\bf x}$  of $\\pi$ at $\\bf x$ have the same image in $\\mathcal V_k$ if they are in the same connected component of the fibre, but --- as that equivalence relation need not give a Hausdorff quotient --- also if there are paths $\\gamma,\\gamma'$ from $(M,\\bf x)$ and $(M',\\bf x)$ to a third point\n$(M'',\\bf x'')$ of $\\mathcal F_k$ which cover the same path from $\\bf x$ to $\\bf x''$ in $\\check V_k$.\n\n\\bigskip\n\n\n\n\n\n\n \n\nTo motivate the  definition of $\\mathcal V_k$ we must enlarge our framework to allow Lorentzian space-times whose time-slices are not compact. The simplest way to do this is to introduce the cobordism category in which a morphism is the part of a  $d$-dimensional allowable submanifold $M$ of $\\mathbb M_{\\mathbb C}$ cut off between two time-symmetric hypersurfaces.\n\nA field theory defined and holomorphic on this category, if it has a Lorentz-invariant vacuum state in a natural sense, will have vacuum expectation values which are holomorphic functions $\\mathcal E_k$ on the spaces $\\mathcal F_k$ of  pairs $(M,{\\bf x})$.  Strictly speaking, $\\mathcal E_k$ is a holomorphic section of a bundle on $\\mathcal F_k$, but we can use the local diffeomorphism $M \\to \\mathbb E$ to trivialize the bundle, giving us a holomorphic function\n$$\\mathcal E_k: \\mathcal F_k \\  \\to  \\ {\\rm Hom}(\\mathcal O^{\\otimes k};\\mathbb C),$$\nwhere $\\mathcal O$ is the space of observables at a point of $\\mathbb E$.\n\nOur much-used Principle 5.1 tells us that the value of the function $\\mathcal E_k$ does not change if, while holding the marked points {\\bf x} fixed in $\\mathbb M_{\\mathbb C}$, we move $M$ smoothly in the allowable class. So in fact we have a holomorphic function on  $\\mathcal F_k$ which is constant on the connected components  of the fibres of $\\mathcal F_{k,\\bf x}$ of the map $\\pi$, i.e. to the isotopy classes of allowable manifolds conatining $\\bf x$.\n\n\\bigskip  \n\nUnfortunately we have no proof that $\\mathcal V_k$ is a domain of holomorphy, but  at least we can assert\n\n\\bigskip\n\n\\noindent {\\bf Proposition 5.4} \\ {\\it $\\mathcal V_k$ contains the Wightman domain $\\mathcal U_k$.}\n\n\\bigskip\n\nFurthermore, we saw in Proposition 2.3 that the holomorphic envelope of $\\mathcal U_k$ contains $\\mathcal V_k^{flat}$, the part of $\\mathcal V_k$ represented by flat affine submanifolds $M \\subset \\mathbb M_{\\mathbb C}$.\n\n\\bigskip\n\n\\noindent{\\it Proof of 5.4} \\ First, $\\mathcal V_k$ is invariant under the complex orthogonal group of $\\mathbb M_{\\mathbb C}$, and under reorderings of the points ${\\bf x} = \\{x_1, \\ldots ,x_k\\}$. So it is enough to consider ${\\bf x}$ such that the imaginary part of $x_{i+1} - x_i$ belongs to the forward light-cone $C \\subset \\mathbb M$ for each $i$.\n\nSmoothing the obvious polygonal path joing the points, we can thus assume that the $x_i$ lie on a curve $x: \\mathbb R \\to \\mathbb M_{\\mathbb C}$ whose derivative  Im$(x'(t))$ belongs to $C$ for all $t$. But then we can choose, smoothly in  $t$, a set of $d-1$ orthonormal vectors $e_j(t)$ in $\\mathbb M_{\\mathbb C}$ which are all orthogonal to $x'(t)$. Let $M_t$ be the \\emph{real} vector subspace of $\\mathbb M_{\\mathbb C}$ spanned by the vectors $e_j(t)$. The points ${\\bf x}$ lie on the $d$-dimensional real ruled manifold $M$ swept out by the  affine $d-1$-planes  $x(t)+ M_t$, and the metric of $M$ is clearly allowable. \\ \\ $\\spadesuit$ \n\n\n\n\n\\bigskip\n\n\n\\noindent{\\bf References}\n\n\\bigskip\n\n\\begin{enumerate}\n\\item[[BF]] Brunetti, R.,   and K. 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+{"text":"\\subsection*{Results}\n\nThe combined UED\/UEDS experiments are based on a pump-probe scheme, and were performed with a radio-frequency compressed ultrafast electron scattering instrument described in detail elsewhere~\\cite{Chatelain2012, Otto2017}. Briefly, \\SI{35}{\\femto\\second} pump pulses at \\SI{800}{\\nano\\meter} (\\SI{1.55}{\\electronvolt}) drive vertical electronic transitions at $t=t_0$, photodoping electrons and holes into the band valleys near $\\Gamma$, $\\frac{2}{3}Y$ and $\\frac{3}{4}Z$~\\cite{Li2015,Zhao2016,Melendez2018,Wei2019} (Fig. \\ref{FIG:structure}c). The effects of this photoexcitation on both the lattice structure and phonon system are measured with UED and UEDS simultaneously at a time $t = t_0 + \\tau$. Time-series of the changes in electron scattering intensity at all scattering vectors, $\\vec{q}$, were assembled by scanning the pump-probe time delay, $\\tau$. The experiments were repeated over a range of pump-fluences at \\SI{300}{\\kelvin}.\n\nThe total measured electron scattering intensity $I(\\vec{q}, \\tau)$ can be decomposed into $I(\\vec{q}, \\tau) = I_0(\\vec{q}, \\tau) + I_1(\\vec{q}, \\tau) + ...$, where $I_0$ corresponds to the Bragg peaks of conventional diffraction and $I_1$ is known as \\emph{single-phonon diffuse scattering}. The intensity of Bragg peaks at scattering vector $\\vec{G}$, $I_0(\\vec{q}=\\vec{G},\\tau)$, reports on the lattice constants, unit cell structure, coherent modulation~\\cite{Chatelain2014, Sie2019} and\/or transient Debye-Waller (DW) suppression~\\cite{Siwick2003, Ernstorfer2009, Waldecker2016} of peak intensity. Diffraction signals can be $10^5$--$10^8$ times more intense than phonon diffuse scattering signals~\\cite{Stern2018,RenedeCotret2019}. The expression for single-phonon diffuse scattering (PDS) is given by:\n\\begin{equation}\n\tI_1(\\vec{q}, \\tau) \\propto \\sum_\\lambda \\left| a_{\\lambda \\vec{k}}\\right|^2 \\left| F_{1\\lambda}\\left(\\vec{q}, \\left\\{ e_{\\lambda\\vec{k}}\\right\\} \\right) \\right|^2 \\label{EQ:diffuse}\n\\end{equation}\nwhere the label $\\lambda$ indicates the specific phonon branch, $\\vec{q}$ is the electron scattering vector, $\\vec{k}$ is the reduced phonon wavevector (i.e. $\\vec{k}$ = $\\vec{q}$ - $\\vec{G}$, where $\\vec{G}$ is the closest Bragg peak), $a_{\\lambda\\vec{k}}$ is the vibrational amplitude of mode $\\lambda$, and $F_{1\\lambda}$ are known as the one-phonon structure factors. $I_1$ provides momentum-resolved information on the nonequilibrium distribution of phonons across the entire Brillouin zone, since $I_1(\\vec{q}, \\tau)$ depends only on phonon modes with wavevector $\\vec{k}$ = $\\vec{q}$ - $\\vec{G}$ (Fig. \\ref{FIG:data}a). The one-phonon structure factors $F_{1\\lambda}$ are geometrical weights that depend sensitively on the phonon mode atomic polarization vectors $\\left\\{ e_{\\lambda\\vec{k}}\\right\\}$~\\cite{RenedeCotret2019}. Most importantly, $F_{1\\lambda}\\left(\\vec{q}, \\left\\{ e_{\\lambda\\vec{k}}\\right\\} \\right)$ are relatively large when the phonon mode $\\lambda$ is polarized parallel to the scattering vector $\\vec{q}$. Terms of higher-order than $I_1$ have lower cross-sections and do not contribute significantly to the interpretation of the low-order Brillouin zone (BZ) signals reported on here~\\cite{Wang2013,Zacharias2021a,Zacharias2021b}. \n\n\\begin{figure*}\n    \\centering\n\t\\includegraphics{diffuse.pdf}\n\t\\caption{Ultrafast electron diffraction and diffuse scattering signals from photodoped SnSe. \\textbf{a} Equilibrium scattering pattern of SnSe oriented along the $[100]$ direction with key vectors, the square BZ ($\\vec{b}^\\star$--$\\vec{c}^\\star$ plane) and high symmetry points indicated. \\textbf{b} Regions of interest for scattering as described in the text, shown around reflection $(00\\bar{2})$ as an example; (1) Bragg intensity, (2) small-wavevector phonons ($\\SI{0.114}{\\per\\angstrom} <|\\vec{k}| < \\SI{0.228}{\\per\\angstrom}$) and (3) larger wavevector phonons ($|\\vec{k}| > \\SI{0.228}{\\per\\angstrom}$)  \\textbf{c} Line cut across the horizontal line shown in panel b). The Bragg peak lineshape is fit with a Voigt profile (solid black line) with a full-width at half-max of \\SI{0.158}{\\per\\angstrom}. \\textbf{d} Transient (photoinduced) ultrafast electron scattering intensity changes in several regions of the BZ shown in a) and b). The decrease of intensity directly on the Bragg peaks shows transient DW decays that are strongly anisotropic. The fast ($\\sim \\SI{300}{\\femto\\second}$) decay component is maximized in Bragg peaks along $\\vec{c}^\\star$ (black), and only the slow components ($\\sim \\SI{4}{\\pico\\second}$) is observed in peaks perpendicular to $\\vec{c}^\\star$ (red). Transient diffuse intensity also shows pronounced anisotropy and $\\vec{k}$-dependence. A fast rise in diffuse intensity (purple) is only observed for $|\\vec{k}| < \\SI{0.228}{\\per\\angstrom}$ (region 2, panel b) in BZs that show the fast DW dynamics. A slow increase in diffuse intensity (orange) is observed at all zone boundary high-symmetry points (see Fig. S4) and elsewhere in the BZ for all reflections $\\vec{G}$ (e.g. region (3), panel b). Error bars represent the standard error in the mean of intensity before time-zero, but are generally smaller than the markers.}\n\t\\label{FIG:data}\n\\end{figure*}\n\nFig. \\ref{FIG:data}a shows an equilibrium diffraction pattern of SnSe at room temperature along the $[100]$ zone axis with the rectangular in-plane ($\\vec{b}^\\star$--$\\vec{c}^\\star$) BZ of the $Pnma$ phase indicated. This pattern contains both Bragg scattering (at zone-center positions) and PDS contributions at all scattering vectors.  Following photoexcitation, Bragg peaks show transient decreases in intensity whose dynamics are well-described by a biexponential decay with time-constants (\\SI{400 \\pm 130}{\\femto\\second} and \\SI{4 \\pm 1}{\\pico\\second}), as shown in Fig. \\ref{FIG:data}d.  The fast component of these dynamics are profoundly anisotropic, with a maximum contribution in the $\\vec{c}^\\star$ direction and below detection along $\\vec{b}^\\star$ (Fig. \\ref{FIG:data}d).  Given the DW factor ($\\exp(-\\frac{1}{2}\\langle \\vec{q} \\cdot \\vec{u} \\rangle^2$), this indicates that there are at least two distinct processes that contribute to increasing the atomic displacement $\\vec{u}$ following photoexcitation.  The ultrafast dynamics of the PDS intensity following photoexcitation provides a clear perspective on these distinct processes. At all high-symmetry BZ boundary positions ($Z$, $Y$ and $T$) we find that PDS intensity increases with a single exponential time constant of \\SI{4 \\pm 1}{\\pico\\second} (Fig. S4) that is the complement of the slow time-constant observed in the Bragg peak dynamics. In fact, within experimental uncertainties an identical time-constant is determined for increases in PDS observed at all BZ positions $|\\vec{k}| > \\SI{0.228}{\\per\\angstrom}$ (i.e. far from Bragg peaks), as is shown in Fig. \\ref{FIG:data}d (yellow) and Fig. \\ref{FIG:polaron}e. \n\nPrevious work has identified a number of soft and strongly-coupled optical phonons in the zone-center region of the $Pnma$ phase~\\cite{Chattopadhyay1986,Li2015,Gong2020,Lanigan2020}.  As $\\vec{k}$ approaches zone-center, the PDS contribution overlaps with the Bragg peak lineshape.  Thus, Bragg peak intensity must be subtracted to accurately determine the differential PDS from small-wavevector phonons following photoexcitation. As part of this analysis we investigated whether photoexcitation resulted in measurable time-dependence of Bragg peak positions and widths.  Fig. S3 demonstrates that the Bragg peak center positions and widths are shot noise limited and neither parameter shows a measurable time-dependence;  i.e. any photoinduced change to in-plane lattice constants (which shift Bragg peak positions) or in-plane long-range strain (which broaden and skew the width of peaks) over the range of delay times investigated here are at a level that is below our signal-to-noise ratio. The Bragg peak lineshapes are effectively constant, but have integrated intensities that vary according to the observed transient Debye-Waller dynamics. Panels b and c of Fig. \\ref{FIG:data} show two regions of interest around every Bragg peak. We define an area at the BZ center associated with strongest Bragg scattering (region (1), $|\\vec{k}| \\leq \\SI{0.114}{\\per\\angstrom}$) and a surrounding region associated with wavevectors in the range $\\SI{0.114}{\\per\\angstrom} < |\\vec{k}| \\leq \\SI{0.228}{\\per\\angstrom}$ (region (2)). By integrating scattered intensity in these areas separately, we assemble time-series that probe both the Bragg (region 1) and the small wavevector ($|\\vec{k}| \\sim \\Gamma$) PDS (region 2) after subtraction of Bragg intensity.  The results for reflections parallel to $\\vec{c}^\\star$ is shown in Fig. \\ref{FIG:data}d (purple), demonstrating that the fast component of the DW dynamics is exclusively associated with a rapid increase in the amplitude of phonons near zone-center. This signal shows the same $\\vec{b}^\\star$\/$\\vec{c}^\\star$ anisotropy as the Bragg peaks (Fig. S6) and is \\emph{insensitive} to the precise definition of the indicated regions, as demonstrated below.  Determination of the relatively small differential PDS signals directly under the Bragg peak (region 1) is obviously subject to large uncertainties and is not reported here. These observations corroborate computational work~\\cite{Caruso2019,Ma2018} which found that the electron-phonon coupling is profoundly anisotropic in SnSe, with carriers coupling very strongly to polar zone-center modes and only much more weakly elsewhere. \n\nThe PDS in intermediate regions of the BZ is a mixture of the fast and slow dynamics shown in Fig. \\ref{FIG:data}d, with the magnitude of these components varying as a function of $|\\vec{k}|$ as shown in Fig. \\ref{FIG:polaron}. In a conventional weakly polar semiconductor like GaAs, these dynamics can be described in terms of a model for the re-equilibration of the photodoped carriers with the phonon-system via intra- and inter-valley inelastic electron-phonon scattering processes based on the electron and phonon bandstructures of the material~\\cite{Sjakste2018}.  The dynamics observed here for SnSe cannot be described in these terms as is explained further in the discussion below.  Polaron formation, however, provides a robust description of these observations.      \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics{polaron.pdf}\n    \\caption{Polaron formation in SnSe visualized with UEDS. \\textbf{a} Wavevector-dependent scattering intensity for a Gaussian displacement field model of the polaron lattice distortion as a function of size (polaron FWHM), as described in the text. \\textbf{b} Measured change in diffuse intensity at \\SI{1}{\\pico\\second} (black triangles) and \\SI{5}{\\pico\\second} (orange circles) fit to the Gaussian displacement model above (solid curves). Best-fit FWHM polaron dimensions are $f=\\SI{18.7 \\pm 0.3}{\\angstrom}$ (\\SI{1}{\\pico\\second}) and $f=\\SI{4.2 \\pm 0.1}{\\angstrom}$ (\\SI{5}{\\pico\\second}).  The larger polaron is consistent with uniaxial displacements along $\\vec{c}^\\star$ and the smaller polaron with uniform displacements in the $\\vec{b}$--$\\vec{c}$ plane. The boundary between regions (2) and (3) from Fig. \\ref{FIG:data}b are indicated. The shaded region represents the standard error in the mean of intensity across the integration region. \\textbf{c} Differential scattering intensity across the BZ due to the large polaron lattice distortion. \\textbf{d} Differential scattering intensity across the BZ due to the small polaron lattice distortion. \\textbf{e} The measured differential diffuse intensity across the BZ at $\\tau=\\SI{5}{\\pico\\second}$ is in excellent agreement with the predicted scattering vector dependence of the model.}\n    \\label{FIG:polaron}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{polaron-realspace.pdf}\n    \\caption{Real-space visualization of the in-plane ($\\vec{b}$ -- $\\vec{c}$) atomic displacement due to the large and small polarons in Fig. \\ref{FIG:polaron}. The unperturbed in-plane dimensions of the unit cell are marked by solid lines. \\textbf{a} Large (\\SI{18.7}{\\angstrom} FWHM) one-dimensional polaron aligned to the $c$-axis \\textbf{b} Small (\\SI{4.2}{\\angstrom} FWHM) three-dimensional polaron. In both subpanels, the blue-shaded background represents the FWHM of the atomic displacement field $\\vec{u}(\\vec{r})$. The magnitude of the atomic displacements has been exaggerated for visual clarity.}\n    \\label{FIG:polaron-realspace}\n\\end{figure}\n\nThe local lattice distortions associated with a polaron in real-space can be modelled as an atomic displacement field $\\vec{u}(\\vec{r})$, where atoms are displaced as $\\vec{r}_j \\to \\vec{r}_j + \\vec{u}(\\vec{r}_j)$ \\cite{Guzelturk2021}. For small displacement fields, the contribution of such a localized lattice distortion to the total scattering amplitude, $f^p$, is given by:\n\\begin{equation}\n    f^p(\\vec{q}) \\approx - i \\sum_j f_{e,j}(\\vec{q}) e^{-i \\vec{q} \\cdot \\vec{r}_j} \\left( \\vec{G} \\cdot \\vec{u}_j \\right)\n\\end{equation}\nwhere $\\left\\{ \\vec{r}_j \\right\\}$ are the atomic positions, $\\left\\{ f_{e,j} \\right\\}$ are the atomic form factors for electron scattering, and $\\vec{G}$ is the Bragg reflection nearest to $\\vec{q}$ as in Fig. \\ref{FIG:data} (see SI). We consider two specific displacement fields due to the anisotropic observations described above; an effectively one-dimensional displacement field directed along the $c$-axis ($\\vec{u}_c(\\vec{r}) \\propto e^{-|\\vec{r}|^2\/r_p^2} ~ \\hat{\\vec{r}} \\cdot \\hat{\\vec{c}}$), and a three-dimensional displacement field ($\\vec{u}(\\vec{r}) \\propto e^{-|\\vec{r}|^2\/r_p^2} \\hat{\\vec{r}}$) (see SI). In both cases, $2 \\sqrt{2 \\ln(2)} ~ r_p$ is the full-width at half-maximum (FWHM) of the local lattice distortion associated with the polaron. The effect of both displacement fields is identical in terms of the impact on the diffuse scattering measured within a BZ, shown as a function of polaron FWHM in Fig. \\ref{FIG:polaron} a).  The one-dimensional displacement field, however, has the same $\\vec{b}^\\star$\/$\\vec{c}^\\star$ anisotropy in scattering intensity as our measurements.\n\nThis model was fit to the measured photoinduced differential PDS signals at \\SI{1}{\\pico\\second} and at \\SI{5}{\\pico\\second}, to capture changes due to the fast and slow dynamics respectively. The fast PDS dynamics are in excellent agreement with the formation of a \\SI{18.7 \\pm 0.3}{\\angstrom} polaron displacement field. The slow PDS dynamics, by contrast, are in excellent agreement with the displacement field associated with a \\SI{4.2 \\pm 0.1}{\\angstrom} FWHM polaron. A qualitative real-space representation of the two polaron modes is presented in Fig. \\ref{FIG:polaron-realspace}. We tentatively assign the larger polaron to the electron and the smaller polaron to the hole, in analogy to the work on Sio \\emph{et al.}~\\cite{Sio2019} on other polar materials. However, these observations are also consistent with electron and hole polarons being a similar size in SnSe (i.e. \\SI{4.2 \\pm 0.1}{\\angstrom} FWHM) and formation occurring in two steps.  We discuss these results within the context of both interpretations below.  \n\n\\subsection*{Discussion}\n\n The carriers generated through photoexcitation in these experiments localize via their interactions with the phonon system.  These interactions create a local potential that minimizes the free energy of the system according to the standard picture of polaron formation ~\\cite{Franchini2021} shown schematically in Fig. 1 d) - f). A polaron quasiparticle is thus formed as a localized carrier dressed by phonons, potentially in several phonon branches and over a range of wavevectors~\\cite{Sio2019}; a local lattice distortion in real-space is equivalent to a distribution of lattice normal modes in reciprocal space. These details are revealed by the UED and UEDS data. As recent ab initio work by Giustino and colleagues has demonstrated in other polar lattices~\\cite{Sio2019}, polarons require the recruitment of phonon modes across the entire BZ when the dimensions of the polaron approaches those of the lattice constants.  In SnSe, we propose that this manifests in the bimodal formation dynamics reported here due to the profoundly anisotropic (momentum-dependent) nature of electron-phonon coupling in the material.  Strong Fr\\\"ohlich coupling to near zone-center polar optical phonons ($A_g$, $B_u$, and $B_g$) results in the rapid ($\\sim\\SI{300}{\\femto\\second}$) formation of relatively large (electron) polarons, since polarons of this size only require the recruitment of strongly coupled small-wavevector phonon modes.  The formation time of the smaller (hole) polarons is an order of magnitude longer ($\\sim \\SI{4}{\\pico\\second}$) due to the relatively weak coupling to the large-wavevector phonons that must be recruited to form polarons of this size~\\cite{Sio2019}. \n\nAlternatively, the observed bimodal polaron formation dynamics are also consistent with a two-step process that has features in common with Onsager's inverse snowball effect, often discussed in the context of the theory of solvation~\\cite{neria1992simulations}.  Here the rapid but weak localization of the charge carriers is provided by the 1D ferroelectric-like lattice distortions along the $c$-axis.  The slower but strong localization is provided by the subsequent 3D distortions.  Polaron formation dynamically proceeds from the outside (long range) inwards (short range), not via layer accumulation from the inside out like a typical snowball.    \n\nThese measurements alone cannot identify precisely how vibrational excitation is distributed over the phonon branches near zone-center by $\\sim \\SI{1}{\\pico\\second}$; however, the increase in mean-square atomic displacements can be determined from the Bragg peak DW decays and can be used to estimate the fraction of excitation energy that has left the carrier system in the form of the polaronic lattice distortion by $\\sim \\SI{1}{\\pico\\second}$. This quantity is shown in Fig. \\ref{FIG:msd}, and is linear with pump fluence over the range investigated.  The observed increase, $\\Delta \\langle u_c^2\\rangle$, is consistent with a nearly complete transfer ($>85\\%$) of the excess photodoped carrier energy to zone center phonons (see SI and Fig. S7). Thus, the coupled electron-phonon system at $\\sim$\\SI{1}{\\pico\\second} is well described as a state in which the photodoped carriers in each valley have reached an equilibrium with only this limited set of strongly coupled small-wavevector phonons.  