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+{"text":"\\section{Introduction}\\label{sec:intro}\nA new liquid crystal equilibrium phase was recently discovered \\cite{cestari:phase,chen:chiral,borshch:nematic,chen:twist-bend}. Any such discovery is \\emph{per se} a rare event, but this was even more striking as in some specific materials an achiral phase which had long been known was shown to conceal a chiral mutant, attainable upon cooling through a weakly first-order transition. This is known as the twist-bend nematic ($\\mathrm{N_{tb}}$) phase.\\footnote{The name \\emph{twist-bend} was introduced by Dozov~\\cite{dozov:spontaneous} together with \\emph{splay-bend} to indicate the two alternative nematic distortions which, unlike pure bend and pure splay, can fill the three-dimensional space, as previously observed by Meyer~\\cite{meyer:structural}.} Molecular bend seems to be necessary for such a phase to be displayed, but it is not sufficient, as most bent-core molecules fail to exhibit it \\cite{jakli:liquid}. In the molecular architecture capable of inducing the $\\mathrm{N_{tb}}$ phase, dimers with rigid cores are connected by sufficiently flexible linkers.\\footnote{Very recently, experimental evidence has also been provided for \\TBN phases arising from \\emph{rigid} bent-core molecules \\cite{chen:twist-bend}.} The molecular effective curvature, while inducing \\emph{no} microscopic \\emph{twist}, allegedly favors a \\emph{chiral} collective arrangement in which bow-shaped molecules uniformly precess along an ideal cylindrical \\emph{helix}.\n\nFigure~\\ref{fig:spiral_arc} sketches the picture envisaged here.\n\\begin{figure}\n  \\centering\n  \\subfigure[]{\\includegraphics[width=.35\\linewidth,angle=0]{arc.eps}}\n  \\hspace{.05\\linewidth}\n  \\subfigure[]{\\includegraphics[width=.2\\linewidth,angle=0]{spiral_new.eps}}\n  \\caption{(a) Molecular achiral model with two symmetry axes, one polar, $\\bm{p}$, and one apolar, $\\bm{m}$. (b) One variant of the helical nematic phase with helix axis $\\twist$. In the other variant, not shown here, the helix winds downwards, in the direction opposite to $\\twist$.}\n\\label{fig:spiral_arc}\n\\end{figure}\nA single bow-shaped molecule exhibits two symmetry axes, represented by the unit vectors $\\bm{p}$ and $\\bm{m}$, polar the former and apolar the latter.\\footnote{Despite a visual illusion caused by the curved arc, in Fig.~\\ref{fig:spiral_arc}(a) the lengths of $\\bm{p}$ and $\\bm{m}$ are just the same.} The local symmetry point-group is $\\mathsf{C}_{2v}$, but, as explained by Lorman and Mettout~\\cite{lorman:unconventional,lorman:theory}, by combining the symmetries broken in the helical arrangement in Fig.~\\ref{fig:spiral_arc}(b), namely, the continuous translations along the helix axis and the continuous rotations around that axis, a symmetry is recovered which involves any given translation along the helix axis, provided it is accompanied by an appropriately tuned rotation. This forbids any smectic modulation in the mass density, rendering the helical phase purely nematic. While the nematic director $\\n$ is defined as the ensemble average $\\n:=\\langle\\bm{m}\\rangle$, no polar order survives in a helical phase, as $\\langle\\bm{p}\\rangle=\\bm{0}$.\\footnote{For this reason, no macroscopic analogue of $\\bm{p}$ will be introduced in the theory, and the phase will be treated as macroscopically \\emph{apolar}.} Fig.~\\ref{fig:spiral_arc}(b) shows only one of the chiral variants that symmetry allows in a \\TBN phase; the other winds in the direction opposite to $\\twist$. In both cases, $\\n$ makes a fixed \\emph{cone angle} $\\ca$ with $\\twist$. For definiteness, we shall call each chiral variant a \\emph{helical} nematic phase. In a different language, each helical nematic phase is precisely the $C$-phase predicted by Lorman and Mettout \\cite{lorman:unconventional,lorman:theory}, which breaks spontaneously the molecular chiral symmetry, producing two equivalent macroscopic variants with opposite chiralities (see Fig.~\\ref{fig:spiral_arc}). This paper is intended to study separately each helical nematic variant hosted in a \\TBN phase. How opposite variants may be brought in contact within a purely elastic theory will be the subject of a forthcoming paper \\cite{virga:double-well}.\n\nThough, in retrospect, a number of experimental studies had already anticipated \\TBN phases (see, for example, \\cite{sepelj:intercalated,imrie:liquid,panov:spontaneous}, to cite just a few), a clear identification of the new phase was achieved in \\cite{cestari:phase} by a combination of methods (see also \\cite{henderson:methylene,cestari:crucial,panov:microsecond,panov:field-induced,beguin:chirality}), and an impressive direct evidence for it has been provided in \\cite{chen:chiral,borshch:nematic,chen:twist-bend} (see also \\cite{copic:nematic}), with accurate measurements of both the helix pitch ($\\approx10\\,\\mathrm{nm}$) and cone angle ($\\approx20\\degree$).