This equilibrium is well described by the formation of polaron quasiparticles (Fig. \\ref{FIG:polaron} b) rather than the simple heating of the phonon system as has been observed in other systems, like graphite where rapid electron cooling through a pre-equilibrium with specific strongly coupled modes has been observed ~\\cite{Stern2018,RenedeCotret2019}.  A similarly rapid coupling of electronic excitation energy to polaronic lattice distortions is thought to be present in methylammonium lead iodide perovskites~\\cite{Niesner2016}.\n\nPolaron formation is the simplest \\emph{self-consistent} explanation of these data. The isotropic slow-rise in diffuse scattering observed across the entire BZ specifically precludes an understanding of these results in terms of the conventional semiconductor picture of carrier relaxation through intervalley scattering mediated by large-wavevector phonons~\\cite{Sjakste2018,Waldecker2017,Stern2018,RenedeCotret2019,Otto2021}.  Based on the electronic dispersion of SnSe calculated by Wei \\emph{et al.}~\\cite{Wei2019}, we modelled the decay of hot electrons and holes, mediated by phonons, via energy-allowed and momentum-conserving pathways. This relaxation mechanism imprints the structure of the electronic dispersion onto the PDS, including a pronounced anisotropy between the $\\vec{b}^\\star$ and $\\vec{c}^\\star$ directions as shown in Fig. S8. This is ruled out by our measurements, which are azimuthally symmetric in reciprocal space (Fig. \\ref{FIG:polaron}e). Neither does the anharmonic decay of phonons provide an explanation for these data, given that phonon lifetimes are estimated to be almost an order of magnitude longer (\\SIrange{15}{30}{\\pico\\second})~\\cite{Chandrasekhar1977,Li2015,Lanigan2020} than the timescales observed herein. Moreover, the anharmonic decay of phonons measured in the time-domain displays an imprint of the phonon dispersion due to energy- and momentum-conservation rules~\\cite{Stern2018,RenedeCotret2019}, which is not seen in our data (Fig. \\ref{FIG:polaron}e).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{displacement.pdf}\n    \\caption{Increase in mean-square-displacement of all atoms along the $c$ axis, $\\Delta \\langle u_c^2 \\rangle$, due to the change in vibrational amplitude of the strongly-coupled zone-center modes exclusively. The associated photocarrier concentration $N_{\\gamma}$ for a sample of dimensions $\\SI{50 x 50 x 0.045}{\\micro\\meter}$ is shown above the plot. Boxes are used to represent error bars along both axes.}\n    \\label{FIG:msd}\n\\end{figure}\n\nThere are a number of important connections between these observations and an understanding of the thermoelectric properties of SnSe.  First, the rate of large polaron formation is very rapid and appears to be at or near the limit imposed by the period of the highest frequency phonons in SnSe ($\\sim \\SI{200}{\\femto\\second}$) due to the strong Fr\\\"ohlich coupling in the polar lattice, even at the high carrier density operating conditions of optimally doped SnSe for thermoelectric device applications~\\cite{Sootsman2009,Zhao2016,Fan2018}.  At this doping level, these results indicate that SnSe is well described as a dense polaron system, since the density of polarons overlapping at a distance equal to their FWHM is  \\SI{3.2e21}{\\per\\cubic\\centi\\meter} for the smaller (hole) polarons, overlapping with this optimal doping range.  \nThe polaronic nature of the charge carriers in SnSe likely plays an important role in preserving the electron-crystal, phonon-glass conditions that are important for high-performance thermoelectric materials~\\cite{Rowe1995}. The dressed charges are better screened from scattering mechanisms that could otherwise deteriorate mobility at high carrier density and temperatures approaching a structural phase transition, where $ZT$ in SnSe is highest.  Several open questions remain regarding the nature of polarons in SnSe. Our measurements directly probe the in-plane lattice distortions, but do not provide information on the inter-layer (or $a$-axis) dependence. There is also the important question regarding the potential ferroelectric nature of polarons in SnSe, as has been discussed in the context of lead halide perovskites~\\cite{Frost2017,Joshi2019}.  Monolayer SnSe has been investigated as a possible platform for ultrathin ferroelectrics~\\cite{Fei2016}, and polar nanodomain formation and interlayer coupling may contribute to the dynamics we have observed. The in-plane polarization within a single layer can be modulated simply through changes in the angle of the Sn-Se bonds relative to the $a$-axis ~\\cite{Fei2016} in a manner that is consistent with the local polaronic displacements shown schematically in Fig. \\ref{FIG:polaron-realspace}.\n\nElectron-phonon coupling is not normally considered to be an important contributor to lattice thermal conductivity, however, previous work has shown that these interactions can play a role in suppressing thermal conductivity and enhancing thermoelectric performance in Si~\\cite{Liao2015} and SiGe~\\cite{Fan2018} at high carrier doping.  While a giant lattice anharmonicity and 3-phonon scattering processes seems to be sufficient to explain the ultralow thermal conductivity of undoped SnSe~\\cite{Zhao2014, Li2015}, a quantitative understanding of electron-phonon interactions and their impact on both electronic and lattice thermal conductivity (in addition to electrical conductivity) in this strongly coupled, high-carrier density regime is important for the further development of thermoelectrics and higher $ZT$.  The rate of electron-phonon scattering with the strongly coupled polar modes is more than an order of magnitude higher than 3-phonon scattering processes at the carrier densities investigated here~\\cite{Lanigan2020}.\n\nWe are currently in an excellent position to make significant progress understanding this complex, strongly-coupled regime across many material classes. The recent parallel development of ab initio approaches for both electron-phonon coupling~\\cite{Ponce2016} and polaron formation~\\cite{Sio2019} and time- and momentum-resolved measurement techniques like ultrafast electron\/xray diffuse scattering that are capable of interrogating electron-phonon interactions in exquisite detail and directly visualizing polaron formation.  Future work that combines these approaches together with more well established ultrafast spectroscopic methods are likely to yield significant insights.\n\n\\subsection*{Synthesis and sample preparation} \n\nA SnSe ingot (\\SI{20}{\\gram}) was synthesized by mixing appropriate ratios of high purity starting materials (Sn chunk, 99.999\\%, American Elements, USA and Se shot, 99.999\\%, 5N Plus, Canada) in \\SI{13}{\\milli\\meter} diameter quartz tube. The tube was flame-sealed at a residual pressure of $\\SI{1e-4}{\\mmHg}$, then slowly heated to \\SI{1223}{\\kelvin} over \\SI{10}{\\hour}, soaked at this temperature for \\SI{6}{\\hour} and subsequently furnace cooled to room temperature. The obtained ingot was crushed into powder and flame-sealed in a quartz tube, which was placed into another, bigger, flame-sealed quartz tube. A crystal with dimensions of $\\sim$\\SI{13}{\\milli\\meter} (diameter) $\\times$ \\SI{20}{\\milli\\meter} (length) was obtained.\n\nSeven samples were used for the ultrafast electron scattering measurements, to ensure reproducibility. Six samples were ultramicrotomed with a \\ang{35} diamond blade, while one sample was mechanically exfoliated. Three of the ultramicrotomed samples were cut from a first SnSe mother flake at a thickness of \\SI{90}{\\nano\\meter}. The remaining three ultramicrotomed samples were cut to a thickness of \\SI{70}{\\nano\\meter} from a different mother flake, synthesized separately from the first. The exfoliated sample was prepared from the second mother crystal, with a final thickness of \\SI{45}{\\nano\\meter} as estimated based on the relative ratio of intensities, compared to thicker ultramicrotomed samples.\n\n\\subsection*{Ultrafast electron scattering experiments} \nElectron scattering experiments employed \\SI{35}{\\femto\\second} pulses of \\SI{800}{\\nano\\meter} light at a repetition rate of \\SI{1}{\\kilo\\hertz}. Part of these light pulses is upconverted to \\SI{266}{\\nano\\meter} pulses via third-harmonic generation, which is then used to generate bunches of $10^6$ -- $10^7$ electrons from a bulk copper photocathode. These electrons are then accelerated to \\SI{90}{\\kilo\\electronvolt} in a DC electric field. Electron bunches are compressed with a radio-frequency cavity to counterbalance space-charge repulsion, resulting in a time-resolution of $\\SI{130}{\\femto\\second}$. Electrons are transmitted through the samples before being collected by an electron camera. The other part of the \\SI{800}{\\nano\\meter} pulses is used to photoexcite the sample almost co-linearly with the electron propagation axis ($\\sim \\ang{5}$). Experiments were repeated for up to \\SI{72}{\\hour} to maximize signal-to-noise, which was made possible by improvements in RF-laser synchronization~\\cite{Otto2017}. Detailed descriptions of this instrument are presented elsewhere~\\cite{Chatelain2012, Otto2017}. The samples were thinner than than the optical depth of $>\\SI{100}{\\nano\\meter}$ at \\SI{800}{\\nano\\meter}~\\cite{Barrios2014, Makinistian2009}, ensuring that the entire sample was photoexcited. The samples were oriented in the $\\langle 100 \\rangle$ direction, giving UEDS measurements a full view of lattice dynamics in the plane spanned by $\\vec{b}$ and $\\vec{c}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn recent years, the studies on black holes in higher dimensions have attracted much attention. \nSome of these studies show that they have much more complicated and richer structure than 4-dimensional ones.\nEspecially, the study of Kaluza-Klein (KK) black holes is important \nin association with our apparent 4-dimensional spacetime. \nThe Gregory-Laflamme instability~\\cite{Gregory:1993vy} \n(see also \\cite{Harmark:2007md} and references therein) and black hole\/black string phase transition \n(see {\\it e.g.} the reviews \\cite{Kol:2004ww}) \nmotivate many studies on thermodynamic aspects of KK black holes. \nNow, the thermodynamic properties and the first law for asymptotically flat KK black holes \nare widely investigated \\cite{Kol:2003if}-\\cite{Kastor:2007wr}. \n\n\nIn 5-dimensional Einstein-Maxwell theory, there is an analytic solution representing an \nelectrically charged black hole with squashed horizons \n\\cite{Ishihara:2005dp} as a generalization of the solution given in \\cite{Dobiasch:1981vh} and \\cite{Gibbons:1985ac}. \nIntriguingly, the spacetime far from the black hole is \nlocally a product of the 4-dimensional Minkowski spacetime and $S^1$. \nIn this sense, the black hole resides in KK spacetime and \nis worth to be named a KK black hole. \n\n\nThe black hole has an interesting property that various definitions of mass take different values, \nwhich means that the black hole gives an opportunity to investigate \nthe differences in various definitions of mass. \nIn this paper, we show some expressions for the first law of black hole thermodynamics \nsatisfied by those masses and discuss the differences from the viewpoint of thermodynamics. \n\n\n\\section{Kaluza-Klein black hole}\n\n\nLet us review the KK black holes  with squashed horizons~\\cite{Ishihara:2005dp}, \nwhich is a solution of the 5-dimensional Einstein-Maxwell theory. \nThe action is given by \n\n \\begin{eqnarray}\n I = \\frac{1}{16\\pi G} \\int_M d^5 x \\sqrt{-g} \n         \\left[ R - F^{\\mu\\nu} F_{\\mu\\nu} \\right]+\\frac{1}{8\\pi G} \\int_{\\partial M} \n K\\sqrt{-h}d^4x, \n \\end{eqnarray}\n\n where $G$ is the 5-dimensional Newton constant, $g_{\\mu\\nu}$ is the metric, $R$ is scalar curvature, \n $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu $ is \n the field strength of the 5-dimensional $U(1)$ gauge field $A_\\mu$. \nThe second term is so-called Gibbons-Hawking term, in which $h$ is determinant of the induced metric and \n $K$ is trace of the extrinsic curvature of the boundary $\\partial M$, respectively. \nThe metric of the black hole is given by \n\\begin{eqnarray}\n ds^2 &=& - V(\\rho) d\\tau^2 + \\frac{B (\\rho)}{V(\\rho)} d\\rho^2 \n + \\rho^2 B (\\rho) d\\Omega^2  \n + \\frac{r_{\\infty}^2}{4 B (\\rho)} \\left(d\\psi + \\cos \\theta d\\phi \\right)^2 ,\n\\label{eq:metric-Trho}\n\\end{eqnarray}\nwhere $d\\Omega^2 = d\\theta^2 + \\sin^2 \\theta d\\phi^2$ is the metric of the unit two-dimensional sphere and \n\\begin{eqnarray}\nV(\\rho)=  \\left( 1- \\frac{\\rho_{+}}{\\rho}\\right)\\left( 1-\\frac{\\rho_{-}}{\\rho} \\right) , \\ \\ \nB(\\rho) =   1+ \\frac{\\rho_0}{\\rho}, \\ \\ \nr_{\\infty} = 2\\sqrt{(\\rho_+ +\\rho_0 )(\\rho_- + \\rho_0 )}. \n\\end{eqnarray}\nHere, the coordinate ranges are \n$0 \\leq \\theta <2\\pi ,\\   0 \\leq \\phi < 2\\pi ,\\  0 \\leq \\psi < 4\\pi $. \nThe gauge potential is given by\n\\begin{eqnarray}\nA = \\mp   \\frac{\\sqrt{3}}{2}  \\frac{\\sqrt{{\\rho_+}{\\rho_-}}}{ \\rho_0}\n  \\left(1+\\frac{\\rho_0}{\\rho} \\right) d\\tau.\n \\end{eqnarray}\n\n\nIt is easy to see the apparent singularity at $\\rho_{+}$ corresponds to the outer horizon of the black hole.\nThe inner horizon $\\rho_{-}$ is analogous to that of the Reissner-Nordstr\\\"om (RN) black holes.\nThe spatial infinity corresponds to a limit $\\rho \\to \\infty$. \nIt should be noted that the shape of the horizon is a squashed sphere \nas was discussed in~\\cite{Ishihara:2005dp}.\nFrom the metric (\\ref{eq:metric-Trho}), \nit is seen that the $S^1$ circle parametrized by a coordinate $\\psi$ has finite size even at the spatial infinity. \nThe non-trivial twisting of the $S^1$ circle fibrated over the $S^2$ base space \nleads a 4-dimensional U(1) gauge field by KK reduction.\nActually, in no horizon limit $\\rho_+, \\rho_- \\to 0$, the black hole spacetime becomes the KK monopole \nspacetime \\cite{Gross-Perry}\\cite{Sorkin}. \nIn the limit $r_\\infty \\to \\infty$, the KK monopole becomes 5-dimensional Minkowski spacetime and \nthe black hole reduces to the 5-dimensional RN black hole.\nWe term this limit the spherically symmetric limit. \n\n\nGiven the metric (\\ref{eq:metric-Trho}), we can calculate various physical quantities.\nThe surface gravity is calculated as\n\\begin{eqnarray}\n   \\kappa_{+} = \\frac{\\rho_{+} -\\rho_{-}}{2\\rho_{+} \\sqrt{{\\rho_{+}}({\\rho_{+} + \\rho_0})}},\n   \\label{eq:surfacegravity}\n\\end{eqnarray}\nwhich gives the Hawking temperature of the black hole $T= \\kappa_+\/2\\pi$ \\cite{Hawking:1974sw}.\nWe assume that the entropy of the black hole is given by the Bekestein-Hawking formula \n\\begin{eqnarray}\nS = \\frac{A_+}{4G} = \\frac{4\\pi^2}{  G}  \\rho_+  (\\rho_+ + \\rho_0) \\sqrt{\\rho_+(\\rho_- + \\rho_0 ) },\n \\label{eq:entropy}\n\\end{eqnarray}\nwhich is consistent with the Wald's entropy formula \\cite{Wald:1993nt} \\cite{Iyer:1994ys}. \nThe electric charge and electrostatic potential of the black hole are also calculated as \\cite{Ishihara:2005dp}\n\\begin{eqnarray}\n  Q = \n    \\pm \\frac{\\sqrt{3} \\pi}{G} r_\\infty \\sqrt{\\rho_{+} \\rho_{-} }, \\quad \n\\Phi \n= \\pm \\frac{\\sqrt{3}}{2} \\sqrt{\\frac{\\rho_-}{\\rho_+} }. \n\\end{eqnarray}\n\n\n\\section{Mass and free energy}\n\nThere are several definitions of mass for the black hole spacetime.\nCai {\\it et al.} \\cite{Cai:2006td} discussed mass of the black hole defined by the counter-term method for\nasymptotically locally flat spacetime \\cite{Mann:1999pc}-\\cite{Kraus:1999di}. \nUsing the counter-term mass $M_{ct}$, they investigated the first law of black hole thermodynamics and \nsuggested the existence of a new work term in the first law. \nThe direct calculation reveals \n\\begin{eqnarray}\n M_{ct} = M_{AD} = \\frac{\\pi}{2G}r_{\\infty} \\left( 2\\rho_+ + 2\\rho_- +\\rho_0 \\right), \n\\label{eq:ct-mass-BH}\n\\end{eqnarray}\nwhere $M_{AD}$ is the Abbott-Deser (AD) definition of mass~\\cite{Abbott:1981ff} \nevaluated on a product $S^1$ bundle over 4-dimensional Minkowski spacetime, which is a completely \nflat spacetime\\footnote{ Cai {\\it et al.} showed that $M_{ct}$ equals the AD mass evaluated on a \"twisted\" \n$S^1$ bundle over 4-dimensional Minkowski spacetime.  \nHowever, this background spacetime is not a solution of the vacuum Einstein equation \nand we can not obtain finite free energy and Hamiltonian of the black hole on it. \nThus, we consider the flat background.\n}.\nHereafter, we term this spacetime flat background, shortly.\nTherefore, $M_{AD}$ satisfies the same equations as $M_{ct}$. \nThe Komar mass is also meaningful as a mass of black holes which possess a timelike Killing vector.\nUsing the timelike Killing vector $\\partial \/\\partial \\tau $ normalized at the spatial infinity, \nwe can calculate the Komar mass for the black hole (\\ref{eq:metric-Trho}) as \n\\begin{eqnarray}\n   M_K  =  \\frac{3\\pi }{4 G} r_{\\infty} \\left( \\rho_{+} + \\rho_{-} \\right) ,\n   \\label{eq:Komar}\n\\end{eqnarray}\nwhere the integral is taken over the 3-dimensional sphere at the spatial infinity.\nSmarr-type formula was shown generally in~\\cite{Gauntlett:1998fz} for the Komar mass \n\\begin{eqnarray}\n M_K -Q\\Phi = \\frac{3}{2} TS,\n\\end{eqnarray}\nwhich is sometimes called integrated expression for the first law. \nThis implies that we would have a differential expression for the first law using the Komar mass.\nClearly, since $M_K$ does not equal $M_{ct}$, then \nthe expressions for the first law satisfied by these masses should be different. \n\n\n\nIn order to deduce the work term suggested by Cai {\\it et al.}, \nwe note the fact that, far from the black hole, the geometry locally looks like the black string. \nIt is known that, in the case of the black p-branes or black string without twisting, \nthe so-called gravitational tension and the size of the extra-dimension \ncontribute to the first law~\\cite{Kol:2003if}\\cite{Townsend:2001rg}\\cite{Harmark:2003eg}\\cite{Traschen:2001pb}. \nOne may think that, also in the case of the squashed KK black hole, \nthese quantities contribute to the first law as a work term. \nThe gravitational tension which can be applied to non-asymptotically flat spacetime was \ndefined by using the Hamiltonian formalism to a foliation of the spacetime \nalong asymptotically translationally-invariant spatial direction~\\cite{Harmark:2004ch}. \nThe definition requires some reference spacetime in order to give finite gravitational tension. \nAs a reference background, we choose the flat background.\nThen, using the definition given in \\cite{Harmark:2004ch}, we can calculate the gravitational tension \nassociated with the direction $\\partial_{\\psi}$ as\n\\begin{eqnarray}\n\\mathcal{T} = \\frac{1}{4 G}\\left(\\rho_+ +\\rho_- +2\\rho_0\\right). \n\\end{eqnarray}\nThe conjugate variable to $\\mathcal{T}$ is \nthe size of the extra-dimension at the spatial infinity, $L:=2\\pi r_{\\infty}$. \nWith these quantities, $M_{AD}$ is related to the entropy and the electric charge \nvia the following expression for the first law: \n\\begin{eqnarray}\ndM_{AD} &=& TdS + \\Phi dQ +\\mathcal{T} dL.\n\\label{eq:first-law-AD}\n\\end{eqnarray}\nThus, the last term $\\mathcal{T}dL$ is  \nthought of as a work term suggested by Cai {\\it et al.}~\\cite{Cai:2006td}. \nThe expression for the first law (\\ref{eq:first-law-AD}) shows that \n$M_{AD}$ is a thermodynamic potential as a function of $(S,Q,L)$. \nIt means that $M_{AD}$ is relevant to the thermodynamic system with natural variables $(S,Q,L)$. \nThe same holds true for $M_{ct}$, because of $M_{ct}=M_{AD}$.\n\n\nThe free energy of the black hole is obtained from the classical Euclidean action $I_E$ as \n\\begin{eqnarray}\nF =T {I_E} = \\frac{\\pi}{2G}r_{\\infty} (\\rho_+ + \\rho_0 ),\n\\end{eqnarray}\nwhich was calculated by the traditional background subtraction method \nwith the flat background~\\cite{Gibbons:1976ue}.\nThe free energy $F$ equals one evaluated by the counter-term method.\nThus, in this case, \nthe counter-term method is equivalent to background subtraction method with the flat background.\nIn the calculation of the free energy by the background subtraction method, we fixed the temperature, \nthe electro-static potential and the size of the extra-dimension at the boundary of the spacetime. \nIt follows that the free energy satisfies the following differential relation: \n\\begin{eqnarray}\ndF = -SdT -Qd\\Phi +\\mathcal{T} dL.\n\\label{eq:first-law-F}\n\\end{eqnarray}\nThus, the free energy $F$ has natural variables $(T, \\Phi,L)$. \nThe equations (\\ref{eq:first-law-AD}) and (\\ref{eq:first-law-F}) suggest the following relation with the AD mass:\n\\begin{eqnarray} \nF=M_{AD} - Q\\Phi -TS,\n\\label{eq:Legendre-F-AD}\n\\end{eqnarray}\nwhich can be easily shown by direct calculation. \nThis relation (\\ref{eq:Legendre-F-AD}) is nothing but Legendre transformation between \n$F$ and $M_{AD}$. \n\n\n\nFor asymptotically flat electrically charged black holes, it is known that Hamiltonian \ndoes not equal ADM mass and these two quantities differ by $Q\\Phi$~\\cite{Hawking:1995ap}.\nThe black hole we consider here is not asymptotically flat but asymptotically locally flat.\nIn spite of the difference in asymptotic structure, \nif we regard the AD mass as a counterpart of ADM mass, \nthe same is true for the case of the squashed black hole; \nThe Hamiltonian of the black hole evaluated on the flat background can be related to the AD mass as\n~\\cite{Hawking:1998jf}\n\\begin{eqnarray}\nH = \\frac{\\pi}{2G} r_{\\infty} \\left( 2\\rho_+ - \\rho_- +\\rho_0 \\right)\n=M_{AD} -Q\\Phi.\n\\label{eq:Hamiltonian}\n\\end{eqnarray}\nOften, Hamiltonian for solution without electric charge is called Hawking-Horowitz (HH) mass \\cite{Hawking:1995fd}.\nIn the case of $Q=0$, the HH mass of the black hole is equal to the AD mass \nas is in the case of asymptotically flat black hole. \nFrom the relation (\\ref{eq:Hamiltonian}), the first law can take the form with the Hamiltonian as \n\\begin{eqnarray}\ndH &=& TdS - Qd\\Phi+\\mathcal{T} dL.\n\\label{eq:first-law-flat-H}       \n\\end{eqnarray}\nEquivalently, \nthe relation (\\ref{eq:Hamiltonian}) is a Legendre transformation between the Hamiltonian and the AD mass. \nThe equation (\\ref{eq:first-law-flat-H}) shows \nthat the Hamiltonian is the thermodynamic potential with natural variables $(S, \\Phi, L)$. \nThe Hamiltonian is also the Legendre transform of the free energy with respect to $TS$; \n$F = H-TS$.\nTherefore, \nthe AD mass and the Hamiltonian are related to the free energy $F$ via Legendre transformations and \ncan be interpreted as different thermodynamic potentials. \n\n\nHowever, it can be shown that the Komar mass does not have $\\mathcal{T}$ or $L$ as a natural variable. \nNow, let us obtain the differential expression for the first law by use of the Komar mass. \nIn order to do so, we require that the Komar mass should be related to the free energy via Legendre transformations.\nIn stead of $(L, \\mathcal{T})$, we introduce a couple of new variables $(\\epsilon, \\Sigma)$ satisfying  \n\\begin{eqnarray}\n F=M_K - TS -Q\\Phi + \\epsilon \\Sigma,\n\\quad \ndM_K = TdS + \\Phi dQ -\\epsilon d \\Sigma.\n\\label{eq:first-law-Komar}\n\\end{eqnarray}\nThen, $(\\epsilon, \\Sigma)$ is determined up to a constant $C$ as\n\\begin{eqnarray}\n\\epsilon = C (2\\pi r_{\\infty})^2=CL^2, \\quad \n\\Sigma =\\frac{1}{16 \\pi GC r_{\\infty}}  \\left(\\rho_+ +\\rho_-+2\\rho_{0}\\right)=  \\frac{\\mathcal{T}}{2CL}. \n\\end{eqnarray}\nThus, the Komar mass has this quantity $\\Sigma$ as a natural variable as well as $S$ and $Q$.\nThe pair of variables $(\\epsilon, \\Sigma)$ contributes not only to the differential relation \nfor the Komar mass \nbut also to the following:\n\\begin{eqnarray}\ndF = -SdT -Qd\\Phi +\\Sigma d\\epsilon,\\ \ndH = TdS -Qd\\Phi +\\Sigma d\\epsilon, \\ dM_{AD} = TdS +\\Phi dQ +\\Sigma d\\epsilon.\n\\label{eq:first-law-H-AD-epsilon}\n\\end{eqnarray}\nTherefore, the expression for the first law including the free energy, \n the Hamiltonian or the AD mass is not unique. \nThese expressions are consistent with the interpretation that $H$ or $M_{AD}$ is the thermodynamic potential \nwith natural variables $(S,\\Phi, L)$ or $(S, Q,L)$, because \nsystem with fixed $L$ \nis equivalent to that with fixed $\\epsilon$ due to the relation $\\epsilon \\propto L^2$.\n\n\n\nIn order to clarify the relation with the case of the 5-dimensional RN black hole,\nlet us consider the spherically symmetric (SS) limit $r_{\\infty}\\to \\infty$.\nHowever, the free energy, the Hamiltonian and the AD mass evaluated on the flat background diverge in the limit.\nAs an alternative background, we consider the KK monopole spacetime. \nThe free energy and the Hamiltonian on the KK monopole background are calculated as \n\\begin{eqnarray}\n\\tilde{F} = \\frac{\\pi}{2G} r_{\\infty} \n \\left(\\rho_+ + \\rho_0 -  \\frac{r_{\\infty}}{2} \\right), \\quad \n\\tilde{H} = \\frac{\\pi}{2G} r_{\\infty} \\left(2\\rho_+ -\\rho_- +\\rho_0 -\\frac{r_{\\infty}}{2} \\right).\n\\end{eqnarray}\nIt is easily checked that the difference between $F$ and $\\tilde{F}$ is the free energy of the KK monopole \non the flat background; $F-\\tilde{F}=\\frac{\\pi}{4G}r_\\infty^2$, \nwhich equals the free energy calculated by the counter-term method \\cite{Mann:2005cx}.\nThe gravitational tension calculated on the KK monopole background is \n\\begin{eqnarray}\n\\tilde{\\mathcal{T}} &=& \\frac{1}{4 G} \\left( \\rho_+ +\\rho_- + 2\\rho_0 -r_{\\infty}\\right).\n\\end{eqnarray}\nWith this tension, \nthe free energy and the Hamiltonian satisfy \n\\begin{eqnarray}\nd\\tilde{F} = -SdT -Qd\\Phi + \\tilde{\\mathcal{T}} dL, \\quad \nd\\tilde{H} = TdS -Qd\\Phi + \\tilde{\\mathcal{T}} dL.\n\\end{eqnarray}\nAs before, $\\tilde{\\mathcal{T}}$ or $L$ can not be a natural variable for the Komar mass.\nAs in the previous case, we can obtain a couple of thermodynamic variables $(\\epsilon, \\tilde{\\Sigma})$ satisfying\n\\begin{eqnarray}\n\\tilde{F}=M_K-TS-Q\\Phi+\\epsilon \\tilde{\\Sigma}, \\quad  dM_K = TdS + \\Phi dQ -\\epsilon d \\tilde{\\Sigma}.\n\\label{eq:first-Komar-KKm}\n\\end{eqnarray}\nThe result is \n\\begin{eqnarray}\n\\epsilon ={C} (2\\pir_\\infty)^2 = {C}L^2, \\quad \n\\tilde{\\Sigma} = \n\\frac{1}{16\\pi G {C} r_{\\infty}}  \\left(\\rho_+ +\\rho_-+2\\rho_{0}-r_\\infty \\right)=\\frac{\\tilde{\\mathcal{T}}}{2{C}L},\n\\end{eqnarray}\nwhere $\\tilde{\\Sigma}$ is different from $\\Sigma$ by a constant $(16 \\pi GC)^{-1}$. \nTherefore, the two quantities $\\Sigma$ and $\\tilde{\\Sigma}$ are essentially the same, \nand the differential expressions (\\ref{eq:first-law-Komar}) \nand (\\ref{eq:first-Komar-KKm}) are equivalent.\nThis is consistent with the fact that the Komar mass does not depend on the choice of background spacetime.\nIn the SS limit, the products $\\epsilon\\tilde{\\Sigma}$ and $L\\tilde{\\mathcal{T}}$ become zero, \nand $M_K$, $\\tilde{F}$ and $\\tilde{H}$ \nbecome those of 5-dimensional RN black hole evaluated on the 5-dimensional Minkowski background.\nThus, this formulation for the squashed black hole \nincludes usual thermodynamic formulation for 5-dimensional RN black hole as a limit.\nIn this sense, it \nis a generalized formulation of thermodynamics for 5-dimensional electrically charged static black holes. \n\n\nBecause $\\tilde{\\Sigma}$ and $\\tilde{\\mathcal{T}}$ vanish when the outer horizon is a perfectly-round three-sphere,\none may think that\nthe thermodynamic variable $\\tilde{\\Sigma}$ or $\\tilde{\\mathcal{T}}$ can be interpreted as \na quantities representing the deformation in the shape of the horizon. \nIn fact, the variables $\\tilde{\\Sigma}$ and $\\tilde{\\mathcal{T}}$ can be rewritten as \n\\begin{eqnarray} \n\\tilde{\\Sigma}  = \\frac{1}{32\\pi G C}\\frac{1}{R_+ \\ell_+} \\left({\\ell_+} - R_+ \\right)^2, \\quad \n\\tilde{\\mathcal{T}} = \\frac{1}{8G}\\frac{ r_{\\infty}}{R_+ \\ell_+} \\left( \\ell_+ - R_+ \\right)^2,\n\\label{eq:Sigma-squashing}\n\\end{eqnarray}\nwhere we have defined new parameters $R_+$ and $\\ell_+$ as follows:\n \\begin{eqnarray}\nR_+ := \\sqrt{\\rho_+(\\rho_+ + \\rho_0 )}, \\quad {\\ell_+} := \\sqrt{\\rho_+ (\\rho_- +\\rho_0)}, \n \\end{eqnarray}\nwhich denote \nthe circumference radius of the $S^2$ base space at the outer horizon and \nthat of the twisted $S^1$ fibre there, respectively. \nIn this way, $\\tilde{\\Sigma}$ and $\\tilde{\\mathcal{T}}$ measure the squashing of the outer horizon.\n\n\n\\section{Summary}\n\n\nAs shown in this paper, \nthe Abbott-Deser mass which equals the counter-term mass, the Komar mass and the Hamiltonian \ncontribute to different expressions for the first law \nand are related to each other by the Legendre transformations. \nEach mass can be interpreted as a thermodynamic potential with its own natural variables.\nThe consistent set of natural variables for each mass has been revealed, \nand we have obtained a more general thermodynamic formulation for electrically charged black holes\nin 5-dimensional Einstein-Maxwell system.\n\n\nNow, we discuss the relation between $(L, \\mathcal{T})$ and \nthe pair of new quantities ($\\epsilon, \\Sigma$).\nLet us begin by considering the case of the free energy. \nIn the evaluation of the free energy, the size of the extra-dimension at spatial infinity $L$ was fixed, \nso that $L$ is a natural variable of the free energy. \nIn general, $L$ can be replaced by any monotonic function of $L$, say $f(L)$, as a thermodynamics variable. \nThis may be trivial because thermodynamic environment characterized by fixing the size $L$ is equivalent to\nthat by fixing $f(L)$. \nOne may write the differential relation for the free energy as \n\\begin{eqnarray}\ndF = -SdT -Qd\\Phi + W d f(L),\n\\label{eq:first-law-fL}\n\\end{eqnarray}\nwhere $W$ is conjugate to $f(L)$. \nSince the last work term can be rewritten as $Wdf(L) =W f'(L) dL$,  \nthe relation (\\ref{eq:first-law-fL}) is equivalent to (\\ref{eq:first-law-F}) \nif the conjugate variable $W$ satisfies \n\\begin{eqnarray}\nW= \\frac{\\mathcal{T}}{f'(L)}.\n\\end{eqnarray}\nIn this way, the work term with the form $\\mathcal{T} dL$ can be replaced by $W d f(L)$.\nThe first law (\\ref{eq:first-law-fL}) is equivalent to (\\ref{eq:first-law-H-AD-epsilon}),\nif $f(L) = CL^2 = \\epsilon$, \nand the relation between $\\Sigma$ and $\\mathcal{T}$ is given as\n\\begin{eqnarray}\n\\Sigma = \\frac{\\mathcal{T}L}{2\\epsilon}\\  \\propto\\  \\frac{\\mathcal{T}}{2L}.\n\\label{eq:Sigma-T-relation}\n\\end{eqnarray}\nThe Hamiltonian and the AD mass are Legendre transforms of the free energy with respect to \n$TS$ or $TS+Q\\Phi$ respectively, so that the expressions in (\\ref{eq:first-law-H-AD-epsilon}) are \nequivalent to (\\ref{eq:first-law-AD}) and (\\ref{eq:first-law-flat-H}). \nTherefore, the work term in the first law for the Hamiltonian and the AD mass is not unique. \n\n\nHowever, the quantities $\\mathcal{T}$ and $\\Sigma$, which are conjugate to $L$ and $\\epsilon$, \nare different in the sense of thermodynamics: \nthermodynamic environment characterized by fixing $\\Sigma$ is one by fixing $\\mathcal{T}\/2L$, as shown in \n(\\ref{eq:Sigma-T-relation}). \nThus, the pair ($\\epsilon, \\Sigma$) is thermodynamically different from ($L, \\mathcal{T}$). \nIt is natural that thermodynamic potentials suitable for different environments are different. \nThe Komar mass is a thermodynamic potential for environment characterized by ($S, Q, \\Sigma$), \nwhile the AD mass is a thermodynamic potential for that by ($S, Q, \\epsilon$) or ($S, Q, L$). \nIn this way, we can interpret the difference of masses from thermodynamical viewpoint. \n\n\nIt is interesting to investigate thermodynamic properties and stability of the black hole in each environment \nand to compare the black hole with black string. \nIt will be reported in a future publication. \n\n\n\n\n\\begin{acknowledgments} \nY.K. was partially supported by the Yukawa memorial foundation and is also supported by \nthe 21st Century COE \"Constitution of wide-angle mathematical basis focused on knots\". \nThis work is supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science and \nCulture of Japan No. 13135208 and No. 14540275. \n\\end{acknowledgments} \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Introduction} \n\nMulticomponent quantum gases constitute an ideal playground for investigating a plethora of many-body (MB) \nprocesses~\\cite{Bloch2008,Cazalilla2011} \nincluding in particular the generation of quasiparticles~\\cite{landau1933bewegung,pekar1946autolocalization} such as polarons. \nQuasiparticle formation can be studied owing to the unprecedented experimental tunability of the impurity-medium \ninteraction strength, via Feshbach \nresonances~\\cite{Ospelkaus2006,Zaccanti2006,chin2010feshbach},  while systems containing few particles \ncan be realized especially in one spatial dimension~\\cite{serwane2011deterministic,wenz2013few}. \nDepending on the quantum statistics of the host, these quasiparticles are known \nas Bose~\\cite{grusdt2015new,rath2013field} and Fermi~\\cite{schmidt2018universal,massignan2014polarons} polarons respectively. \nTheir existence and a variety of their properties have already been experimentally probed in both \nBose~\\cite{jorgensen2016observation,hu2016bose,yan2020bose} and \nFermi~\\cite{schirotzek2009observation,kohstall2012metastability,koschorreck2012attractive,cetina2016ultrafast,scazza2017repulsive} \ngases, e.g. via employing injection spectroscopy~\\cite{cetina2016ultrafast,jorgensen2016observation,hu2016bose}. \nThe progress regarding the understanding of the quasiparticles features has also been corroborated by an extensive \ntheoretical activity revealing different aspects of their underlying dressing mechanism such as their effective \nmass~\\cite{grusdt2017bose,ardila2015impurity}, lifetime~\\cite{kohstall2012metastability}, induced \ninteractions~\\cite{dehkharghani2018coalescence,mistakidis2020many,mistakidis2019repulsive}, and bound \nstates termed bipolarons~\\cite{camacho2018bipolarons,schmidt2018universal} or trimerons~\\cite{nishida2015polaronic,alhyder2020impurity,naidon2018two}. \n\nAccordingly, the interaction of the impurities with their surrounding leads to deformations of the latter in the vicinity of the former being manifested \nas impurity-medium bound states~\\cite{ardila2019strong} for strong attractions as well as sound wave emission~\\cite{marchukov2020shape} \nand phase-separation~\\cite{mistakidis2020many,mistakidis2019quench} for repulsive interactions. \nThese phenomena are a direct imprint of the inevitable entangled nature of these systems~\\cite{mistakidis2019quench} \nwhose non-equilibrium dynamics is far less appreciated~\\cite{skou2020non}. \nThe impurity dynamics holds the premise of unveiling even more complex \nprocesses that will shape our understanding on these settings and may be exploited in future technological applications. \nTo date remarkable demonstrations of the impurities' non-equilibrium dynamics include the spontaneous generation \nof nonlinear excitations~\\cite{grusdt2017bose,mistakidis2019correlated}, collision induced pattern \nformation~\\cite{mistakidis2019dissipative,burovski2014momentum,gamayun2018impact,knap2014quantum,tajima2019collisional}, their \nmediated correlations~\\cite{kwasniok2020correlated,mistakidis2020many,mistakidis2020induced} and relaxation \nprocesses~\\cite{mistakidis2020pump,lausch2018prethermalization,boyanovsky2019dynamics}, as well as their \ntransport properties in optical lattices~\\cite{johnson2011impurity,theel2020entanglement,keiler2020doping,keiler2019interaction}. \nIt is also important to emphasize that the above-mentioned investigations have predominantly considered a single impurity \nwhilst the effect of larger impurity concentrations leading to enhanced correlation-induced phenomena is until now largely unexplored. \nFor these latter settings, the interplay of the quantum statistics between the impurities and the host is of \nimportance especially for the induced impurity-impurity correlations. \n\nIn this context, a very promising candidate is a fermionic environment containing two bosonic impurities which can \ninteract via direct $s$-wave scattering. \nIndeed, most of the experimental and theoretical endeavors of Fermi polarons have been mainly focused on the \nlimiting case of a strongly spin imbalanced Fermi gas ~\\cite{Schmidt2012,Vlietinck2013,Mora2009,Trefzger2012,Massignan2013,Pilati2010,Schmidt2011,Sanner2012}, while the situation of bosonic impurities in a Fermi sea is arguably much less studied~\\cite{De2014,Fratini2012,cetina2016ultrafast,huber2019medium}. \nIn this setting it is very interesting to reveal the presence and nature of induced impurity-impurity interactions which are known to be suppressed for fermionic impurities~\\cite{mistakidis2019correlated,dehkharghani2018coalescence,mistakidis2019repulsive}. \nThe study of the competition between induced and direct $s$-wave interactions, with the latter being naturally absent for fermionic impurities, is \nan intriguing prospect. \nAn additional perspective is the possible emergence of impurity-impurity and impurity-medium bound states for strong attractions. \nCertainly, the identification of the above properties in the dynamical response of the system e.g. subjected to an \nimpurity-medium interaction quench~\\cite{volosniev2015real,mistakidis2019effective} as well as the characterization of the respective pattern formation especially of the host is desirable. \nIn order to address these questions we consider, as a paradigmatic setup, a one-dimensional harmonically trapped Bose-Fermi (BF) mixture consisting of two bosonic impurities immersed in a few-body fermionic environment. \nTo track the stationary properties and importantly the quantum dynamics of this impurity setting we resort to the multi-layer multi-configuration time-dependent Hartree method for atomic mixtures (ML-MCTDHX)~\\cite{cao2017unified,cao2013multi} being a variational approach that allows us to capture all the relevant correlations of the BF mixture. \n\nFor the ground state we find that the impurities and the fermionic environment \nphase-separate for strong impurity-medium repulsions~\\cite{mistakidis2019correlated,Viverit2000,lous2018probing}, while \nthey exhibit a localization \ntendency close to the trap center for large attractions. \nInterestingly, attractive induced impurity-impurity interactions~\\cite{huber2019medium} mediated by the fermionic host \nare revealed for the case of non-interacting bosons for increasing impurity-medium repulsion and attraction. \nHowever, for repulsively interacting impurities we unveil that the \ninduced interactions dominate the direct $s$-wave ones for increasing impurity-medium attractions. \n\nA quench from zero to finite repulsive impurity-medium interactions triggers a breathing motion~\\cite{boudjemaa2020breathing,kiehn2019spontaneous}, \nin each component, with an interaction dependent frequency and amplitude for the impurities. \nMoreover, a dynamical impurity-bath phase-separation takes place for quenches to strong repulsions. \nImportantly, induced impurity-impurity correlations mediated by the host are identified during the evolution of two \nnon-interacting impurities and become more pronounced for quenches to stronger repulsions. \nHowever, in the case of repulsively interacting impurities a competition of induced and direct interactions is evident \nwith the latter dominating the former and enforcing the impurities to reside in a two-body superposition. \n\nQuenching to attractive impurity-medium interactions gives rise to \na beating pattern~\\cite{mistakidis2020many} on the single-particle level which originates from the participation of two breathing \nfrequencies in the dynamics of the impurities due to the dominant presence of their attractive induced interactions. \nThe impurities show a spatial localization tendency around the trap center leading to a density accumulation of the Fermi sea at \ntheir instantaneous location. \nThe strength of the attractive induced interactions is larger compared to the reverse quench scenario and it is possible to overcome \nthe direct impurities coupling for large post-quench attractions~\\cite{mistakidis2020induced,mistakidis2020many}. \nIn all cases, we show that the degree of impurity-medium entanglement is appreciable, and exhibits a hierarchy. \nFor instance, it is larger for fixed impurity interaction and increasing \nquench amplitude. \n\n\nThis work is structured as follows. \nSection~\\ref{theory} introduces the setup under consideration [Sec.~\\ref{setup}], the employed many-body variational approach [Sec.~\\ref{method}] \nand the main observables [Sec.~\\ref{observables}] utilized for the characterization of the ground state and the dynamics of the BF mixture. \nIn section~\\ref{Ground state} we address the ground state properties of the BF mixture \nwith a particular focus on the impurity-impurity induced interactions [Sec.~\\ref{corel_ground}]. \nThe non-equilibrium dynamics upon considering a quench of the impurity-medium coupling to either repulsive [Sec.~\\ref{repulsive_quench}] \nor attractive [Sec.~\\ref{attractive quench}] interaction regimes is discussed in Sec.~\\ref{quench_dynamics}. \nThe emergent entanglement dynamics is presented in Sec.~\\ref{entanglemet_dynamics}. \nWe summarize our results and give an outlook in Sec.~\\ref{conclusion}. \nAppendix~\\ref{convergence} elaborates further on the details of our variational method and \ndelineates the convergence of the presented results exemplarily. \n\n\n\n\\section{Theoretical Background}\\label{theory} \n\n\\subsection{Setup and Hamiltonian}\\label{setup}\n\nWe consider a particle-imbalanced ultracold BF mixture containing $N_B=2$ bosonic impurities and $N_F=6$ spin-polarized fermions constituting the environment. \nThe mixture is assumed to be mass-balanced i.e. $M_B=M_F\\equiv M$ and both species are confined in the same one-dimensional (1D) harmonic trap namely $\\omega_B=\\omega_F\\equiv \\omega=0.1$. \nThis 1D geometry can be experimentally realized by imposing a strong transverse confinement ($\\omega_{\\perp}$) compared to the longitudinal ($\\omega_{\\parallel}$) \none obeying $\\omega=\\omega_{\\parallel}\/\\omega_{\\perp} \\ll 1$~\\cite{katsimiga2020observation,serwane2011deterministic,wenz2013few}. \nThe individual species of such an approximately mass-balanced BF mixture correspond, for instance, to bosonic and fermionic isotopes of the same element e.g. $^{7}$Li-$^{6}$Li~\\cite{Kempen2004, Delehaye2015} or $^{171}$Yb-$^{172}$Yb~\\cite{Honda2002}. \nThe underlying MB Hamiltonian of the above-described system reads  \n\\begin{equation}\\label{1}\n\\begin{split}\n&H = \\sum_{\\sigma = F, B}^{}\\sum_{i = 1}^{N_\\sigma} \\bigg [ -\\frac{\\hbar^2}{2M}\\bigg (\\frac{\\partial}{\\partial x^{\\sigma}_i} \\bigg)^2  \n+   \\frac{1}{2}M\\omega^2(x^\\sigma_i)^2 \\bigg ] \\\\ &  + g_{BB}\\sum_{ i < j }^{} \\delta(x^{B}_i - x^{B}_j) + g_{BF}\\sum_{ i = 1 }^{N_F} \n\\sum_{j = 1}^{N_B}\\delta(x^{F}_i - x^{B}_j).\n\\end{split}\n\\end{equation}\nOperating in the ultracold regime, $s$-wave scattering constitutes the dominant two-body interaction process and hence interparticle interactions can be modeled by a short-range contact potential~\\cite{olshanii1998atomic}. \nNote that for the spin-polarized fermions $s$-wave scattering is forbidden due to the Pauli exclusion principle~\\cite{pethick2008bose,pitaevskii2003international} and therefore \ntheir intraspecies interactions vanish. \nAccordingly, the boson-boson and boson-fermion (alias impurity-medium) 1D effective coupling constants~\\cite{olshanii1998atomic} are $g_{BB} = 4 \\hbar^2a_{BB}\/(M a^2_{\\perp}) [1 - C a_{BB}\/a^2_{\\perp,B}]^{-1}$ and $g_{BF} = 4 \\hbar^2a_{BF}\/(M a^2_{\\perp}) [1 - Ca_{BF}\/a^2_{\\perp,B}]^{-1}$ respectively. \nHere, $a_{BB}$ ($a_{BF}$) is the three-dimensional boson-boson (boson-fermion) $s$-wave scattering length and $C \\approx 1.4603$. \nThe parameter $a_\\perp = \\sqrt{\\hbar\/M\\omega_\\perp}$ denotes the transversal confinement length scale, with $\\omega_{\\perp}$ being the transversal trapping frequency. \nImportantly, the boson-boson and boson-fermion interaction strengths $g_{BB}$ and $g_{BF}$ can be experimentally tuned \neither by means of ${a^s_{BB}}$, ${a^s_{BF}}$ using Feshbach resonances~\\cite{kohler2006production,chin2010feshbach} or \nvia adjusting ${{\\omega_\\bot}}$ by employing confinement-induced resonances~\\cite{olshanii1998atomic}. \n\nBelow, we rescale the MB Hamiltonian of Eq.~(\\ref{1}) in terms of $\\hbar \\omega_\\perp$. \nAs a consequence, the length, time and interaction strengths are expressed in units of $\\sqrt{\\hbar\/M\\omega_\\perp}\\equiv a_{\\perp}$, $\\omega^{-1}_\\perp$ and $\\sqrt{\\hbar^3 \\omega_\\perp\/M}$, respectively. \nIt is also worth mentioning that a BF mixture with $N_B \\ll N_F$, as the one considered herein, features supressed \nthree-body recombination particle losses, since their rate is known \nto be proportional to $N^2_B N_F$~\\cite{Helfrich2010}. \n\nIn the following, we characterize the ground-state properties of the highly particle-imbalanced BF mixture particularly focusing on the emergent correlation patterns and unveiling, for instance, phase-separation processes as well as identify impurity-impurity induced interactions for varying boson-boson and impurity-medium interaction strengths, see Sec.~\\ref{Ground state}. \nRecall that in the absence of an external confinement the two species are miscible by means that they spatially overlap when $g^2_{BF} < g_{BB}$, otherwise they phase-separate~\\cite{mistakidis2018correlation,Viverit2000,Roth2002,lous2018probing,mistakidis2019correlated}. \nIn the presence of an external trap and also away from the thermodynamic limit the above-mentioned relation is modified, i.e. $g_{BF}$ should \nbecome substantially larger than $g_{BB}$ in order to achieve the phase-separation. \nSubsequently, we trigger the non-equilibrium dynamics of the BF mixture by applying a quench of the impurity-medium interaction strength ($g_{BF}$) from zero to either repulsive [Sec.~\\ref{repulsive_quench}] or attractive [Sec.~\\ref{attractive quench}] couplings. \nImportantly, within these latter post-quench interaction regimes impurity-impurity correlations are finite whilst they vanish for the initial state. \nThus, the system is driven towards regions of finite impurity-impurity interactions aiming at exploring their dynamical fate, the consequent pattern formation and the associated build-up of correlations.   \n\n\n\n\\subsection{Variational wavefunction ansatz and quantum dynamical approach} \\label{method} \n\nTo investigate the ground-state and most importantly the quench dynamics of the particle-imbalanced BF mixture we solve the underlying MB Schr{\\\"o}dinger \nequation using the variational ML-MCTDHX approach~\\cite{cao2017unified,cao2013multi}.   \nIt is based on expanding the MB wavefunction in terms of a time-dependent and variationally optimized basis. \nThis asset enables us to capture both the inter- and the intraspecies correlations of the binary system in a computationally efficient manner compared to methods relying on a time-independent basis set.  \n\nThe MB wavefunction, $\\Psi_{\\rm MB}$, is initially expressed in the form of a truncated Schmidt decomposition of rank $D$~\\cite{Horodecki2009}. \nNamely \n\\begin{equation} \\label{4}\n\\Psi_{\\rm MB}(\\vec{x}^B, \\vec{x}^F;t) = \\sum_{k = 1}^{D} \\sqrt{\\lambda_k(t)}\\Psi^B_k(\\vec{x}^B;t)\\Psi^F_k(\\vec{x}^F;t). \n\\end{equation}\nThe values of the Schmidt coefficients, $\\lambda_k(t)$, characterize the degree of entanglement of the binary system. \nIn decreasing order they are also known as natural species populations of the $k$-th species function. Evidently, the \nsystem is entangled~\\cite{roncaglia2014,Horodecki2009,mistakidis2018correlation} in the case that more than a single \ncoefficients $\\lambda_k(t)$ exhibit an non-zero population. \nThen, the many-body state [Eq.~(\\ref{4})] is a superposition of the respective species states instead of being a direct product of only two states (non-entangled case). \n\nAs a next step, each of the above-mentioned species functions is expanded in terms of the determinants and permanents of $d_\\sigma$ distinct time-dependent fermionic and bosonic single particle functions (SPFs) respectively. \nTherefore, each $\\Psi^{\\sigma}_k(\\vec{x}^{\\sigma};t)$ reads  \n\\begin{equation}\\label{5}\n\\begin{split}\n&\\Psi_k^{\\sigma}(\\vec x^{\\sigma};t) = \\sum_{\\substack{l_1,\\dots,l_{d_{\\sigma}} \\\\ \\sum l_i=N}} C_{k,(l_1,\n\\dots,l_{d_{\\sigma}})}(t)\\sum_{i=1}^{N_{\\sigma}!} \\big ( {\\rm sign} (\\mathcal{P}_i) \\big ) ^{\\zeta} \\\\ & \\times \\mathcal{P}_i\n \\left[ \\prod_{j=1}^{l_1} \\varphi_1^{\\sigma}(x_j;t) \\cdots \\prod_{j=1}^{l_{d_{\\sigma}}} \\varphi_{d_{\\sigma}}^{\\sigma}(x_{K(d_{\\sigma})+j};t) \\right]. \n  \\end{split}\n\\end{equation} \nIn this expression, $C_{k,(l_1,....., l_{d_{\\sigma}})}(t)$ denote the time-dependent expansion coefficients of a particular \ndeterminant for fermions or permanent for bosons and $n_i(t)$ is the occupation number of the SPF, $\\varphi_i(x;t)$. \nThe index $\\zeta = 0, 1 $ for bosons and fermions respectively and $\\mathcal{P}$ is the permutation operator exchanging the particle configuration $x_{\\nu}^{\\sigma}$, $\\nu=1,\\dots,N_{\\sigma}$ within the SPFs. \nAlso, $\\rm sign(\\mathcal{P}_i)$ is the sign of the corresponding permutation and $K(r)\\equiv l_1+l_2+\\dots+l_{r-1}$,\nwhere $l_{r}$ is the occupation of the $r$-th SPF and $r\\in\\{1,2,\\dots,d_{\\sigma}\\}$. \nWe remark that the bosonic subsystem is termed intraspecies correlated if more than one SPF is occupied otherwise it is fully coherent~\\cite{lode2020colloquium}, see also the discussion below. \nOn the other hand, the fermionic species exhibit beyond non-trivial Hatree-Fock correlations when more than $N_F$ eigenvalues possess a macroscopic population~\\cite{mistakidis2019correlated,erdmann2019phase}. \n\nThe time-evolution of the $(N_B+N_F)$-body wavefunction obeying the MB Hamiltonian of Eq.~(\\ref{1}) is determined by calculating the corresponding ML-MCTDHX equations of motion~\\cite{cao2013multi}. \nThe latter are found by performing e.g. the Dirac-Frenkel variational principle~\\cite{Dirac193,Frenkel1934} for the MB ansatz provided by Eqs.~\\eqref{4} and \\eqref{5}. \nAs a result we obtain a set of $D^2$ linear differential equations of motion for the coefficients $\\lambda_k(t)$ being coupled to $D[$ ${N_B+d_B-1}\\choose{d_B-1}$+${d_F}\\choose{N_F}$] non-linear integro-differential equations for the species functions and $d_F+d_B$ integro-differential equations for the SPFs. \nFinally, let us mention in passing that the variational ML-MCTDHX ansatz can be easily reduced to different levels of approximation. \nAs a case example, the corresponding mean-field wavefunction ansatz of the BF mixture corresponds to the case of $D = d_B = 1$ and $d_F = N_F$ while the respective mean-field equations of motion are retrieved \nby following a variational principle, see e.g. for details \\cite{lode2020colloquium,kohler2019dynamical}. \n\n\n\\subsection{Observables and analysis}\\label{observables}\n\nIn the following, we briefly introduce the basic observables that will be employed in the remainder of our work in order to characterize \nboth the stationary properties and the non-equilibrium dynamics of the BF mixture. \nA particular emphasis is paid on the impurities subsystem. \nTo visualize the spatial distribution of the $\\sigma=B,F$ species, i.e. the impurities and the medium respectively, on the single-particle \nlevel we invoke the corresponding one-body reduced density matrix \n\\begin{equation}\\label{1BD}\n\\rho^{(1)}_{\\sigma}(x, x';t)=\\langle \\Psi_{\\rm MB}(t)| \\hat{\\Psi}_\\sigma^{\\dagger}(x) \\hat{\\Psi}_{\\sigma}(x')  | \\Psi_{\\rm MB}(t) \\rangle.\n\\end{equation}\nHere, $\\hat{\\Psi}_{B}(x)$ [$\\hat{\\Psi}_{F}(x)$] is the so-called bosonic [fermionic] field operator acting on position $x$ and satisfying the \nstandard commutation [anti-commutation] relations~\\cite{pethick2008bose,pitaevskii2003international}. \nThe diagonal of $\\rho^{(1)}_{\\sigma}(x, x';t)$ is the well-known one-body density of the $\\sigma$-species \ni.e. $\\rho^{(1)}_{ \\sigma}(x;t) = \\rho^{(1)}_ {\\sigma}(x, x' = x;t)$~\\cite{lode2020colloquium}. \nThe latter is accessible in ultracold atom experiments using the single-shot absorption imaging technique~\\cite{sakmann2016single,Bloch2008} \nand especially for few atoms can be retrieved by averaging over a sample of single-shots~\\cite{mistakidis2018correlation,Klein2005,mistakidis2019dissipative}. \nWe remark that the eigenfunctions of the $\\sigma$-species one-body reduced density matrix are known as the $\\sigma$-species natural orbitals, namely $\\phi^{\\sigma}_i$. \nIn this sense, when more than $N_F$ (one) fermionic (bosonic) natural orbitals are significantly populated the corresponding subsystem is called \nfragmented or intraspecies correlated~\\cite{mistakidis2018correlation,lode2020colloquium}. \nAccordingly, the underlying degree of fragmentation can be quantified via measuring $F_F(t) = 1 -\\sum_{i = 1}^{N_F}n^F_i(t)$ and $F_B(t) = 1 - n^B_1(t)$ for the fermionic and the bosonic subsystems respectively. \nHere we consider that the population of the total number of orbitals are normalized to unity i.e. \n$\\sum_{i=1}^{d_F}n^{F}_i(t) = 1$ and $\\sum_{i=1}^{d_B}n^{B}_i(t)=1$. \nRecall that in the MF limit of the BF mixture~\\cite{pitaevskii2003international,mistakidis2019correlated,karpiuk2004soliton} where \n$\\Psi_{\\rm MB}(\\vec{x}^F, \\vec{x}^B;t) \\rightarrow \\Psi_{MF}(\\vec{x}^{F}, \\vec{x}^B ; t) $ the natural populations \nof the fermionic and the bosonic species satisfy the constraints $\\sum_{i=1}^{N_F}n^{F}_i(t) = 1$, $n^F_{i > N_F}(t) = 0$, \nand $n^{B}_1(t) = 1$, $n^{B}_{i >1}(t) = 0$. \n\nThe emergence of impurity-medium entanglement can be identified by calculating the Schmidt coefficients, $\\lambda_k(t)$, participating in the MB wavefunction ansatz as described by Eq. (\\ref{4}). \nIndeed, in the case that more than one coefficients are populated, i.e. $\\lambda_{k>1}(t)\\neq 0$, then the MB wavefunction is not a single product state and the system is entangled~\\cite{Horodecki2009,mistakidis2018correlation}. \nThe Schmidt coefficients are essentially the eigenvalues of the species reduced density matrix namely  \n$\\rho^{N_{\\sigma}} (\\vec{x}^{\\sigma}, \\vec{x}'^{\\sigma};t)=\\int d^{N_{\\sigma'}} x^{\\sigma'} \\Psi^*_{\\rm MB}(\\vec{x}^{\\sigma}, \n\\vec{x}^{\\sigma'};t) \\Psi_{\\rm MB}(\\vec{x}'^{\\sigma},\\vec{x}^{\\sigma'};t)$, with $\\vec{x}^{\\sigma}=(x^{\\sigma}_1, \\cdots,x^{\\sigma}_{N_{\\sigma-1}})$, and $\\sigma\\neq \\sigma'$. \nConsequently, in order to determine the degree of the impurity-medium entanglement we use the Von-Neumann entropy~\\cite{Catani2009, Horodecki2009} given by \n\\begin{equation}\\label{VN}\nS_{VN}(t) = - \\sum_{k = 1}^{D} \\lambda_k(t) \\ln[\\lambda_k(t)].\n\\end{equation}\nIt becomes apparent that $S_{VN}(t) \\geq 0$ only when $\\lambda_{k>1}(t)\\neq 0$ meaning that entanglement is present. \nFor instance, in the mean-field limit where $\\lambda_1 (t) = 1$, $\\lambda_{k>1}(t)= 0$ and entanglement is absent it holds that $S_{VN}(t)=0$. \n\nTo infer the role of impurity-impurity and fermion-fermion two-body correlation processes in the ground state as well as \nin the dynamics of the BF mixture in a spatially resolved manner we resort to the diagonal of the two-body reduced \ndensity matrix~\\cite{mistakidis2018correlation,lode2020colloquium,sakmann2008reduced}\n\\begin{equation}\\label{2BD}\n\\begin{split}\n \\rho^{(2)}_{\\sigma \\sigma}(x, x';t) = & \\langle \\Psi_{\\rm MB}(t)| \\hat{\\Psi}^{\\dagger}_{\\sigma}(x')\n \\hat{\\Psi}^{\\dagger}_{\\sigma}(x) \\hat{\\Psi}_{\\sigma}(x') \\\\ &\n \\times \\hat{\\Psi}_\\sigma(x) | \\Psi_{\\rm MB}(t) \\rangle.\n \\end{split}\n\\end{equation} \nThis measure refers to the probability of detecting simultaneously one impurity $\\sigma=B$ (fermionic, $\\sigma=F$) particle located at $x$ and another one at $x'$. \nIn that light, it reveals the occurrence of impurity-impurity (fermion-fermion) two-body correlations and thus provides insights on how \nthe two bosons (fermions) behave with respect to one another~\\cite{mistakidis2019correlated,erdmann2019phase,mistakidis2020many,mistakidis2020induced}.  \n\nTo estimate the strength of the effective interactions between the two bosonic impurities \nwe utilize their relative distance~\\cite{mistakidis2019correlated,mistakidis2020many,mistakidis2019repulsive} defined as\n\\begin{equation} \\label{7}\n\\mathcal{D}_{\\rm rel}(t) = \\frac{\\int dx_1 dx_2 \\abs{x_1 -x_2} \\rho^{(2)}_{BB} (x_1, x_2; t)}{\\langle \\Psi_{\\rm MB}(t) | \\hat{N}_B(\\hat{N}_B -1) | \n\\Psi_{\\rm MB}(t) \\rangle}.\n\\end{equation}\nHere, $\\hat{N}_B$ is the bosonic number operator and $\\rho^{(2)}_{BB} (x_1, x_2; t)$ denotes the \ntwo-body density matrix [Eq.~(\\ref{2BD})] of the bosonic impurities subsystem. \nThe relative distance can be experimentally accessed using {\\it in-situ} spin-resolved single-shot measurements~\\cite{bergschneider2018spin}, \nwhere in particular the actual shape of $\\mathcal{D}_{\\rm rel}(t)$ can be retrieved by averaging over a sample of the individually obtained images. \n\n\n\n\\section{Ground state properties of two bosonic impurities in a fermionic environment}\\label{Ground state} \n\nWe consider $N_B = 2$ bosonic impurities in a fermionic finite-sized medium composed of $N_F=6$ spin-polarized fermions. \nRecall that an one-dimensional Fermi sea with $N_F>5$ atoms approaches the behavior of a many-body fermionic environment, see for instance \nRef.~\\cite{wenz2013few} for a corresponding experimental verification. \nIn our setting we have checked that our results, to be presented below, regarding both the ground state and the dynamics remain qualitatively the same also for $N_F=8$ (not shown here for brevity). \nThe system is mass-balanced and both species are trapped in the same harmonic oscillator of frequency $\\omega = 0.1$, unless it is stated otherwise. \nBelow, we examine the ground state characteristics of the composite system with a particular focus on the impurities properties for attractive and repulsive impurity-medium interactions. \nIn order to discriminate between direct and effective impurity-impurity interaction effects we analyze both the cases of non-interacting and interacting impurities. \nThe impact of the impurities mass on their induced interactions mediated by the fermionic environment is also discussed. \nAnother objective of our analysis is to unveil the spatial distributions of each species, discuss possibly emerging phases of the BF mixture as well as their associated correlation properties for varying impurity-medium interactions. \nTo obtain the ground state of the BF mixture governed by Eq.~\\eqref{1} we employ either the imaginary-time propagation or the improved relaxation \nmethod within ML-MCTDHX~\\cite{cao2017unified}.  \n\n\n\\subsection{Single-particle density distribution}\\label{ground_state_density} \n\nLet us first inspect the spatial configuration of the ground state of the bosonic impurities and the fermionic sea for varying impurity-medium interaction strength $g_{BF}$. \nFor this reason, we employ the corresponding single-particle densities $\\rho^{(1)}_{F}(x)$ and $\\rho^{(1)}_{B}(x)$ with respect to $g_{BF}$ [Fig.~\\ref{spd_g}] \nfor both the cases of two non-interacting $g_{BB}=0$ [Figs.~\\ref{spd_g}($a_1$), ($b_1$)] and two repulsively interacting \nwith $g_{BB}=1$ [Figs. \\ref{spd_g} ($a_2$), ($b_2$)] bosonic impurities. \nOverall, we observe that the behavior of both the impurities and the medium depends strongly on the value of $g_{BF}$. \nAlso $\\rho^{(1)}_{F}(x)$ exhibits six shallow local density maxima [Figs.~\\ref{spd_g}($c$)-($h$)] almost irrespectively of $g_{BF}$, which \nindicates the presence of six fermions~\\cite{kwasniok2020correlated}, see \nalso the remark in \\footnote{Note that for increasing attraction, i.e. $g_{BF}<-2.5$, a major portion of $\\rho^{(1)}_{F}(x)$ resides around \n$x=0$ and its two central local maxima come very close to each other and eventually merge for very strong attractions.}. \nInterestingly, the shape of $\\rho^{(1)}_{F}(x)$ for fixed $g_{BF}$ remains almost unchanged between the $g_{BB}=0$ and the $g_{BB}=1$ cases, \nsee Figs.~\\ref{spd_g}($a_1$) and ($a_2$) as well as Figs.~\\ref{spd_g}($c$)-($h$). \nOn the other hand, $\\rho^{(1)}_{B}(x)$ at a certain value of $g_{BF}$ is affected by the direct impurity-impurity interactions since \nfor $g_{BB}=1$ it becomes slightly broader than for $g_{BB}=0$ especially when $-1.5<g_{BF}<1.5$, see Figs.~\\ref{spd_g}($d$)-($g$). \nAs we shall argue below, this difference is caused by the presence of attractive induced impurity-impurity interactions mediated by the Fermi sea and become more \npronounced when $g_{BB}=0$. \nA similar effect has also been discussed in the context of two bosonic impurities in a BEC bath~\\cite{mistakidis2020many,mistakidis2020pump}. \nThe impact of $g_{BB}$ on the behavior of the impurities and the interplay between direct and  attractive impurity-impurity induced interactions \nwill be discussed in detail in Sec.~\\ref{corel_ground}.  \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig1.eps}\n\\caption{Ground state single-particle density of ($a_1$), ($a_2$) the six fermions and ($b_1$),($b_2$) the two bosons for varying impurity-medium interaction strength $g_{BF}$ when [($a_1$), ($b_1$)] $g_{BB}=0$ and [($a_2$), ($b_2$)] $g_{BB}=1$. \n($c$)-($h$) Density profiles $\\rho^{(1)}_{B}(x)$ and $\\rho^{(1)}_{F}(x)$ at specific $g_{BF}$ values and for both non-interacting and repulsively interacting impurities (see legend). \nThe BF mixture consists of $N_B = 2$ bosonic impurities and $N_F = 6$ fermions both trapped in a harmonic oscillator with $\\omega = 0.1$.}  \n\\label{spd_g}\n\\end{figure}\n\nIn particular, for $-1.5<g_{BF}<1.5$ the Fermi sea exhibits a broad spatial distribution as identified by $\\rho^{(1)}_{F}(x)$ enclosing the \nbosonic impurities whose $\\rho^{(1)}_{B}(x)$ is located around the trap center [Figs.~\\ref{spd_g}($d$)-($f$)]. \nTherefore, for these impurity-medium interactions the bosons and the Fermi sea show a miscible behavior~\\cite{mistakidis2019correlated} independently of $g_{BB}$, \nsee Figs. \\ref{spd_g} ($a_1$), ($a_2$). Turning to attractive $g_{BF}<-1.5$ we observe a drastically altered behavior of both the impurities \nand the fermionic ensemble. \nMore specifically, the impurities are strongly localized in the vicinity of the trap center since $\\rho^{(1)}_{B}(x)$ shows a peaked structure \nat $-1<x<1$ having a sech-like shape [Fig.~\\ref{spd_g}($b_1$), ($b_2$)]. \nThis spatial localization tendency of the impurities signals the formation of an attractively bound pair~\\cite{mistakidis2020many,mistakidis2020pump} \nas a consequence of the presence of strong attractive induced impurity-impurity interactions \\cite{mistakidis2020many,mistakidis2020induced}, \nsee for more details Sec.~\\ref{corel_ground}. \nSimultaneously $\\rho^{(1)}_{F}(x)$ majorly accumulates at the same location by developing a density hump, see for instance \nFigs.~\\ref{spd_g}($a_1$), ($a_2$) and ($c$), while its background is significantly reduced when compared to smaller $g_{BF}$ values. \nThe latter behavior of $\\rho^{(1)}_{F}(x)$ indicates that the fermions of the medium lie very close to the impurities suggesting \nthat for even larger attractions a bound state between an impurity and an atom of the Fermi sea may be formed~\\cite{mistakidis2019dissipative}, a phenomenon which is not further examined herein. \nOf course, the fermions of the medium can not reside exactly at the same location due to the Pauli exclusion \nprinciple~\\cite{pethick2008bose,pitaevskii2003international}, see also the discussion in Sec.~\\ref{corel_ground}. \n\nOn the other hand, for strong repulsive impurity-bath interactions such that $g_{BF}>1.5$ the spatial configuration of the system and especially of the Fermi sea is significantly changed compared to smaller values of $g_{BF}$. \nIndeed, a local density dip builds upon $\\rho^{(1)}_{F}(x)$ around $x\\approx 0$ [Fig.~\\ref{spd_g}($g$)] which becomes more pronounced for increasing $g_{BF}$ and for $g_{BF}>2$ $\\rho^{(1)}_{F}(x)$ is segregated into two fragments residing in the left and right side with respect to $x=0$ [Figs. \\ref{spd_g} ($a_1$), ($h$)]. \nNote that each of the fragments has three local density maxima indicating that predominantly three fermions populate each of them and also reflects the fact that the first six lowest-lying single-particle eigenstates of the harmonic trap majorly contribute to the fermionic MB wavefunction. \nThe impurities density $\\rho^{(1)}_{B}(x)$ lies in between the two fragments of $\\rho^{(1)}_{F}(x)$ and therefore an impurity-medium phase-separation process takes place~\\cite{mistakidis2018correlation}, see e.g. Figs. \\ref{spd_g} ($a_1$), ($a_2$) and ($h$). \nThis procedure is identified by the small spatial overlap among the components~\\cite{mistakidis2018correlation} which becomes suppressed for increasing $g_{BF}$. \nWe remark that the phase-separation region is shifted to larger $g_{BF}$ values when $g_{BB}$ is finite, compare in particular Figs. \\ref{spd_g} ($a_1$) and ($a_2$). \nIndeed, phase-separation occurs when the interspecies interaction energy overcomes the intraspecies one and thus a larger $g_{BF}$ is required for \nincreasing $g_{BB}$~\\cite{mistakidis2018correlation} in order to accomplish this process. \nIt is also worth mentioning at this point that a system of two fermionic impurities immersed in a bosonic bath exhibits a similar phase-separation behavior at repulsive $g_{BF}$ but in this case the impurities reside at the edges of \nthe bosonic medium~\\cite{mistakidis2019correlated}. \n\nThe above-described phase-separation process as well as the localization tendency of the components taking place at large repulsive and attractive impurity-medium interactions respectively can be intuitively understood in terms of an effective potential approach~\\cite{mistakidis2020many,mistakidis2019quench,kiehn2019spontaneous}. \nFor this picture one can consider an effective potential for the impurities [Fermi sea] constructed by superimposing the single-particle density of the Fermi sea [impurities] to the external harmonic trap, namely $V_{\\rm eff}^{B}(x) = \\frac{1}{2}m \\omega^2 x^2 + g_{BF}\\rho^{(1)}_{F}(x)$ [$V_{\\rm eff}^{F}(x) = \\frac{1}{2}m \\omega^2 x^2 + g_{BF}\\rho^{(1)}_{B}(x)$]. \nReferring to the impurities subsystem at strong repulsive $g_{BF}$ their effective potential $V_{\\rm eff}^{B}(x)$ corresponds to a deformed harmonic trap due to $\\rho^{(1)}_{F}(x)$, see e.g. Fig. \\ref{spd_g} ($h$). \nIn this sense the impurities reside around the trap center possessing a Gaussian-like spatial distribution [Fig.~\\ref{spd_g}($h$)]. \nOn the other hand, for $g_{BF}>0$ the corresponding $V_{\\rm eff}^{F}(x)$ has a double-well like structure where the role of the potential barrier at $x=0$ is played by $\\rho^{(1)}_{B}(x)$. \nIn turn, this $V_{\\rm eff}^{F}(x)$ enforces the splitting of $\\rho^{(1)}_{F}(x)$ into two fragments, see e.g. Fig.~\\ref{spd_g}($h$). \nNote also here that for $g_{BB}=1$ the maximum of $\\rho^{(1)}_{B}(x)$ is smaller compared to the $g_{BB}=0$ case [Fig.~\\ref{spd_g}($g$)]. \nThis gives rise to a shallower double-well effective potential for fixed $g_{BF}$ and thus the barrier height that allows for phase-separation is achieved for larger values of $g_{BF}$ when $g_{BB}$ is finite. \nA similar argumentation can also be applied for attractive $g_{BF}$ where, for instance, the aforementioned localization tendency of $\\rho^{(1)}_{B}(x)$ is essentially determined by the hump structure building upon $\\rho^{(1)}_{F}(x)$ [Fig. \\ref{spd_g} ($c$)] and vice versa due to back-action. \nFor more details on the range of applicability of this effective potential picture we refer the interested \nreader to Refs.~\\cite{mistakidis2020many,mistakidis2019dissipative,mistakidis2019quench,kiehn2019spontaneous}. \n\n\n\n\\subsection{Impurity-impurity induced interactions}\\label{corel_ground}\n\nThe impurities being immersed in the Fermi sea are dressed by its excitations forming quasiparticles, herein Fermi \npolarons~\\cite{schmidt2018universal,massignan2014polarons,mistakidis2019repulsive}. \nAn intriguing property of the generated quasiparticles is the emergence of attractive induced interactions among them mediated by their host~\\cite{dehkharghani2018coalescence,mistakidis2020many,mistakidis2019repulsive} and that they can possibly form a bound pair for strong impurity-medium attractions~\\cite{schmidt2018universal,massignan2014polarons}. \nTo identify such quasiparticle related mechanisms in the ground state of the BF mixture we subsequently inspect the relative \ndistance $\\mathcal{D}_{\\rm rel}$ [Eq.~(\\ref{7})] and the spatially resolved two-body reduced density matrix $\\rho^{(2)}_{BB}(x_{1}, x_2)$ of \nthe bosonic impurities~\\cite{mistakidis2020induced} for different impurity-medium interaction strengths, see \nFigs.~\\ref{Relative_dis}, \\ref{2bd_g0} and \\ref{2bd_g1}. \nRecall that $\\rho^{(2)}_{BB}(x_1,x_2)$ measures the probability of finding a boson at position $x_1$ while the second one is located at $x_2$. \nImportantly, the combination of the behavior of $\\mathcal{D}_{\\rm rel}$ and $\\rho^{(2)}_{BB}(x_{1}, x_2)$ enables us to infer the presence \nand strength of the attractive induced interactions as well as the spatial configuration of the impurities~\\cite{mistakidis2020many}.   \nBelow, we discuss the cases of both non-interacting ($g_{BB}=0$) and repulsively interacting ($g_{BB}=1.0$) impurities as well as the \neffect of a mass-imbalance between the impurities and their bath. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig2.eps}\n\\caption{Relative distance $\\mathcal{D_{\\rm rel}}$ between the two bosonic impurities in the ground state of the BF mixture with \nrespect to the impurity-medium interaction strength $g_{BF}$. The relative distance is presented for the cases of two \nnon-interacting ($g_{BB} = 0$), and  interacting ($g_{BB} = 1$) impurities in a mass-balanced as well as a mass-imbalanced system (see legend). \nThe medium consists of $N_F = 6$ fermions, while the BF mixture is confined in a harmonic trap with frequency $\\omega = 0.1$.}  \n\\label{Relative_dis}\n\\end{figure}\n\nThe corresponding relative distance $\\mathcal{D}_{\\rm rel}$ between the impurities and their two-body density $\\rho^{(2)}_{BB}(x_{1}, x_2)$ \nfor a mass-balanced BF system containing non-interacting impurities are presented in Fig.~\\ref{Relative_dis} and \nFigs.~\\ref{2bd_g0}($a_1$)-($a_5$) respectively as a function of the impurity-medium interaction strength $g_{BF}$ ranging \nfrom attractive to repulsive values. \nIt becomes apparent that $\\mathcal{D}_{\\rm rel}$ gradually decreases, when compared to its value for $g_{BF}=0$, as $\\abs{g_{BF}}$ is \nincreased towards the attractive or the repulsive interaction regime. This overall decreasing behavior of $\\mathcal{D}_{\\rm rel}$ for \nlarger $\\abs{g_{BF}}$ indicates that for finite attractive or repulsive impurity-medium interactions the impurities move close to each other as compared to the $g_{BF}=0$ scenario. \nThe latter tendency suggests the emergence of attractive impurity-impurity induced interactions~\\cite{huber2019medium,mistakidis2020many}. \nInterestingly, $\\mathcal{D}_{\\rm rel}$ tends to approach a constant value which is different for strong repulsions ($g_{BF}>3$) and \nattractions ($g_{BF}<-3$). \nIndeed, the saturation value of $\\mathcal{D}_{\\rm rel}$ for strong repulsions is somewhat larger when compared to the corresponding \nvalue for strong attractions. \nThis means that for attractive $g_{BF}$ the impurities are substantially closer with respect to one another than in the repulsive case. \nConcluding, $\\mathcal{D}_{\\rm rel}$ signals the presence of induced impurity-impurity interactions, which are manifested to be attractive \nin general, irrespectively of the sign of the impurity-medium coupling~\\cite{huber2019medium,mistakidis2020many}. \n\n\nTo confirm the existence of attractive impurity-impurity induced interactions when $g_{BB}=0$ we next rely on the impurities two \nparticle density $\\rho^{(2)}_{BB}(x_1, x_2)$, see Figs.~\\ref{2bd_g0} ($a_1$)-($a_5$). \nThis quantity allows us to explicitly identify the spatial distribution of impurities. \nAs it can be seen, irrespectively of the value of $g_{BF}$ the two non-interacting bosons prefer to reside together close to the \ntrap center since $\\rho^{(2)}_{BB}(x_1, x_2)$ shows a maximum value in the vicinity of $x_1=0$, $x_2=0$, see Figs.~\\ref{2bd_g0}($a_1$)-($a_5$). \nIn particular, for $g_{BF} = 0$ [Fig.~\\ref{2bd_g0}($a_3$)] $\\rho^{(2)}_{BB}(x_1, x_2)$ has a circularly symmetric shape in the ($x_1,x_2$)-plane while showing a peak around $x_1=x_2=0$. \nThis can be understood by the fact that in the absence of any correlation with the majority species there is no induced interaction among the bosons. \nHence, the probability of finding the two bosons together at $x_1=x_2$ or one at $x_1$ and the other at $x_2=-x_1$ is the same and becomes maximal at the trap minimum i.e. at $x_1=x_2=0$. \nHowever, for a finite $g_{BF}$ the shape of $\\rho^{(2)}_{BB}(x_1, x_2)$ is significantly altered when compared to the $g_{BF}=0$ case \nsince predominantly the diagonal $\\rho^{(2)}_{BB}(x_1,x_2=x_1)$ is populated. \nIn fact, $\\rho^{(2)}_{BB}(x_1, x_2)$ becomes more elongated along the diagonal ($x_1 = x_2$) with increasing $\\abs{g_{BF}}$, while it \nshrinks across its anti-diagonal ($x_2 = -x_1$), see Figs.~\\ref{2bd_g0}($a_4$)-($a_5$) and Figs.~\\ref{2bd_g0}($a_1$)-($a_2$). \nThis means that the probability of detecting the two bosons at two different positions is substantially smaller than that of being \nclose together. \nTherefore, an effective attractive interaction between the impurities is established and occurs for both attractive and repulsive \nimpurity-medium interactions. \nImportantly, within the attractive $g_{BF}$ regime the shrinking of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1, x_2)$ is much more \npronounced than on the repulsive $g_{BF}$ side, compare in particular Figs.~\\ref{2bd_g0}($a_1$)-($a_2$) with Figs.~\\ref{2bd_g0}($a_4$)-($a_5$). \nThe latter observation supports our previous statement in terms of $\\mathcal{D}_{\\rm rel}$ of a much stronger effective attraction \nbetween the impurities for negative than positive $g_{BF}$, a result that also holds for Bose polarons as it has been \ndemonstrated in Ref.~\\cite{mistakidis2020many}. \nMoreover, the pronounced elongation along the diagonal accompanied by the strong suppression of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1, x_2)$ e.g. for $g_{BF}=-3$ is indicative of a bound state being formed between the impurities known as a bipolaron state~\\cite{mistakidis2020many,camacho2018bipolarons,schmidt2018universal}.     \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig3.eps}\n\\caption{Reduced two-body ($a_1$)-($a_5$) boson-boson $\\rho^{(2)}_{BB}(x_1, x_2)$, and ($b_1$)-($b_5$) fermion-fermion $\\rho^{(2)}_{FF}(x_1, x_2)$ \ndensity in the ground state of the BF mixture for selective impurity-medium interaction strengths (see legend).  \nThe system contains $N_B =2$ non-interacting ($g_{BB}=0$) bosonic impurities and $N_F = 6$ fermions. \nIt is further confined in a harmonic trap of $\\omega = 0.1$. }  \n\\label{2bd_g0}\n\\end{figure}\n\nNext, we turn our attention to repulsively interacting bosonic impurities with $g_{BB}=1$ aiming to investigate the competition between attractive induced interactions and direct $s$-wave ones. \nTo this end, we measure the impurities relative distance [Fig.~\\ref{Relative_dis}] and their two-particle density [Fig.~\\ref{2bd_g1}($a_1$)-($a_5$)] for distinct values of $g_{BF}$. \nAs expected, in the absence of direct impurity-impurity interactions i.e. $g_{BB}=0$ the impurities distance $\\mathcal{D}_{\\rm rel}$ is in general smaller than the corresponding for two repulsively interacting ones with $g_{BB} = 1$. \nThis difference becomes maximal for zero impurity-medium interactions, namely $g_{BF}=0$. \nIndeed, $\\mathcal{D}_{\\rm rel}$ decreases for a larger positive or negative $g_{BF}$ tending to approach a constant value which is smaller for attractive $g_{BF}$ interactions. \nConsequently, also the difference $\\mathcal{D}_{\\rm rel}(g_{BB} = 1)-\\mathcal{D}_{\\rm rel}(g_{BB}=0)$ gradually decreases and becomes constant for increasing $\\abs{g_{BF}}$. \nA direct comparison between $\\mathcal{D}_{\\rm rel}(g_{BB}=1)$ and $\\mathcal{D}_{\\rm rel}(g_{BB}=0)$ reveals that the \ndistance saturates at relatively larger (smaller) positive (negative) $g_{BF}$ values when $g_{BB}=1$. \nFor instance, the decreasing rate of $\\mathcal{D}_{\\rm rel}$ in the attractive $g_{BF}$ regime is much larger in the $g_{BB}=1$ scenario before showcasing a saturation tendency around $g_{BF}=-4$ towards $\\mathcal{D}_{\\rm rel} \\approx 0.5$. \nThe above-described overall behavior of $\\mathcal{D}_{\\rm rel}$ for varying $g_{BF}$ suggests the occurence of attractive induced interactions for a finite $g_{BF}$ despite the existence of direct $s$-wave ones. \nNevertheless, as we shall explicate in the following the direct interactions dominate the induced ones at least for repulsive impurity-medium \ncouplings, a result which reveals that the strength of induced interactions is larger in the negative $g_{BF}$ regime.   \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig4.eps}\n\\caption{Reduced two-body ($a_1$)-($a_5$) impurity-impurity $\\rho^{(2)}_{BB}(x_1, x_2)$, and ($b_1$)-($b_5$) fermion-fermion $\\rho^{(2)}_{FF}(x_1,x_2)$ \ndensity in the ground state of the BF mixture for different values of the boson-medium coupling constant $g_{BF}$ (see legend). \nThe system consists of $N_B =2$ interacting with $g_{BB}=1$ bosonic impurities and $N_F = 6$ fermions while it is trapped in a harmonic oscillator with frequency $\\omega = 0.1$.}  \n\\label{2bd_g1}\n\\end{figure}\n\nIndeed by inspecting the impurities two-body density $\\rho^{(2)}_{BB}(x_1,x_2)$, illustrated in Figs.~\\ref{2bd_g1}($a_1$)-($a_5$), the following conclusions can be immediately drawn. \nThe circularly symmetric pattern of $\\rho^{(2)}_{BB}(x_1,x_2)$ occurring for $g_{BB}=0$ when $g_{BF}=0$ [Fig.~\\ref{2bd_g0}($a_3$)] is completely modified for repulsively interacting impurities [Fig.~\\ref{2bd_g1}($a_3$)]. \nThis modification favors a pattern whose diagonal is depleted, giving rise to a correlation hole~\\cite{erdmann2019phase,kwasniok2020correlated}, whilst the anti-diagonal develops two symmetric lobes with respect to $x_1=x_2$ and it is predominantly populated. \nThis is an explicit imprint of the direct $s$-wave interaction among the impurities and means that the probability of finding two bosons exactly at the same position is vanishingly \nsmall in contrast to the situation where each boson resides on a separate side in terms of the trap center. \nSwitching on $\\abs{g_{BF}}$ introduces deformations in the shape of $\\rho^{(2)}_{BB}(x_1, x_2)$ and in particular in the position of its anti-diagonal lobes which suggests that the induced interactions set in. \nReferring to repulsive impurity-medium interactions [Figs.~\\ref{2bd_g1}($a_4$-($a_5$)] it is apparent that for increasing $g_{BF}$ the anti-diagonal lobes of $\\rho^{(2)}_{BB}(x_1, x_2)$ approach the diagonal. \nTherefore, the two bosons get closer due to the presence of their attractive induced interactions mediated by the fermionic environment. \nNotice, however, that the two lobe structure of $\\rho^{(2)}_{BB}(x_1, x_2)$ is maintained also at $g_{BF}=3$ [Fig.~\\ref{2bd_g1}($a_5$)], indicating that the $s$-wave interactions dominate the induced ones. Turning to the attractive $g_{BF}$ regime we observe that for weak $g_{BF}$ values the anti-diagonal of $\\rho^{(2)}_{BB}(x_1, x_2)$ shrinks and thus the bosons come closer when compared to the $g_{BF}=0$ case due to the existence of attractive induced interactions. \nImportantly, this behavior of $\\rho^{(2)}_{BB}(x_1, x_2)$ is drastically changed for large attractive impurity-medium interactions. \nMore precisely, the two-lobed anti-diagonal structure related to the dominant repulsive contact interaction is changed into a circularly symmetric pattern, see e.g. $\\rho^{(2)}_{BB}(x_1,x_2)$ at $g_{BF}=-3$ depicted in Fig.~\\ref{2bd_g1}($a_1$). \nRecall that the appearance of such a circularly symmetric structure in $\\rho^{(2)}_{BB}(x_1, x_2)$ occurs in the case of zero effective interactions between the two bosons when $g_{BB}=0$ and $g_{BF}=0$ [Fig.~\\ref{2bd_g0}($a_3$)]. \nThis observation suggests that the attractive induced interactions nullify the direct repulsive contact ones for large attractive impurity-medium couplings, a phenomenon that is absent in the repulsive $g_{BF}$ regime. \nSummarizing, attractive impurity-medium couplings lead to stronger induced interactions than repulsive ones. \n\nSubsequently, we study the impact of the impurities mass on the strength of the induced interactions by  invoking as an appropriate measure the impurities relative distance [Fig. \\ref{Relative_dis}]. \nFor this investigation we consider a harmonically trapped mass-imbalanced BF mixture consisting of a $^{40}$K fermionic environment and two $^{87}$Rb bosonic impurities~\\cite{Fratini2012}. \nEvidently, the overall phenomenology of $\\mathcal{D}_{\\rm rel}$ for varying $g_{BF}$ is similar to the mass-balanced scenario for both the $g_{BB}=0$ and the $g_{BB}=1$ cases. \nMoreover, $\\mathcal{D}_{\\rm rel}$ is always reduced when compared to the mass-balanced system, thus suggesting that heavier impurities prefer to stay closer to each other than lighter ones~\\cite{kwasniok2020correlated,mistakidis2020many}. Accordingly, we can deduce that an \nincreasing impurity mass allows for stronger attractive induced interactions. \n\n\n\\subsection{Two-body correlations of the fermionic medium}\\label{corel_ground_bath} \n\nHaving explicated the existence of attractive induced impurity-impurity correlations we then analyze \nthe two-particle distributions of the fermionic environment for different impurity-medium interactions. \nOur main objective here is to expose the back-action of the impurities onto their host~\\cite{mukherjee2020pulse,mistakidis2019dissipative}. \nRegarding the system with two non-interacting impurities ($g_{BB}=0$) $\\rho^{(2)}_{FF}(x_1,x_2)$ is presented in Figs. \\ref{2bd_g0} ($b_1$)-($b_5$) for specific values of $g_{BF}$. \nA depleted diagonal is observed irrespectively of $g_{BF}$ due to the Pauli exclusion principle, namely two fermions cannot occupy the same spatial region. \nAt $g_{BF}=0$ two fermions can be found at any two distinct positions within the interval $[-10, 10]$, see Fig.~\\ref{2bd_g0}($b_3$), possessing a slightly larger probability to reside close to the trap center either on the same side or at different ones with respect to $x=0$. \nInterestingly, even the presence of a very small number of bosonic impurities is able to significantly alter the properties of the Fermi sea if $g_{BF}\\neq 0$. \nFor repulsive $g_{BF}$, the fermions exhibit a tendency to stay away from the trap center, \ne.g. $\\rho^{(2)}_{FF}(x_1=5,x_2=-5)\\approx 0.14$ at $g_{BF} = 1$ in Fig. \\ref{2bd_g0} ($b_4$). \nThis behavior is manifested by the appearance of relatively low density stripes along the lines $x_1=0$ and $x_2=0$ e.g. for $g_{BF}=1$ [Fig.~\\ref{2bd_g0}($a_4$)] which are transformed into completely depleted density regions e.g. for $g_{BF} = 3$ [Fig. \\ref{2bd_g0}($a_5$)]. \nTurning to the attractive $g_{BF}$ regime [Figs.~\\ref{2bd_g0}($b_1$)-($b_2$)], the distribution of $\\rho^{(2)}_{FF}(x_1,x_2)$ is changed significantly. \nIndeed, the probability of finding two fermions at different positions in the vicinity of the trap center is the dominant contribution to $\\rho^{(2)}_{FF}(x_1,x_2)$ especially for larger attractions. \nFor instance, at $g_{BF} = -1$, the two particle density shown in Fig.~\\ref{2bd_g0}($b_2$) is higher close to the trap \ncenter [e.g. $\\rho^{(2)}_{FF}(x_1=1,x_2=-1)\\approx 0.16$] compared to the edges [e.g. $\\rho^{(2)}_{FF}(x_1=5,x_2=-14)\\approx 0.09$]. \nAlso, for $g_{BF} = -3$, the spatial region apart from the one close to the trap center is almost completely depleted [see Fig.~\\ref{2bd_g0}($b_1$)], as identified by the relevant cross-like pattern building upon $\\rho^{(2)}_ {FF}(x_1,x_2)$. \n\nComparing now $\\rho^{(2)}_{FF}(x_1, x_2)$ between the cases of $g_{BB}=1$ [Figs.~\\ref{2bd_g1}($b_1$)-($b_5$)] and $g_{BB}=0$ [Figs.~\\ref{2bd_g0}($b_1$)-($b_5$)] we can easily deduce that their shapes at specific $g_{BF}$ values are to a great extent similar. \nA slight difference occurs for moderate repulsive interactions e.g. $g_{BF}=1$ where the two-body density stripes imprinted along the lines $x_1=0$ and $x_2=0$ for $g_{BB}=0$ [Fig.~\\ref{2bd_g0}($b_4$)] \nare not noticeable for $g_{BB}=1$ [Fig.~\\ref{2bd_g1}($b_4$)]. \nAlso, for attractive impurity-medium interactions $g_{BF}<0$ the regions away from the trap center are relatively \nstronger populated for $g_{BB}=1$, compare Figs. \\ref{2bd_g1} ($b_1$)-($b_2$) with Figs. \\ref{2bd_g0} ($b_1$)-($b_2$). \nAs a case example, for $g_{BF}=-3$ it holds that $\\rho^{(2)}_{FF}(x_1=4,x_2=4) \\approx 0.1$ when $g_{BB} = 0$ \nwhile $\\rho^{(2)}_{FF}(x_1=4,x_2=-4)\\approx 0.12$ for $g_{BB} = 1$, see Fig.~\\ref{2bd_g0}($b_1$) and \nFig. ~\\ref{2bd_g1}($b_1$) respectively.  \n\n \n\\section{Quench Dynamics}\\label{quench_dynamics}\n\nUp to now we have discussed the ground state properties of the harmonically trapped particle imbalanced BF mixture \nwith $N_F=6$ and $N_B=2$ for different impurity-medium interaction strengths ranging from attractive to repulsive values. \nImportantly, we have identified the presence of attractive induced interactions for the non-interacting impurities and \nanalyzed the competition between the direct $s$-wave repulsive interactions with the induced ones. \nAlso, in all cases we have quantified the back-action of the impurities to their fermionic environment. \n\nBelow, we explore the corresponding non-equilibrium dynamics of the impurities and the Fermi sea. \nThe mixture is prepared in its ground-state configuration, as already discussed in Sec.~\\ref{Ground state}, with zero impurity-medium coupling strength. \nThe dynamics is triggered by applying a quench of this coupling towards either the repulsive [Sec.~\\ref{repulsive_quench}] or the \nattractive [Sec.~\\ref{attractive quench}] regime of interactions~\\cite{volosniev2015real,mistakidis2019effective}. \nOur main objective is to inspect the dynamical emergence of induced impurity-impurity correlations and the pattern formation of the fermionic environment as a result of the impurities motion. \nIn the subsequent analysis we first study the dynamics of two non-interacting impurities and then contrast our findings to the case of two repulsively interacting ones. \n\n\n\\subsection{Quench to repulsive interactions}\\label{repulsive_quench} \n\nWe focus first on the correlated dynamics of the BF mixture induced by a quench from a vanishing to repulsive impurity-medium interactions. \nThe emergent dynamics is firstly analyzed by employing the corresponding single-particle density evolution of the participating \ncomponents [Sec.~\\ref{density_evol_repul}] and then by inspecting their two-body density matrix [Sec.~\\ref{two_body_evol_repul}] in \nthe course of the evolution. \nThese observables enable us to gain an overview of the dynamical evolution and importantly shed light on \nthe existence of impurity-impurity and bath correlations respectively. \n\n\n\\subsubsection{Single-particle density evolution}\\label{density_evol_repul}\n\nTo gain an overview of the spatially resolved quench-induced dynamics of the BF mixture we show the corresponding single-particle density evolution of the impurities $\\rho^{(1)}_{B}(x;t)$ and the Fermi sea $\\rho^{(1)}_{F}(x;t)$ in Fig.~\\ref{spd_dr} for different impurity-medium and impurity-impurity interaction strengths. \nNaturally, we commence our discussion on the system containing non-interacting impurities which provides the most clear signatures of induced correlations. \nReferring to weak post-quench interactions namely $g_{BF}=0.8$ the fermionic environment performs an overall \nbreathing motion~\\cite{huang2019breathing,boudjemaa2020breathing} manifested as a small amplitude periodic expansion and contraction of its cloud, see Fig.~\\ref{spd_dr}($a_1$). \nThe frequency of this global breathing mode is $\\omega_F^{br}\\approx 0.193\\approx2\\omega$ which is indeed in accordance to the corresponding theoretical predictions~\\cite{bauch2009quantum,abraham2012quantum}. \nMoreover $\\rho^{(1)}_{F}(x;t)$ exhibits at each time-instant of the evolution six in total shallow local maxima, namely three on the left and other three on the right side with respect to $x=0$, while a shallow density dip occurs around the trap center $x=0$. \nThese local maxima are indicative of the six fermions present in the system and also the fact that majorly the first six single-particle eigenstates of the trap participate in the dynamics. \nOn the other hand, the shallow dip of $\\rho^{(1)}_{F}(x=0;t)$ is caused by the presence of the impurities at the same location. \nThe impurities density $\\rho^{(1)}_{B}(x;t)$ undergoes a very weak amplitude breathing dynamics [see Fig.~\\ref{spd_dr}($a_3$)] characterized by a predominant \nfrequency $\\omega_{B}^{br}\\approx 0.24$. \nNotice here that $\\omega_B^{br}$ is slightly larger than $\\omega_F^{br}$ since the impurities experience an effective potential, created by the external trap and the density of the Fermi sea, which possesses a larger than the trap frequency. \nMoreover, $\\rho^{(1)}_{B}(x;t)$ completely overlaps with $\\rho^{(1)}_{F}(x;t)$ throughout the time-evolution, thus indicating the miscible character of the dynamics. \nRecall that the two components are also miscible in the ground state of the system for $g_{BF}=0.8$ as discussed in Sec.~\\ref{ground_state_density} and demonstrated in Figs. \\ref{spd_g} ($a_1$),($b_1$). \n\nTurning to repulsively interacting impurities with $g_{BB}=1$ for the same quench amplitude i.e. from $g_{BF} = 0$ to $g_{BF} = 0.8$ we observe that a qualitatively similar to the \nabove-described dynamics takes place when $g_{BB}=1$ for both the fermionic environment [Fig.~\\ref{spd_dr}($a_5$)] and the \nbosons [Fig.~\\ref{spd_dr}($a_7$)]. \nA notable difference is that $\\rho^{(1)}_{B}(x;t)$ is broader in the $g_{BB}=1$ case due to the presence of the direct $s$-wave interaction, compare in particular Fig.~\\ref{spd_dr}($a_3$) with Fig.~\\ref{spd_dr}($a_7$). \nNote that this broadening of $\\rho^{(1)}_{B}(x;t)$ for finite $g_{BB}$ occurs also in the ground state of the system, see Figs.~\\ref{spd_g}($b_1$), ($b_2$). \nAs expected, also the breathing amplitude of the impurities is larger when $g_{BB}=1$ while the frequency of this motion remains almost \nthe same ($\\omega_{B}^{br}\\approx 0.22$) with the $g_{BB}=0$ case. \nThis small deviation in the value of $\\omega_{B}^{br}$ is attributed to interaction effects~\\cite{schmitz2013quantum,kiehn2019spontaneous}. Consequently, due to the broader $\\rho^{(1)}_{B}(x;t)$ when $g_{BB}=1$ the central dip in $\\rho^{(1)}_{F}(x=0;t)$ occurring for $g_{BB}=0$ [Fig.~\\ref{spd_dr}($a_1$)] becomes very shallow and almost disappears for $g_{BB}=1$ [Fig.~\\ref{spd_dr}($a_3$)]. \nOtherwise, the inclusion of direct $s$-wave impurity-impurity interactions does not alter the characteristics of the system's dynamics at least on the single-particle level. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig5.eps}\n\\caption{One-body density evolution of [($a_1$), ($a_2$), ($a_5$), ($a_6$)] the Fermi sea $\\rho^{(1)}_{F}(x,t)$ and [($a_3$), ($a_4$), ($a_7$), ($a_8$)] the bosonic impurities $\\rho^{(1)}_{F}(x,t)$ upon considering an impurity-medium interaction quench from $g_{BF}=0$ to a finite repulsive value (see legend). \nThe impurities are ($a_1$)-($a_4$) either free i.e. $g_{BB} = 0$ or ($a_5$)-($a_8$)  repulsively interacting with $g_{BB} = 1$. \nThe system is confined in a harmonic trap with $\\omega=1$ and comprises of $N_B = 2$ bosons immersed in Fermi sea of $N_F = 6$ fermions. \nIt is initialized in its ground state with $g_{BF}=0$ and either $g_{BB}=0$ or $g_{BB}=1$.}  \n\\label{spd_dr}\n\\end{figure} \n\nIncreasing the post-quench interaction strength e.g. to $g_{BF}=2.5$ gives rise to a much more intricate dynamics for both the non-interacting impurities [Fig.~\\ref{spd_dr}($a_4$)] and the fermionic medium [Fig.~\\ref{spd_dr}($a_2$)] when compared to the $g_{BF}=0.8$ quench amplitude. \nWe remark that such a difference is already expected from the ground state properties of the system since for $g_{BF}=0.8$ the components are spatially overlapping (miscible) and become immiscible for $g_{BF}=2.5$, see also Figs. \\ref{spd_g} ($a_1$), ($b_1$). \nIn particular, the cloud of the Fermi sea exhibits a breathing oscillation with almost the same frequency $\\omega_F^{br}\\approx0.195$ \nas for the quench to $g_{BF}=0.8$. \nHowever, the amplitude of the contraction and expansion dynamics of $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_2$)] is larger when compared to the smaller $g_{BF}=0.8$ [Fig.~\\ref{spd_dr}($a_1$)] leading to a comparatively more excited medium in the former case. \nAccordingly, $\\rho^{(1)}_{F}(x;t)$ appears to be in general wider for $g_{BF}=2.5$, a result that can again be traced back to the ground state density of the Fermi sea [Fig.~\\ref{spd_g}($a_1$)]. \nMoreover, the local density humps building upon $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_2$)] are found to be shallower (deeper) during the expansion (contraction) of the fermionic cloud for $g_{BF}=2.5$ than for $g_{BF}=0.8$. \nImportantly, the density dip of $\\rho^{(1)}_{F}(x;t)$  around the trap center is substantially deeper when $g_{BF}=2.5$. \nThis is a direct consequence of the emergent phase-separation being anticipated already from the ground state of the system for such strongly repulsive impurity-medium interactions, see also Figs.~\\ref{spd_g} ($a_1$), ($b_1$). \n\nOf course, most of the above-described features of $\\rho^{(1)}_{F}(x;t)$ are intimately connected with the corresponding behavior of the single-particle density of the bosons $\\rho^{(1)}_{B}(x;t)$ [Fig.~\\ref{spd_dr}($a_4$)] since the two components are inevitably interdependent due to their mutual finite coupling $g_{BF}$. \nSpecifically, the impurities density $\\rho^{(1)}_{B}(x;t)$ shows a relatively larger localization tendency [Fig.~\\ref{spd_dr}($a_4$)] than for $g_{BF}=0.8$ [Fig.~\\ref{spd_dr}($a_3$)] which is expected due to the aforementioned phase-separated behavior among the components. \nMoreover, $\\rho^{(1)}_{B}(x;t)$ exhibits a weaker amplitude and \nlarger frequency $\\omega_{B}^{br}\\approx 0.36$ breathing motion for $g_{BF}=2.5$ than for $g_{BF}=0.8$. \nThe alteration of the impurities breathing frequency for $g_{BF}=2.5$ can in turn be explained within an effective potential picture. \nIndeed, as already argued in the ground state of the system the impurities can be viewed as trapped in the potential formed by \nthe harmonic trap with a superimposed density of their Fermi sea. \nSince $\\rho^{(1)}_{F}(x;t)$ is wider for increasing $g_{BF}$ also the impurities effective trapping frequency being r\nelated to the breathing one is larger. \n\nThe dynamics of interacting impurities with $g_{BB}=1$ following a quench to $g_{BF}=2.5$ as captured by  $\\rho^{(1)}_{F}(x;t)$ \n[Fig.~\\ref{spd_dr}($a_6$)] and $\\rho^{(1)}_{B}(x;t)$ [Fig.~\\ref{spd_dr}($a_8$)] is more involved than the $g_{BB}=0$ case, especially for long evolution times $t>40$. \nEvidently, the impurities exhibit a significantly broader density distribution for $g_{BB}=1$ [Fig.~\\ref{spd_dr}($a_8$)] than \nfor $g_{BB}=0$ [Fig.~\\ref{spd_dr}($a_4$)], while performing a breathing motion of a larger amplitude and smaller \nfrequency $\\omega_{B}^{br}\\approx 0.33$ in the former case. \nFurthermore, $\\rho^{(1)}_{B}(x;t)$ initially ($t=0$) having a Gaussian profile deforms already within the initial stages of the dynamics ($t>5$) \nby developing three shallow humps being more pronounced during expansion and coming very close at the contraction points [Fig.~\\ref{spd_dr}($a_8$)]. \nThis behavior of $\\rho^{(1)}_{B}(x;t)$ essentially indicates that for $t<5$ the interacting impurities dominantly occupy the lowest-lying \nsingle-particle eigenstate of their external potential while as time evolves the contribution of higher-lying eigenstates becomes significant \nand excitations are formed. \nThis statement is also supported by the population of the individual bosonic orbitals $\\phi_i^{B}(x,t)$ with $i=1,2,\\dots,12$ \n(see also the discussion following Eq.~(\\ref{1BD})) from which the first eight have a non-negligible population during the evolution (results not shown). \nThe above-mentioned differences, regarding mainly the breathing mode and the structure of $\\rho^{(1)}_{B}(x;t)$ for $g_{BB}=1$ compared to \nthe $g_{BB}=0$ case are attributed to the presence of the direct $s$-wave repulsive interaction between the impurities. \nAs expected, the features of $\\rho^{(1)}_{B}(x;t)$ are also imprinted to a certain extent in the density of the fermionic environment \n$\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_6$)] due to the finite $g_{BF}$. \nNotably, $\\rho^{(1)}_{F}(x;t)$ shows an arguably suppressed central dip compared to the $g_{BB}=0$ case due to \nthe broader distribution of the impurities for $g_{BB}=1$. \nThis in turn gives rise to an almost vanishing degree of phase-separation in the latter $g_{BB}=1$ case. \nOther properties, such as the amplitude and the frequency of the breathing mode remain almost the same as in the $g_{BB}=0$ scenario. \n\nConcluding this section, it is important to emphasize that the quench dynamics of non-interacting and interacting bosonic impurities \ndiffers noticeably already on the single-particle level, especially for large post-quench interaction strengths. \nThis impact of the direct $s$-wave interaction of the impurities is also imprinted in the Fermi sea leading to changes in its \npattern formation. \nAs we shall explicate below, the origin of the above-mentioned differences is the presence of impurity-impurity induced interactions. \n\n\n\\subsubsection{Dynamics of impurity-impurity correlations}\\label{two_body_evol_repul} \n\nTo track the spatially resolved dynamics of the two bosonic impurities with respect to one another we next invoke their \ntwo-particle density, $\\rho^{(2)}_{BB}(x_1,x_2);t)$ [Eq.~(\\ref{2BD})], which essentially provides the probability of measuring \nsimultaneously one particle at position $x_1$ and the other at $x_2$. \nAs a complementary measure of the impurities position we also calculate their relative distance $\\mathcal{D}_{\\rm rel}(t)$ [Eq.~(\\ref{7})] during the time-evolution. \nThis observable will allow us to identify whether the impurities interact among each other via induced correlations mediated by their host or \nthey move independently~\\cite{mistakidis2019correlated,mistakidis2020many}. \nSnapshots of $\\rho^{(2)}_{BB}(x_1, x_2;t)$ at specific time-instants of the evolution upon considering a quench from $g_{BF}=0$ to $g_{BF}=2.5$ for the cases of $g_{BB}=0$ and $g_{BB}=1$ are presented in Figs.~\\ref{2B_den_r}($a_1$)-($a_4$) and Figs.~\\ref{2B_den_r}($c_1$)-($c_4$) respectively. \nMoreover, the corresponding $\\mathcal{D}_{\\rm rel}(t)$ when $g_{BB}=0$ [Fig.~\\ref{2B_den_r}($d$)] and $g_{BB}=1$ [Fig.~\\ref{2B_den_r}($e$)] is demonstrated for different post-quench interactions providing an overview of the impurity-impurity correlations. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig6.eps}\n\\caption{Snapshots of the two-body reduced density of ($a_1$)-($a_4$) two non-interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$, ($b_1$)-($b_4$) two fermions of the medium $\\rho^{(2)}_{FF}(x_1,x_2)$ and ($c_1$)-($c_4$) two repulsively interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$ at specific time-instants of the evolution (see legends). \nThe system consists of $N_F=6$ fermions and $N_B=2$ bosons while it is confined in a harmonic trap with $\\omega=0.1$. \nIt is initialized in its ground state with $g_{BF}=0$ and either $g_{BB}=0$ or $g_{BB}=1$. \nTo trigger the dynamics an interaction quench is performed from $g_{BF} = 0$ to $g_{BF} = 2.5$. \nTime-evolution of the relative distance $\\mathcal{D}_{\\rm rel}(t)$ between the two bosonic impurities with ($d$) $g_{BB} = 0$ and ($e$) $g_{BB} =1$ at distinct post-quench $g_{BF}$ values (see legend).}  \n\\label{2B_den_r}\n\\end{figure}\n\nFor non-interacting bosonic impurities, $\\rho^{(2)}_{BB}(x_1,x_2;t=0)$ has a circular shape in the ($x_1,x_2$)-plane [Fig.~\\ref{2B_den_r}($a_1$)] with a peak around $x_1,x_2\\in [-2,2]$. \nTherefore, the bosons are likely to reside in this spatial region close to the trap center. \nHowever, in the course of the dynamics this shape of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is drastically altered exhibiting an elongated diagonal and a suppressed anti-diagonal, see Figs.~\\ref{2B_den_r}($a_2$)-($a_4$). \nNote that the anti-diagonal of the two-particle density of the impurities dictates their relative distance $\\mathcal{D}_{\\rm rel}(t)$ [Eq.~(\\ref{7})]. \nThe latter is illustrated in Fig.~\\ref{2B_den_r}($d$) for a variety of post-quench $g_{BF}$ values. \nAs it can be seen, in all cases $\\mathcal{D}_{\\rm rel}(t)$ undergoes a decaying amplitude oscillatory motion characterized by \ntwo dominantly participating frequencies which essentially correspond to the center-of-mass and relative coordinate \nbreathing modes~\\cite{schmitz2013quantum}, e.g. $\\omega_1\\approx 0.19$, $\\omega_2\\approx 0.24$ for $g_{BF}=0.8$ and \n$\\omega_1\\approx 0.19$, $\\omega_2\\approx 0.36$ when $g_{BF}=2.5$. \nIndeed, the oscillatory behavior of $\\mathcal{D}_{\\rm rel}(t)$ reflects the breathing motion of the impurities cloud already identified in \nthe dynamics of $\\rho^{(1)}_{B}(x;t)$ [Fig.~\\ref{spd_dr}($a_3$)]. \nThis is also imprinted in the modulated shape of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ [Figs.~\\ref{2B_den_r}($a_2$-($a_4$)], e.g. \nthe anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is more expanded at $t=55$ compared to $t=130$. \nAlso, the oscillation amplitude of $\\mathcal{D}_{\\rm rel}(t)$ and as a consequence of the breathing mode is enhanced \nfor a larger post-quench $g_{BF}$, see also Figs.~\\ref{spd_dr}($a_3$), ($a_7$). \nImportantly the decaying amplitude in time of $\\mathcal{D}_{\\rm rel}(t)$, and thus the elongated shape \nof $\\rho^{(2)}_{BB}(x_1,x_2;t)$ across its diagonal, implies that the impurities tend to approach \neach other during the dynamics and since they are non-interacting ($g_{BB}=0$) they experience an effective \nattraction mediated by the fermionic environment. \n\nProceeding we examine the role played by the direct $s$-wave repulsive contact interaction between the impurities and its competition with the induced interactions on the quench dynamics. \nDue to the finite impurity-impurity repulsion, herein $g_{BB}=1$, the bosons initially ($t=0$) reside one in the left ($x<0$) and the other in the right ($x>0$) side of the trap, see the pronounced anti-diagonal distribution of $\\rho^{(2)}_{BB}(x_1,x_2=-x_1;t=0)$ in Fig.~\\ref{2B_den_r}($c_1$). \nIn contrast, after the quench ($t>0$) three distinct segments develop in $\\rho^{(2)}_{BB}(x_1,x_2;t)$ [Figs.~\\ref{2B_den_r}($c_2$)-($c_4$)]. \nNamely the impurities are either very close to each other around the trap center [see the diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$] or they remain spatially separated with one of them located in the left and the other in the right side of the trap with respect to $x=0$ [see the anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$]. \nThis two-body superposition is a consequence of the competition between the direct repulsive and induced \nattractive interactions~\\cite{mistakidis2020many,mistakidis2020induced}. \nInspecting now $\\mathcal{D}_{\\rm rel}(t)$ for different post-quench values of $g_{BF}$ [Fig.~\\ref{2B_den_r}($d$)] we can \nreadily see that it performs oscillations possessing two predominant frequencies, for instance \n$\\omega_1\\approx 0.22$, $\\omega_2\\approx 0.19$ when $g_{BF}=0.8$ and $\\omega_1\\approx 0.33$, $\\omega_2\\approx 0.19$ if $g_{BF}=2.5$. \nThese frequencies are again related to the center-of-mass and relative coordinate breathing modes respectively. \nInterestingly, the oscillation amplitude of $\\mathcal{D}_{\\rm rel}(t)$ e.g. for $g_{BF}=0.8$ and $g_{BF}=2.5$ is almost constant while for $g_{BF}=4$ it shows a decaying tendency. \nThis means that in the latter case the induced attraction tends to surpass the impurities direct repulsion. \nFinally, we remark that the oscillation amplitude (decay rate) of $\\mathcal{D}_{\\rm rel}(t)$ for fixed $g_{BF}$ is in general larger (smaller) when $g_{BB}=1$ [Fig.~\\ref{2B_den_r}($e$)] compared to $g_{BB}=0$ [Fig.~\\ref{2B_den_r}($d$)], thus evincing the presence of the direct impurities repulsion. \n\n\n\\subsubsection{Correlations of the fermionic environment}\\label{two_body_evol_repul_bath} \n\nTo complement our study we then investigate the correlation patterns of the Fermi sea as encoded in its two-body \ndensity $\\rho^{(2)}_{FF}(x_1,x_2;t)$ shown in Figs.~\\ref{2B_den_r}($b_1$)-($b_4$) at specific time-instants after a \nquench from $g_{BF}=0$ to $g_{BF}=2.5$ for $g_{BB}=0$. \nA correlation hole occurs along the diagonal of $\\rho^{(2)}_{FF}(x_1,x_2;t)$ throughout the evolution due to the Pauli exclusion principle. \nAlso, an expansion [Fig.~\\ref{2B_den_r}($b_2$)] and contraction [Fig.~\\ref{2B_den_r}($b_4$)] of the anti-diagonal \nof $\\rho^{(2)}_{FF}(x_1,x_2;t)$ \ntakes place which manifest the collective breathing motion of the fermionic cloud~\\cite{kwasniok2020correlated}, see \nalso Fig.~\\ref{spd_dr}($a_2$). \nMoreover, a depletion along the $x_1=0$ and $x_2=0$ spatial regions is observed indicating that it is more likely for \none fermion to be located in the vicinity of a density hump at $x<0$ and the other one being symmetrically placed \nwith respect to the trap center, see also Fig.~\\ref{spd_dr}($a_2$). \n\n\n\n\\subsection{Quench to attractive interactions}\\label{attractive quench} \n\nIn the following, we shall study the dynamical response of the impurities and the fermionic bath after a quench from $g_{BF}=0$ to \nthe attractive ($g_{BF}<0$) impurity-medium interaction regime. \nTo quantify the arising distinctive dynamical features we analyze the single-particle density [Sec.~\\ref{density_evol_attract}] \nand the corresponding two-body density [Sec.~\\ref{two_body_evol_attract}] evolution of the participating components. \nAs in the previous section, we first discuss the time-evolution of two non-interacting ($g_{BB} = 0$) impurities and subsequently \ncompare to the case of two repulsively interacting ($g_{BB} = 1.0$) ones. \n\n\n\\subsubsection{Density evolution}\\label{density_evol_attract}\n\nThe spatio-temporal evolution of $\\rho^{(1)}_{F}(x;t)$ and $\\rho^{(1)}_{B}(x;t)$ after a quench from $g_{BF}=0$ to $g_{BF}=-0.8$ \nfor $g_{BB}=0$ is presented in Fig.~\\ref{s_den_a} ($a_1$) and Fig.~\\ref{s_den_a} ($a_3$) respectively. \nAs a consequence of the interaction quench both the impurities and the fermionic clouds undergo a collective weak amplitude \nbreathing motion identified by their contraction and expansion dynamics~\\cite{huang2019breathing,mistakidis2019correlated}. \nThe breathing frequency of the fermionic bath is $\\omega_F^{br}\\approx 0.208\\approx2\\omega$ while for the impurities it corresponds to \n$\\omega_B^{br} \\approx 0.251$ since they experience a modified external potential composed of the harmonic oscillator and the density of their host, see for details Refs.~\\cite{mistakidis2020many,kiehn2019spontaneous}. \nImportantly, the attractive impurity-medium coupling results in the formation of a shallow density hump in $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{s_den_a}($a_1$)] at the instantaneous location of the impurities [Fig.~\\ref{s_den_a}($a_3$)] i.e. around the trap center. \nTurning to repulsively interacting impurities where $g_{BB}=1$ we observe that a similar to the above-described dynamics takes place, see Figs.~\\ref{s_den_a} ($a_5$), ($a_7$). \nHowever, the expansion amplitude of $\\rho^{(1)}_{B}(x;t)$ is larger and the breathing frequency $\\omega_B^{br}\\approx 0.226$ is slightly smaller compared to the $g_{BB}=0$ scenario due to the inclusion of direct $s$-wave repulsive interactions. \nAlso, since $\\rho^{(1)}_{B}(x;t)$ is relatively wider than for $g_{BB}=0$ the density hump developed in $\\rho^{(1)}_{F}(x;t)$ in \nthe latter case almost disappears for $g_{BB}=1$.   \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig7.eps}\n\\caption{Time-evolution of the one-body density of [($a_1$), ($a_2$), ($a_5$), ($a_6$)] the fermionic medium $\\rho^{(1)}_{F}(x,t)$ and [($a_3$), ($a_4$), ($a_7$), ($a_8$)] the bosonic impurities $\\rho^{(1)}_{B}(x,t)$ after an interaction quench of the boson-fermion coupling constant from $g_{BF}=0$ to different attractive values (see legend). \nThe impurities are considered to be ($a_1$)-($a_4$) free i.e. $g_{BB} = 0$ and ($a_5$)-($a_8$) repulsively interacting with $g_{BB} = 1$. \nThe harmonically trapped mixture with $\\omega=1$ consists of $N_F = 6$ fermions and $N_B = 2$ bosons while it is prepared in its ground state with $g_{BF}=0$ and either $g_{BB}=0$ or $g_{BB}=1$.} \n\\label{s_den_a}\n\\end{figure} \n\nFollowing a quench to stronger impurity-medium interactions, e.g. $g_{BF}=-2.5$, leads to a more intricate response of both components than for $g_{BF}=-0.8$, see Figs.~\\ref{s_den_a}($a_2$), ($a_4$), ($a_6$), ($a_8$). \nReferring to the system containing non-interacting impurities [Figs.~\\ref{s_den_a}($a_2$), ($a_4$)] we observe that $\\rho^{(1)}_{B}(x;t)$ has a pronounced spatial localization tendency around the trap center while performing an ``irregular'' weak amplitude breathing dynamics. \nThe latter is characterized by two dominant frequencies, namely $\\omega_{B_1}^{br}=0.234$ and $\\omega_{B_2}^{br}=0.263$ corresponding to the \ncenter-of-mass and relative coordinate breathing mode respectively. \nSince these frequencies are close the dynamics of $\\rho^{(1)}_{B}(x;t)$ is reminiscent of a beating pattern. \nWe remark that a similar time-evolution takes place also for bosonic impurities immersed in a bosonic background~\\cite{mistakidis2020many}.\nAccordingly, as a result of this sharply peaked distribution of $\\rho^{(1)}_{B}(x;t)$ in the vicinity of $x=0$ there is an accumulation \nof the fermionic density at the same location due to the finite $g_{BF}$. \nIndeed, a prominent density hump builds upon $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{s_den_a}($a_2$)] which otherwise exhibits collective \nbreathing oscillations of a frequency $\\omega_F^{br}\\approx 0.211$. \nThe dynamical response is somewhat changed when considering interacting impurities ($g_{BB}=1$) as depicted in Figs.~\\ref{s_den_a}($a_6$), ($a_8$). \nThe impurities possess a comparatively wider density distribution than for $g_{BB}=0$ as a consequence of their finite repulsion, $g_{BB}=1$. \nAlso, the amplitude of their breathing motion is slightly larger compared to the one of $g_{BB}=0$ and the participating frequencies \n$\\omega_{B_1}^{br}=0.355$ and $\\omega_{B_2}^{br}=0.376$ are very close thereby producing a beating pattern (hardly visible \nin Fig.~\\ref{s_den_a}($a_8$)) manifested by the periodic \nappearance of sharp peaks in $\\rho^{(1)}_{B}(x;t)$ as a result of its contraction. \nThe differences in the dynamics of non-interacting and interacting impurities for fixed post-quench $g_{BF}$ will be further analyzed in the next section. \nThis beating pattern is also imprinted in $\\rho^{(1)}_{F}(x;t)$ which as a back-action develops humps which follow the location of the impurities. \nOtherwise, $\\rho^{(1)}_{F}(x;t)$ performs a breathing mode of almost the same amplitude and equal \nfrequency $\\omega_{F}^{br}\\approx0.2$ compared to the $g_{BB}=0$ case. \n\n\n\n\\subsubsection{Evolution of impurity-impurity correlations}\\label{two_body_evol_attract} \n\nTo identify and consequently characterize the nature of the impurity-impurity correlations in the course of the dynamics we monitor the two-body density $\\rho^{(2)}_{BB}(x_1,x_2;t)$ and relative distance $\\mathcal{D}_{\\rm rel}(t)$ of the impurities depicted in Fig.~\\ref{2B_den_a} for different quench amplitudes. \nFigures~\\ref{2B_den_a}($a_1$)-($a_4$) show snapshots of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ upon considering a quench of two non-interacting impurities from $g_{BF}=0$ to $g_{BF}=-2.5$. \nInitially, $t=0$, the impurities lie in the vicinity of the trap center since $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is non-zero within the spatial region $x_1,x_2\\in [-2,2]$ [Fig.~\\ref{2B_den_a}($a_1$)]. However, as time evolves, the two bosons start to occupy a smaller spatial region and therefore approach each other, a tendency that becomes evident by the gradual shrinking of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ across its diagonal accompanied by the depression of its anti-diagonal, see Figs.~\\ref{2B_den_a}($a_2$)-($a_4$). \n\nAs it has already been discussed in Sec.~\\ref{two_body_evol_repul} the shape of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is well captured by $\\mathcal{D}_{\\rm rel}(t)$ which is presented in Fig.~\\ref{2B_den_a}($d$) for distinct post-quench values of $g_{BF}$. \nEvidently $\\mathcal{D}_{\\rm rel}(t)$ oscillates irrespectively of $g_{BF}$, a behavior that corresponds to the breathing motion of the impurities and can also be inferred from the weak expansion [Fig.~\\ref{2B_den_a}($a_3$)] and contraction [Fig.~\\ref{2B_den_a}($a_2$)] of $\\rho^{(2)}_{BB}(x_1,x_2=-x_1;t)$. \nIts evolution contains a multitude of frequencies whose number and value depend on $g_{BF}$ and refer to the underlying \nbreathing motion, e.g. for $g_{BF}=-2.5$ the dominantly involved frequencies are $\\omega_1=0.234$ and $\\omega_2=0.263$ respectively. \nAlso, the oscillation amplitude of $\\mathcal{D}_{\\rm rel}(t)$ is smaller for a larger $\\abs{g_{BF}}$ which is in accordance to the localization tendency of the impurities for quenches to stronger impurity-medium attractions. \nImportantly, $\\mathcal{D}_{\\rm rel}(t)$ exhibits a decaying tendency in time which is more pronounced for increasing $\\abs{g_{BF}}$ and shows a saturation behavior for quite strong attractions, e.g. $g_{BF}=-4$ here, and long evolution times $t>80$. \nThis latter decaying behavior is again a manifestation of the presence of attractive induced impurity-impurity interactions. \nIt is also worth commenting at this point that the suppression of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ or equivalently \nthe decay of $\\mathcal{D}_{\\rm rel}(t)$ is significantly more pronounced for quenches towards the attractive interaction regime than \nin the repulsive one, e.g. compare Fig.~\\ref{2B_den_r}($d$) and Fig.~\\ref{2B_den_a}($d$). \nTherefore, we can infer the generation of stronger attractive induced interactions for quenches in the attractive than in the repulsive \nimpurity-medium interaction regime~\\cite{mistakidis2020induced}, a result that also holds for the ground state of the system as \nexplicated in Sec.~\\ref{corel_ground}. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig8.eps} \n\\caption{Instantaneous profiles of the two-body reduced density of ($a_1$)-($a_4$) two non-interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$, ($b_1$)-($b_4$) two fermions of the bath $\\rho^{(2)}_{FF}(x_1,x_2)$ and ($c_1$)-($c_4$) two repulsively interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$. \nThe BF mixture contains $N_F = 6$ fermions and $N_B =  2$ bosons. \nIt is confined in a harmonic trap with $\\omega = 0.1$ and it is prepared in its ground state with $g_{BF} = 0$ and either $g_{BB}=0$ or $g_{BB}=1$. \nThe dynamics is triggered upon considering an impurity-medium interaction quench from $g_{BF} = 0$ to $g_{BF} = -2.5$. \nTemporal-evolution of the boson-boson relative distance $\\mathcal{D}_{\\rm rel}(t)$ for ($d$) $g_{BB} = 0$ and ($e$) $g_{BB} =1$ at specific post-quench $g_{BF}$ couplings (see legend).}  \n\\label{2B_den_a}\n\\end{figure} \n\nOn the other hand, the two-body dynamics of two repulsively interacting impurities, here $g_{BB}=1$, subjected to a quench \nfrom $g_{BF}=0$ to $g_{BF}=-2.5$ showcases quite different characteristics from the $g_{BB}=0$ case, compare in particular \nFigs.~\\ref{2B_den_a}($c_1$)-($c_4$) with Figs.~\\ref{2B_den_r}($c_1$)-($c_4$). \nIndeed, even for the ground state of the system ($t=0$) the impurities, since $g_{BB}$ is finite, are spatially separated with the one residing \nat $x<0$ and the other at $x>0$ as can be inferred from the correlation hole of $\\rho^{(2)}_{BB}(x_1,x_2=x_1;t=0)$ in Fig.~\\ref{2B_den_a}($c_1$). \nAfter the quench, they oscillate between two distinct configurations. \nNamely they either stay separated, see the elongated anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ in Figs.~\\ref{2B_den_a}($c_2$), ($c_3$), or they move close to each other, see their bunching tendency in the region $x_1,x_2\\in [-2,2]$ in Fig.~\\ref{2B_den_a}($c_4$). \nThis behavior is caused by the competition of their inherent repulsive contact interaction and the induced attraction mediated by the fermionic environment~\\cite{huber2019medium}. \n\nTo understand better the aforementioned competing mechanism we present the time-evolution of the impurities relative distance $\\mathcal{D}_{\\rm rel}(t)$ for a variety of post-quench interactions $g_{BF}$ in Fig.~\\ref{2B_den_a}($e$). \nAs it can be directly seen, the response of $\\mathcal{D}_{\\rm rel}(t)$ depends crucially on $g_{BF}$. Indeed for quenches to weak attractions, e.g. $g_{BF}=-0.8$, $\\mathcal{D}_{\\rm rel}(t)$ oscillates with an almost constant amplitude and a \ndominant frequency $\\omega_1=0.226$. \nHowever, by increasing the quench amplitude e.g. to $g_{BF}=-2.5$ $\\mathcal{D}_{\\rm rel}(t)$ performs ``irregular'' oscillations characterized by multiple frequencies and importantly a decaying amplitude. \nThis decay is more pronounced for larger attractions e.g. $g_{BF}=-4$ where $\\mathcal{D}_{\\rm rel}(t)$ drops at the early stages of the dynamics and saturates to a fixed value for $t>50$. \nAs a consequence, we can deduce that the attractive induced interactions become stronger for quenches towards larger impurity-medium attractions and gradually dominate with respect to the impurities direct repulsion. \nNote here that such a mechanism is also present for quenches to repulsive impurity-medium interactions, see Fig.~\\ref{2B_den_r}($e$), but it is apparently less effective compared to the attractive $g_{BF}$ quench scenario. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig9.eps}\n\\caption{Temporal-evolution of the Von-Neumann entropy $S_{VN}(t)$ for different post-quench ($a$)-($b$) attractive and ($c$)-($d$) repulsive impurity-medium interaction strengths (see legends). \nThe bosonic impurities are considered to be either [($a$), ($c$)] non-interacting $g_{BB}=0$ or [($b$), ($d$)] interacting with $g_{BB}=1$. \n($e$) Time-averaged Von-Neumann entropy $\\bar{S}_{VN}$ for distinct post-quench $\\abs{g_{BF}}$ attractive and repulsive values (see legend) when $g_{BB}=0$. \nThe solid and dashed lines provide a guide to the eye. \nThe BF mixture consists of $N_B=2$ bosons and $N_F=6$ fermions with equal masses and are confined in the same harmonic trap with $\\omega = 0.1$.}  \n\\label{fig:vn_ent}\n\\end{figure*}\n\n\n\\subsubsection{Correlations of the Fermi bath}\\label{two_body_evol_bath_attract}\n\nTurning to the Fermi sea we observe the appearance of completely different correlation patterns building upon $\\rho^{(2)}_{FF}(x_1,x_2;t)$ when compared to the repulsive quench scenario [Figs.~\\ref{2B_den_r}($c_1$)-($c_4$)] as demonstrated in Figs.~\\ref{2B_den_a}($c_1$)-($c_4$) for a quench to $g_{BF}=-2.5$ in the system with two non-interacting ($g_{BB}=0$) impurities. \nAs expected, a correlation hole exists for the entire time-evolution due to the fermionic character of the bath~\\cite{erdmann2019phase,kwasniok2020correlated}. \nInitially, $t=0$, the fermions are symmetrically placed with respect to $x=0$ and predominantly reside one at $x>0$ and the other \nat $x<0$, see the bright spots close to the diagonal in Fig.~\\ref{2B_den_a}($c_1$). \nFollowing the quench a cross-like correlation pattern appears in $\\rho^{(2)}_{BB}(x_1,x_2;t)$ which becomes more elongated across the spatial regions lying in the vicinity of $x_1=0$ and $x_2=0$ [Figs.~\\ref{2B_den_a}($c_3$), ($c_4$)]. \nThis cross-like correlation pattern is the two-body analogue of the accumulation of the Fermi density $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_2$)] around the trap center and more precisely in the vicinity of the position of the impurities due to the attractive $g_{BF}$. \nThus it is a direct imprint of the impurities motion into their host, see also the elongated diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ in Figs.~\\ref{2B_den_a}($a_2$)-($a_4$), evincing that for strong attractive $g_{BF}$ the fermions move close to the trap center, a mechanism that competes with their inherent Fermi pressure. \n\n\n\\subsection{Entanglement dynamics}\\label{entanglemet_dynamics} \n\nTo further unveil the degree of impurity-medium correlations during the quench dynamics of the BF mixture we employ the time-evolution of the Von-Neumann entropy $S_{VN}(t)$ [Eq.~\\eqref{VN}]. \nThis quantity provides a measure of the overall build up of the impurity-medium entanglement~\\cite{mukherjee2020pulse,theel2020entanglement,kwasniok2020correlated} and also reveals the complexity of the time-evolved post-quench state of the system. \nThe dynamics of $S_{VN}(t)$ after a quench to attractive (repulsive) interactions for the system of two non-interacting and interacting impurities is shown in Figs.~\\ref{fig:vn_ent}($a$) and ($b$) [Figs.~\\ref{fig:vn_ent}($c$) and ($d$)] respectively. \n\nFocusing on the attractive post-quench interaction regime [Figs.~\\ref{fig:vn_ent}($a$), ($b$)] we observe that independently of the inclusion \nof direct $s$-wave impurity-impurity interactions $S_{VN}(t=0)=0$ and hence the components are initially non-entangled. \nHowever, directly after the quench an appreciable impurity-medium entanglement generation takes place in all cases \nsince $S_{VN}(t)\\neq 0$~\\cite{mistakidis2019correlated,mistakidis2019quench,mukherjee2020pulse}. \nMore precisely, an almost ballistic linear growth of $S_{VN}(t)$ is manifested at the very early stages of the dynamics ($t<5$) accompanied by a fluctuating behavior of $S_{VN}(t)$ around a fixed $g_{BF}$-dependent value at later evolution times. \nNotice that for both $g_{BB}=0$ and $g_{BB}=1$ the response of $S_{VN}(t)$ shows a hierarchy in terms of $g_{BF}$, namely it acquires \nlarger values for stronger attractions. \nAlso, the temporal fluctuations of $S_{VN}(t)$ deep in the evolution are suppressed for quenches to weak attractions, e.g. compare $S_{VN}(t)$ for $g_{BF}=-0.8$ and $g_{BF}=-2.5$ in Figs.~\\ref{fig:vn_ent}($a$), ($b$). \nThe latter means that for larger post-quench attractions the system is in a more complicated many-body superposition involving a \nlarger amount of states [see also Eq.~(\\ref{4})] than for smaller negative $g_{BF}$ values. \nThis situation holds equal for fixed $g_{BF}$ but increasing $g_{BB}$. \nIndeed, it becomes apparent by inspecting $S_{VN}(t)$ for fixed $g_{BF}$ between the $g_{BB}=0$ and $g_{BB}=1$ cases that in the latter case the temporal fluctuations of $S_{VN}(t)$ are enhanced, especially for a larger $\\abs{g_{BF}}$. \nWe remark that the saturating tendency of $S_{VN}(t)$ for long times can be attributed to the finite size of the system~\\cite{Calabrese_2005}, i.e. if the system would have been infinite then $S_{VN}(t)$ should increase linearly in time throughout the time-evolution. \n\nA similar to the above-described phenomenology regarding the entanglement dynamics takes place also during the unitary evolution of the system for quenches towards the repulsive impurity-medium interaction regime for both non-interacting [Fig.~\\ref{fig:vn_ent}($c$)] and interacting [Fig.~\\ref{fig:vn_ent}($d$)] impurities. \nIndeed, the sudden increase of $g_{BF}$ leads to entanglement formation since $S_{VN}(t)\\neq 0$ while $S_{VN}(t=0)=0$. \nAs for $g_{BF}<0$, here also $S_{VN}(t)$ increases linearly for $t<5$ and subsequently oscillates around a mean value, see Figs.~\\ref{fig:vn_ent}($c$), ($d$). \nInterestingly, the degree of entanglement is larger for quenches to the repulsive than the attractive interaction regimes, e.g. compare Fig.~\\ref{fig:vn_ent}($a$) with Fig.~\\ref{fig:vn_ent}($c$). \nThis fact evinces that a larger amount of dynamical impurity-medium entanglement is established in the repulsive interaction regime. \nTo support our argument we exemplarily showcase the time-averaged Von-Neumann entropy, defined as $\\bar{S}_{VN} = (1\/T)\\int_{0}^{T}dtS_{VN}(t)$ with $T$ being the considered evolution time, in Fig.~\\ref{fig:vn_ent}($e$) for varying post-quench repulsive ($g_{BF}>0$) and attractive ($g_{BF}<0$) impurity-medium interactions in the system containing the non-interacting impurities. \nAs it can be readily seen, irrespectively of the quench direction $\\bar{S}_{VN}$ increases monotonously with increasing magnitude of $g_{BF}$. \nHowever, it is also apparent that $\\bar{S}_{VN}$ is in general slightly larger for quenches to repulsive than to attractive interactions at a specific post-quench $\\abs{g_{BF}}$. \n\n\n\\section{Conclusions}\\label{conclusion} \n\nWe have unraveled the role of induced correlations and pattern formation in the ground state and the non-equilibrium quantum dynamics of two bosonic impurities embedded in a fermionic environment. \nThe one-dimensional Bose-Fermi mixture is harmonically trapped and the time-evolution is initiated upon considering a quench of the \nimpurity-medium coupling from a vanishing towards the repulsive or the attractive interaction regime. \nInspecting both one- and two-body observables enables us to expose correlation-induced phenomena mediated by the host, analyze the \ncompetition of induced interactions and direct $s$-wave ones, the emergent phase-separation processes and the underlying entanglement dynamics. \n\nReferring to the ground state of two non-interacting bosonic impurities it is shown that on the single-particle level they phase-separate \nwith the Fermi sea for strong repulsions and accumulate at the trap center together with their environment for large attractions, otherwise they are miscible. \nIn the system of two repulsively interacting impurities the boundaries of the aforementioned regions are shifted to larger interactions. \nImportantly, we identify the presence of induced impurity-impurity interactions mediated by the fermionic environment, in the system with non-interacting bosons, for either increasing impurity-medium repulsion or attraction. \nFor repulsively interacting impurities we elaborate on the competition of induced and direct interactions with the latter (former) dominating for \nrepulsive (attractive) impurity-medium couplings, evincing that the strength of induced interactions is larger for attractive impurity-bath interactions. \nInspecting the two-body correlation function of the Fermi sea we showcase that two fermions are likely to remain far apart (approach each other) for larger impurity-medium repulsions (attractions). \n\nWe trigger the dynamics by suddenly changing the impurity-medium interaction strength from zero to finite repulsive or attractive values. \nA quench to repulsive interactions induces in both components a collective breathing motion. \nThe impurities breathing frequency and amplitude depend on the post-quench coupling and their interacting nature. \nMoreover, a dynamical phase-separation occurs for quenches to large repulsions with the impurities residing at the origin and the \nfermionic environment splitting into two symmetric density branches with respect to the trap center. \nHere, two fermions are likely to lie one on the left and the other on the right density branch. \nInterestingly, induced impurity-impurity correlations mediated by the host are manifested in the course of the evolution of two \nnon-interacting impurities and become more pronounced for quenches to stronger repulsions. \nOn the other hand, monitoring the dynamics of repulsively interacting impurities we showcase the competition of induced and \ndirect interactions with the latter prevailing and enforcing the impurities to be in a two-body superposition. \nThe impact of induced interactions is also captured by the decaying amplitude in time of the impurities relative distance, which is \nclearly more prominent for non-interacting bosonic impurities. \n\nFor quenches to attractive impurity-medium interactions both components perform an overall breathing motion, whose amplitude and frequency regarding the impurities are impacted by the considered impurity-impurity and post-quench impurity-medium couplings. \nRemarkably, a beating pattern appears on the single-particle level stemming from the involvement of two nearly resonant breathing frequencies in the dynamics of the impurities due to the dominant nature of their attractive induced interactions. \nFurthermore, the impurities exhibit a spatial localization tendency around the trap center causing a density accumulation of the Fermi sea at their instantaneous location. \nThis mechanism becomes more pronounced for quenches to larger attractions and it is imprinted as a cross-like correlation pattern in the Fermi sea and dictates the dominant presence of attractive induced interactions whose strength is enhanced for quenches to larger attractions. \nIndeed they can even gradually surpass the direct impurity-impurity repulsive coupling, a result that is also evident by the prominent decaying amplitude of the impurities relative distance during the time-evolution. \n\nMoreover, by measuring the Von-Neumann entropy we explicate that in all cases the impurity-medium entanglement rises in a linear manner at the initial stages of the dynamics and afterwards it exhibits a fluctuating \nbehavior around a constant value. \nAlso, the entanglement exhibits a hierarchy by means that it is larger for fixed impurity (post-quench impurity-medium) interaction \nand increasing quench amplitude (impurity coupling). \n\nThere are several research directions that can be pursued in future endeavors. \nAn intriguing perspective is to study the robustness of the discussed phenomena in the presence of finite \ntemperature effects~\\cite{tajima2019thermal} and in particular their impact on the impurities induced interactions. \nCertainly, the generalization of our findings to higher dimensional settings as well as to larger impurity concentrations is desirable. \nAnother interesting direction would be to consider an additional long-range interparticle interaction potential~\\cite{kain2014polarons} \nand unravel the corresponding quench induced dynamics. \nThe emergent quasiparticle properties~\\cite{schmidt2018universal,mistakidis2019repulsive} such as the lifetime, residue, \neffective mass and induced interactions are of particular interest. \n\n\n\n\\begin{acknowledgments} \nK.M. acknowledges a research fellowship (Funding ID no 57381333) from the Deutscher Akademischer Austauschdienst (DAAD). \nS. I. M. gratefully acknowledges financial support in the framework of the Lenz-Ising Award of the University of  Hamburg. \nP.S. is grateful for financial support by the Deutsche Forschungsgemeinschaft (DFG) in the framework \nof the SFB 925 ``Light induced dynamics and control of correlated quantum systems''. \n\\end{acknowledgments} \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection{Formulation}\n\nUnlike pulsed lasers, pulsation in optical microresonators requires neither active nor passive mode locking elements (e.g. modulators or saturable absorbers)\\cite{haus2000mode, kutz2006mode}. Rather, pulsed states arise naturally from a simple damped driven nonlinear Schroedinger equation known as the Lugiato-Lefever equation (LLE) \\cite{lugiato1987spatial, haelterman1992dissipative, matsko2011mode, coen2013modeling, chembo2013spatiotemporal}. Two categories of stable pulsed state solutions have been identified for the LLE: stable modulational instability (MI, also known as hyper-parametric oscillations or Turing rolls) and stable cavity solitons \\cite{matsko2012hard, erkintalo2014coherence}. Turing rolls arise from the intra-cavity equilibrium field through modulational instability of vacuum fluctuations and usually have multiple-FSR (free spectral range) spacing between their adjacent teeth, while the word soliton is used to refer to coherent combs with single-FSR spacing.  Experimental and theoretical studies have suggested that soliton states are not accessible from the continuous wave (CW) intra-cavity field without seeding \\cite{taheri2015soliton, lobanov2015generation}, changing the pump frequency or power \\cite{matsko2012hard, lamont2013route, herr2014temporal}, or a suitable input pulse \\cite{leo2010temporal}. Owing to the low phase noise and exceedingly stable frequency spacing of the comb teeth in Turing rolls and solitons, chip-scale pure low-phase-noise radio frequency (RF) sources \\cite{liang2015high} and coherent communication with speeds in excess of 100 Gbit\/s per comb line have been demonstrated \\cite{pfeifle2014coherent, pfeifle2015optimally}.\\newline\n\\indent In addition to the generation of a frequency comb with equidistant teeth, temporal pulse generation requires mutual phase locking of the complex amplitudes. Phase locking in optical microresonators has been studied in terms of the cascaded emergence of phase-locked triplets \\cite{coillet2014robustness}. Injection locking of overlapping bunched combs has been explained using the Adler equation \\cite{del2014self}. Few-mode models have explained the phase offset between the pumped mode and the rest of the comb teeth \\cite{loh2014phase, taheri2015anatomy}. More recently, Wen \\emph{et al.} \\cite{wen2014self} have emphasized the link between oscillator synchronization---most famously described by the Kuramoto model \\cite{strogatz2000kuramoto}---and the onset of pulsing behavior. However, while stable ultrashort pulses have been demonstrated in a variety of microresonator platforms \\cite{herr2014temporal, saha2013modelocking, brasch2016photonic, vahala2015soliton}, the underlying phase locking mechanism is still unknown. As a result, features of microcomb phase spectra revealed in recent measurements \\cite{del2015phase} are yet not understood.\\newline\n\\indent In this paper, we introduce a reduced phase model which governs the nonlinear mode interactions responsible for spontaneous creation of pulsed states in the LLE which result from a balance between Kerr nonlinearity, dispersion (or, in the spatial case, diffraction), parametric gain, and cavity loss \\cite{grelu2012dissipative}.  The model interactions are \\emph{ternary} (that is, they involve three-variable combinations) rather than binary, as in typical phase models.  Our model admits attracting solutions which, interpreted in the context of nonlinear optical cavities, correspond to stable cavity solitons and Turing patterns, and provides an explanation of recent observations of phase steps in optical frequency combs. Moreover, our model clarifies the role of MI and chaos in the generation and stability of Turing rolls and solitons. \\newline\n\\indent The LLE is a nonlinear partial differential equation in time and the azimuthal angle around the whispering-gallery mode resonator \\cite{chembo2013spatiotemporal}, or, equivalently, in a slow and a fast time variable \\cite{haelterman1992dissipative,coen2013modeling}. Equivalently, a set of coupled nonlinear ordinary differential equations (ODEs) can be used to study resonator-based optical frequency comb generation \\cite{chembo2010modal}. The generalized spatiotemporal LLE in normalized form\n\\begin{equation}\\label{eq:LLE}\n\\pdifdisp{\\psi}{\\tau}=-(1+\\imi\\alpha)\\psi-\\imi \\frac{d_2}{2} \\frac{\\partial^2\\psi}{\\partial\\theta^2}+\\imi|\\psi|^2\\psi+F,\n\\end{equation}\nand its corresponding ODEs\n\\begin{equation}\\label{eq:CNODE}\n\\difdisp{\\tilde{a}_\\eta}{\\tau}=-(1+\\imi\\alpha)\\tilde{a}_\\eta+\\imi\\frac{d_2}{2}\\eta^2\\tilde{a}_\\eta+\\imi\\sum_{l,\\, m,\\, n}\\tilde{a}_l \\tilde{a}_m^*\\tilde{a}_n \\, \\delta_{\\eta_{lmn}\\eta}+\\tilde{F}_\\eta,\n\\end{equation}  \nare Fourier transform pairs, the conjugate variables of the transform being the azimuthal angle around the resonator $\\theta$ and comb mode number $\\eta$, see the Supplemental Material (SM). The number of ODEs equals the number of modes comprising the frequency comb; each equation follows the temporal evolution of the complex amplitude (magnitude and phase) of a single mode. In the ODEs picture, each optical comb tooth can be thought of as a nonlinear oscillator coupled to other oscillators (comb teeth). In Eq.~(\\ref{eq:LLE}), $\\psi(\\theta, \\tau)$ is the normalized field envelope, $\\tau = t\\Delta\\omega_0\/2$ is the normalized time with $\\Delta\\omega_0$ the resonance linewidth for the cavity mode closest to the pump (the pumped resonance) and $t$ the laboratory time, $\\alpha = -2(\\omega_{\\mathrm{P}}-\\omega_0)\/\\Delta\\omega_0$ is the normalized detuning between the pump laser frequency $\\omega_{\\mathrm{P}}$ and the cold-cavity pumped resonance frequency $\\omega_0$, $d_2 = -2D_2\/\\Delta\\omega_0$ is the normalized second-order dispersion parameter, $D_2$ being the cavity second-order dispersion coefficient, and $F$ is the normalized pump amplitude. The field envelop $\\psi$ and the pump amplitude $F$ are normalized to the sideband generation threshold such that the comb generation threshold in Eq.~(\\ref{eq:LLE}) is equal to unity \\cite{chembo2010modal}. In Eq.~(\\ref{eq:CNODE}), $\\tilde{a}_\\eta=a_\\eta(\\tau) \\exp[\\imi\\phi_\\eta(\\tau)]$ is the complex-valued comb tooth amplitude for mode $\\eta$ with magnitude $a_\\eta(\\tau)$ and phase $\\phi_\\eta(\\tau)$, $\\tilde{F}_\\eta(\\tau)$ is the Fourier transform of $F$ and equals $\\delta_{0\\eta}F_{\\mathrm{P}}\\exp(\\imi\\phi_{\\mathrm{P}})$ for CW pumping, $\\delta_{pq}$ (for integers $p$ and $q$) is the Kronecker delta, and $\\eta_{lmn}=l-m+n$. All mode numbers $\\eta$ are define relative to the pumped mode. For a soliton, $\\mathord{\\eta\\in\\{0, \\pm 1, \\dots, \\pm N\\}}$ while for Turing rolls $\\mathord{\\eta\\in\\{0, \\pm\\mu, \\pm2\\mu, \\dots, \\pm N\\mu\\}}$, where the integer $\\mu \\geq 1$ is the mode number at which MI gain peaks.\\newline\n\\indent When driven by a CW pump, experiments and numerical simulations suggest that for stable solutions, the magnitude of the pumped mode is much larger than that of the other modes and that in the absence of third- and higher-order dispersion, the magnitude spectrum of these solutions are symmetric with respect to the pumped mode $\\eta=0$ \\cite{saha2013modelocking,herr2014temporal} (see, e.g., the inset curves $a_\\eta^2$ vs. mode number in Fig.~\\ref{fig:prelim:phasealigned}). Therefore, we exploit the symmetry of the magnitude spectrum, adopt a perturbative approach (with $a_\\eta$ for $\\eta\\ne0$ as the small perturbation parameters), and following Ref.~\\cite{wen2014self} simplify Eq.~(\\ref{eq:CNODE}) by keeping only terms with at least one contribution from the pumped mode $a_0$ in the triple summations \\cite{taheri2015anatomy}. The magnitude and phase equations for the pumped mode include no linear contributions from $a_{\\eta\\ne0}$ (corrections are proportional to $a_\\eta^2$, $\\eta\\ne0$), and their solutions settle on a fast time scale to the equilibrium intra-cavity field $\\psi_\\mathrm{e}=a_0\\exp(\\imi\\phi_0)$; subsequently, $a_0$ and $\\phi_0$ can be treated as constants (\\cite{taheri2015anatomy}, also see SM).\\newline\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{phasealigned}\n\\caption[Phase alignment in (a) solitons and (b) Turing rolls.]{\\label{fig:prelim:phasealigned}Phase alignment in (a) solitons and (b) Turing rolls seen in the steady-state solutions of Eqs.~(\\ref{eq:LLE}) and (\\ref{eq:CNODE}). The inset curves in red (top corners) show the spatiotemporal waveforms and those in black (bottom corners) are the frequency spectra. For both solitons and rolls the phases lie on straights lines of arbitrary slope. Parameter values are (a) $\\alpha=2,\\, d_2=-0.0124,\\, F=1.41$, and (b) $\\alpha=0,\\, d_2=-0.0124,\\, F=1.63$. The phase profile has been unwrapped in (b).}\n\\end{figure}\n\\indent Equations of motion for the magnitudes $a_\\eta(\\tau)$ and phases $\\phi_\\eta(\\tau)$ are readily found from Eq.~(\\ref{eq:CNODE}). The equation for the temporal evolution of the centered phase averages $\\zeta_\\eta=\\bar{\\phi}_\\eta-\\phi_0$, where the phase average $\\bar{\\phi}_\\eta=(\\phi_\\eta+\\phi_{-\\eta})\/2$ is centered to the pumped mode phase $\\phi_0$, can be found using the equations for $\\phi_{\\pm\\eta}$ and $\\phi_0$. To lowest non-zero order in $a_{\\eta\\ne 0}$, this equation can be integrated directly to give\n\\begin{equation}\\label{eq:antisym}\n\\tan{\\zeta_\\eta}=\\sqrt{\\abs*{\\frac{C+2}{C}}}\\tanh[\\sqrt{|C(C+2)|}a_0^2(\\tau-\\tau_0)].\n\\end{equation}\nHere $C=d_2\\eta^2\/2a_0^2-F_\\mathrm{P}\\sin(\\phi_\\mathrm{P}-\\phi_0)\/a_0^3$, $\\phi_\\mathrm{P}$ and $F_\\mathrm{P}$ are the phase and normalized magnitude of the pump, and $\\tau_0$ accounts for constants of integration (or initial conditions). Equation~(\\ref{eq:antisym}) holds when $|2a_0^2-\\alpha+d_2\\eta^2\/2|<a_0^2$, a condition that is automatically satisfied when MI gain exists (see SM).  Because the hyperbolic function approaches unity as $\\mathord{\\tau\\to\\infty}$, $\\bar{\\phi}_\\eta$ reaches the same constant irrespective of the initial conditions.  Since $\\bar{\\phi}_\\eta$ is fixed, each pair of phases $\\phi_{\\pm \\eta}$ must take values symmetrically located relative to the same value. We will refer to this as ``\\emph{anti-symmetrization}'' of the phases, following the terminology used in \\cite{wen2014self}.  Once established, phase anti-symmetrization means each centered phase average $\\zeta_\\eta$ can be treated as a constant to first order in $a_{\\eta\\ne0}$. \\newline\n\\indent The equations of motion for the phase differences $\\Delta_\\eta=(\\phi_\\eta-\\phi_{-\\eta})\/2$, \n\\begin{equation}\\label{eq:pl}\n\\dot{\\Delta}_\\eta=\\frac{a_0}{a_\\eta}\\sum\\nolimits_l K(l,\\eta) \\sin(\\Delta_l+\\Delta_{\\eta-l}-\\Delta_\\eta),\n\\end{equation}\nare found by combining the equations for each $\\pm\\eta$ mode pair (see SM). In Eq.~(\\ref{eq:pl}), the over-dot indicates time derivative, and $K(l,\\eta)=a_l a_{\\eta-l} \\{2 \\sin(\\zeta_\\eta-\\zeta_{\\eta-l}+\\zeta_l)+\\sin(\\zeta_\\eta-\\zeta_{\\eta-l}-\\zeta_l)\\}$ is the coupling coefficient for the pump--non-degenerate interaction of comb teeth $\\tilde{a}_0$, $\\tilde{a}_l$, $\\tilde{a}_{\\eta-l}$, and $\\tilde{a}_\\eta$. A family of fixed point solutions of Eq.~(\\ref{eq:pl}) is $\\Delta_\\eta=s_0\\eta+k\\pi$, where $s_0$ is a constant and $k$ an integer.  These solutions imply that the phases have aligned:  the slope of the line passing through the phases of any pair of comb teeth $\\eta$ and $-\\eta$ will be the same and equal to $s_0$, that is $(\\phi_\\eta-\\phi_{-\\eta})\/2\\eta=s_0$. Figure~\\ref{fig:prelim:phasealigned} shows two examples, in solitons and Turing rolls, where Eq.~(\\ref{eq:LLE}) has been integrated numerically using the split-step Fourier transform method for a typical microresonator.\n\nNext, we consider the stability of these states.  For simplicity, we take $k=0$.  (Stability analysis for $k\\ne0$ follows in a similar way.)  The linear stability of a frequency comb with $2N+1$ phase-locked teeth, with mode numbers $\\mathord{\\eta\\in\\{0,\\pm 1,\\pm 2,\u2026,\\pm N\\}}$, is found using Eq.~(\\ref{eq:pl}). We temporarily ignore the dependence of the comb teeth magnitudes on the mode number and take $a_\\eta=a$. (The effect of the mode number dependence of the comb teeth magnitudes will be included shortly.) After phase locking, the centered phase averages $\\zeta_\\eta$ reach a steady-state value which is independent of mode number $\\eta$ since the phases $\\phi_\\eta$ lie on a straight line. Therefore, the coupling coefficients in Eq.~(\\ref{eq:pl}) are all equal, i.e., $K(l,\\eta)=K > 0$. The Jacobian matrix $\\bm{\\mathrm{J}}$ and its eigenvalues can be expressed in closed form for any $N$ (see SM). Except for one zero eigenvalue forced by the rotational symmetry of the LLE, all of the eigenvalues are negative and real, indicating asymptotic stability of the synchronized state. Figure~\\ref{fig:prelim:eigs}(a) shows the non-zero eigenvalues of the equilibrium for increasing comb span ($2N+1$). It is seen that the eigenvalue closest to zero grows more negative with increasing comb span. Hence, for the constant comb amplitude case, a wider comb demonstrates superior stability.\n\nTo investigate the effect of a non-constant comb amplitude profile, we set $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$. This profile assumes a linear decay (in logarithmic scale) of the comb teeth magnitude with slope $-20k_0$ dB per increasing mode number by unity, (see, e.g., the insets $a_\\eta^2$ vs. mode number in Fig.~\\ref{fig:prelim:phasealigned}).  Though not analytically tractable, we find numerically that the eigenvalues of $\\bm{\\mathrm{J}}$ all have negative real part (except for the single zero eigenvalue forced by symmetry). Figure~\\ref{fig:prelim:eigs}(b) shows the eigenvalue spectrum vs. increasing combs span for $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$. Note that as the comb span increases, the smallest magnitude eigenvalue becomes bounded and almost independent of $N$ (black curve in Fig.~\\ref{fig:prelim:eigs}(b)). Therefore, the stability of the comb \\emph{does not improve}---nor does it degrade---with increasing comb span when the mode number dependence of the comb teeth magnitude is taken into account. Pfeifle \\emph{et al}. \\cite{pfeifle2015optimally} showed that in the presence of pump magnitude and frequency noise, solitons are less robust than Turing rolls in the same microresonator with comparable pump power. Our results suggest that the superior stability of Turing rolls does not originate from their smaller number of comb teeth compared to solitons. Rather, it is linked to the presence of MI gain, which is responsible for the generation of Turing rolls from vacuum fluctuations. We note that Eq.~(\\ref{eq:pl}) does not explicitly include the effect of MI gain; this influence is reflected through the coupling coefficients $K(l,\\eta)$.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{eigs}\n\\caption{\\label{fig:prelim:eigs}Non-zero eigenvalues of the equilibrium (the Jacobian matrix $\\bm{\\mathrm{J}}$) versus comb span for Eq.~(\\ref{eq:pl}) for (a) uniform and (b) mode-number--dependent comb teeth magnitude profile of $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$, ($k_0=0.1$). The negative eigenvalue of smallest magnitude (black curve) increases in size with increasing comb span for constant magnitudes, but reaches a constant for the realistic comb magnitude profile.}\n\\end{figure}\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{numsol}\n\\caption[Numerical solutions of Eq.~(\\ref{eq:pl}) for uniform as well as mode-number--dependent comb teeth magnitude spectra, and the emergence of phase steps.]{\\label{fig:prelim:numsol}Numerical solutions of Eq.~(\\ref{eq:pl}) for uniform and mode-number--dependent comb teeth magnitude spectra, and the emergence of $\\pi$ phase steps. (a) Sample steady-state solution of Eq.~(\\ref{eq:pl}) for a comb with 101 teeth, with uniform magnitudes and random initial conditions for the phase differences (shown in (e)). (b) The temporal evolution of the phase differences shown in (a). Most of the phase differences settle to integer multiples of $\\pi$, but some deviations may arise (dotted red circles). (c,d) Same as (a,b) but for comb magnitude profile of $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$, ($k_0=0.1$). All of the phase differences $\\Delta_\\eta$ settle to integer multiples of $\\pi$. In (a,c), only the $\\pi$ phase steps are physically significant. The $2\\pi$ steps have not been removed (e.g., through unwrapping the phases) to better illustrate the correspondence of (a,c) with (b,d). (e) The phase differences at the onset of integration (initial conditions) for (a-d). (f) Schematic illustrating $\\pi$ steps in the phase profile $\\phi_\\eta$ of a comb resulting from steps in the phase differences $\\Delta_\\eta$. Comb teeth symmetrically positioned around the pumped mode ($\\eta$ and $-\\eta$) will show $\\pi$ phase steps, with one phase increasing as its counterpart decreases. Such phase steps do not change the phase averages $\\bar{\\phi}_\\eta$.}\n\\end{figure*}\n\nWe consider next numerical solutions of Eq.~(\\ref{eq:pl}). Our numerous runs of numerical integration for different comb spans ($N$ from 5 to 1000) and random initial phase differences taken from a uniform distribution over the range $(-\\pi, \\pi]$ and for uniform comb magnitude spectrum, typically lead to $\\Delta_\\eta=k\\pi$, ($k$ an integer). While a steady-state is always obtained, other steady-state solutions are also possible. A mode-number-dependent comb magnitude spectrum, in contrast, always leads to phase differences equal to integer multiples of $\\pi$, even for those cases in which steady-state phase differences for uniform comb magnitude spectra are not equal to integer multiples of $\\pi$. The reason is that a non-uniform comb magnitude profile places more strict constraints on the steady-state solution of Eq.~(\\ref{eq:pl}). In Figs.~\\ref{fig:prelim:numsol}(a-d), we show sample solutions found by numerically integrating Eq.~(\\ref{eq:pl}) for $N=50$ phase differences (a comb with $2N+1=101$ teeth) for constant comb teeth magnitudes (Fig.~\\ref{fig:prelim:numsol}(a,b)) and the non-constant magnitude profile of $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$ (Fig.~\\ref{fig:prelim:numsol}(c,d)). We show the steady-state solutions $\\Delta_\\eta$ at the end of the simulation time vs. mode number as well as the the evolution of the phase differences with time. The initial values of $\\Delta_\\eta$, $\\mathord{\\eta\\in\\{1,2,\u2026,50\\}}$, in both cases is shown in Fig.~\\ref{fig:prelim:numsol}(e). While most of the steady-state phase differences in Fig.~\\ref{fig:prelim:numsol}(a) are integer multiples of $\\pi$, some of them deviate from these values (the dotted red circles). For the non-uniform magnitude spectrum, however, steady-state phase differences are all integer multiples of $\\pi$, as seen in Fig.~\\ref{fig:prelim:numsol}(c). \\newline\n\\indent The $\\pi$ phase steps in the phase differences $\\Delta_\\eta$ imply similar steps in the phases $\\phi_\\eta$. To show this, we assume a set of solutions $\\Delta_\\eta=s_0\\eta$ is known and try to find another set based on it. It may seem that any constant $x$ can be added to the phases of comb teeth symmetrically positioned with respect to the pumped mode (i.e., $\\phi_{\\pm\\eta}\\to\\phi_{\\pm\\eta}+x$) without affecting the solution. Unfortunately, this alters the phase averages $\\bar{\\phi}_\\eta$ and so invalidates the stability analysis presented earlier. However, we can generate new stable phase-locked solutions by considering anti-symmetric changes of the phases, i.e., $\\phi_\\eta\\to\\phi_\\eta\\pm x$ and $\\phi_{-\\eta}\\to\\phi_{-\\eta}\\mp x$, which means $2\\Delta_\\eta=\\phi_\\eta-\\phi_{-\\eta}\\pm 2x=2s_0\\eta +2k\\pi$, and hence $x=\\pm \\pi$ (recall that $k$ is an integer). This demonstrates that the appearance of $\\pi$ steps in the phase spectrum of stable phase-locked frequency combs is permissible, as shown schematically in Fig.~\\ref{fig:prelim:numsol}(f). Indeed, such phase steps have been observed experimentally \\cite{del2015phase} (see, e.g., Fig.~\\ref{fig:prelim:steps}(a)) but have, to the best of our knowledge, remained unexplained until now. \\newline\n\\indent Besides $\\pi$ phase steps, $\\pi\/2$ phase steps \\cite{del2015phase} have also been observed in experiments. These phase steps also can be explained within the framework of our model, as follows. A comb with $\\pi\/2$ phase steps is in fact two interleaved \\emph{non-interacting} combs, each of which has $\\pi$ steps in its phase spectrum \\cite{del2015phase}. These two combs do not interact as a result of the $\\pi\/2$ offset between their phase spectra because this phase offset causes the coupling coefficients between their comb teeth, $K(l, \\eta)$, to vanish. To clarify this point, we refer to Fig.~\\ref{fig:prelim:steps}(c,d) where comb teeth labeled with $\\mathord{\\eta_\\mathrm{A}\\in\\{0,2,4,6,\u2026,14\\}}$ (red) share a constant phase, while those with mode numbers $\\mathord{\\eta_\\mathrm{B}\\in\\{1,3,5,\u2026,13\\}}$ (blue) share another phase, different from that of the former group by $\\pi\/2$, such that $\\zeta_1=\\bar{\\phi}_1-\\phi_0=\\pi\/2$ and $\\zeta_{\\eta_\\mathrm{A}}-\\zeta_{\\eta_\\mathrm{B}}=\\pi\/2$ (recall that the value of $\\zeta_{\\eta}$ is independent of the mode number for a stable comb, cf. Fig.~\\ref{fig:prelim:numsol}(f)). As a result, the coupling coefficient $K(\\eta, \\eta+1)$ is zero because $\\zeta_{\\eta+1}\\pm\\zeta_{\\eta}=\\pm\\pi\/2$, and so there is no coupling between modes $\\eta$ and $\\eta+1$ of the comb for any $\\eta$ (see Eq.~(\\ref{eq:pl})). It is worth noting that the frequency combs with phase steps in \\cite{del2015phase} were obtained through tuning the laser pump into resonance, which alters the MI gain profile and sweeps its peak.\\newline\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{steps}\n\\caption[Experimental data showing steps in the measured phase spectra of optical frequency combs.]{\\label{fig:prelim:steps}Experimental data showing steps in the measured phase spectra of optical frequency combs. (a,c) are the phase spectra while (b,d) depict the power profiles of the combs. (a,b) $\\pi$-steps in the phase spectrum of a stable comb; The green dot corresponds to the pumped mode phase. (c,d) $\\pi$\/2-steps in the phase spectrum of a stable comb; This comb is recognized as two interleaved combs (red and blue) with phases offset by $\\pi\/2$, each exhibiting $\\pi$-phase steps as well (cf. Fig.~\\ref{fig:prelim:numsol}(f)). The $\\pi\/2$-phase offset leads to the decoupling of the two combs, indicated by vanishing coupling coefficients, i.e., $K(l, \\eta)=0$ in Eq.~(\\ref{eq:pl}). The green dots correspond to the phases of the stronger comb teeth. (These plots are reproduced using data originally presented in~\\cite{del2015phase}.)}\n\\end{figure}\nIt has been argued that passing through the chaotic state is necessary for microcomb soliton formation \\cite{lamont2013route}. The foregoing analysis suggests a way of understanding this: passage through the chaotic state serves to provide the system with a large pool of initial conditions, which increases the odds of getting peaks that will then grow into solitons. Numerical simulations of Eq.~(\\ref{eq:pl}) suggest that with increasing comb span and non-uniform comb magnitude spectrum, chances of getting groups of phase-locked comb teeth, or weak pulses, will increase. These weak pulses will then grow into the modes of the nonlinear system (i.e., the solitons). \\newline\n\\indent Finally, although we have compared our theoretical results with microresonator-based frequency comb experiments, they should also apply to mode-locked laser systems.  In 2002, Gordon and Fisher developed a many-body statistical mechanical theory to describe the onset of laser pulsations as a first order phase transition, treating the modes as the elementary degrees of freedom \\cite{gordon2002phase}.  Their ordered collective state is analogous to our synchronized dynamical attractor.  Now, Eq.~(\\ref{eq:pl}) roots in the cubic nonlinear term in the LLE, and the same nonlinearity appears in the master equation for passive mode locking based on a saturable absorber, which approximates the absorber with a cubic nonlinearity \\cite{haus2000mode}. We therefore expect the same dynamical mechanism to be responsible for the creation of sharp pulses in passively mode-locked lasers, despite the different physical source of optical gain (population inversion and stimulated emission rather than parametric amplification).  What matters is the fundamental link between spatiotemporal pulse formation and mode synchronization. \\newline\n\\indent The generic reduced nonlinear oscillator model introduced in this work clearly demonstrates the fundamental link between mode synchronization and spatiotemporal pulse formation in Kerr-nonlinear media. This model admits attracting fixed point solutions corresponding to stable cavity solitons and Turing patterns, permits analyzing their stability in a unified scheme, and explains phase jumps observed in recent phase measurements of stable optical frequency combs. It also provides insight into the role of chaos and parametric gain in the generation of solitons and Turing rolls. This insight can be utilized towards devising novel techniques for controlled formation of robust pulses in optical microresonators.\n\\begin{acknowledgments}\nK.W. and H.T. thank Brian Kennedy for useful discussions.  K.W. also thanks Henry Wen and Steve Strogatz for generously discussing the details of their results reported in \\cite{wen2014self}.  H.T. was supported by the Air Force Office of Scientific Research Grant No.~2106DKP.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}