\nTheoretically, $\\mathrm{N_{tb}}$ phases had already been predicted by Meyer~\\cite{meyer:structural} and Dozov~\\cite{dozov:spontaneous}, from two different perspectives, the former inspired by the symmetry of polar electrostatic interactions (a line of thought recently further pursued in \\cite{shamid:statistical}) and the latter starting from purely elastic (and steric) considerations. A helical molecular arrangement was also seen in the molecular simulations of Memmer~\\cite{memmer:liquid}, who considered bent-core Gay-Berne molecules with \\emph{no} polar electrostatic interactions, though featuring an effective shape polarity.\n\nDozov's $\\mathrm{N_{tb}}$ theory requires a \\emph{negative} bend elastic constant, which is compatible with the boundedness (from below) of the total energy only if appropriate \\emph{quartic} corrections are introduced in the energy density. However, a large number of these terms are allowed by symmetry \\cite{dozov:spontaneous}, and the theory may realistically be applied only by choosing just a few of them and neglecting all the others \\cite{meyer:flexoelectrically}. Moreover, recent experiments \\cite{adlem:chemically} have reported an increase in the (positive) bend constant measured in the nematic phase near the transition to the $\\mathrm{N_{tb}}$ phase. In an attempt to justify the negative elastic bend constant required by Dozov's theory, a recent study has replaced it with an \\emph{effective} bend constant resulting from the coupling with the polarization characteristic of flexo-elasticity \\cite{shamid:statistical}. As a consequence, however, the twist-bend modulated phase envisaged in \\cite{shamid:statistical} is locally \\emph{ferroelectric}, whereas, as explained by the symmetry argument of Lorman and Mettout~\\cite{lorman:theory,lorman:unconventional}, each helical nematic phase is expected to be \\emph{apolar}.\n\nThe theories in \\cite{dozov:spontaneous} and \\cite{shamid:statistical} are not in contradiction with one another, the only difference being that the latter justifies a negative bend elastic constant as commanded by the very bend-polarization coupling that gives rise to a modulated polar phase. There is still a conceptual difference between these theories: the former is purely elastic but quartic, whereas the latter is quadratic but flexo-electric. Being highly localized, the ferroelectricity envisaged in \\cite{shamid:statistical} does not produce any average macroscopic polarization, and so it would be compatible with the current experimental observations which have not found so far any trace of macroscopic ferroelectricity. This, however, leaves the question unanswered as to whether an intrinsically quadratic, purely elastic theory could also explain \\TBN phases.\nIn this work, we propose such a theory; it will feature the same elastic constants as Frank's classical theory \\cite{frank:theory}, with a positive bend constant and an extra director field. This theory reduces to Frank's for ordinary nematics, for which the extra director becomes redundant.\n\nThis paper is organized as follows. In Section~\\ref{sec:helical}, we shall formally introduce a helical nematic phase, defined as each \\TBN variant with a prescribed \\emph{helicity} in its ground state.\\footnote{See \\eqref{eq:helicity} for a precise definition.} A quadratic elastic free energy will be considered for each helical nematic variant, under the (temporary) assumption that they can be thought of as separate manifestations of one and the same \\TBN phase. For a given sign of the helicity, negative for definiteness, we shall apply the proposed elastic theory to two classical instabilities, namely, the helix unwinding first encountered in chiral nematics (in Section~\\ref{sec:helix}), and the Freedericks transition that has long made it possible to measure the elastic constants of ordinary nematics (in Section~\\ref{sec:freedericks}). Both these applications will acquire some new nuances within the present theory. In Section~\\ref{sec:Double-Well}, we first derive the quadratic elastic energy density for a helical nematic phase with \\emph{positive} helicity. The helical nematic variants with both helicities are then combined together in a \\TBN phase; the corresponding elastic energy density need to attain one and the same minimum in two separate \\emph{wells}. There are several ways to construct such an energy, which by necessity will \\emph{not} be \\emph{convex}; we shall build upon the quadratic elastic energy for a single helical phase arrived at in Section~\\ref{sec:helical}. In the two ways that we consider in detail, the elastic energy density for a \\TBN phase features only four elastic constants. Finally, in Section~\\ref{sec:conclusion}, we collect the conclusions reached in this work and comment on some possible avenues for future research.\n\n\n\\section{Helical nematic phases}\\label{sec:helical}\nA \\TBN differs from classical nematics in its ground state, the state relative to which the elastic cost of all distortions is to be accounted for. The ground state of a classical nematic is the uniform alignment (in an arbitrary direction) of the nematic director $\\n$. The $\\mathrm{N_{tb}}$ ground state is a \\emph{heliconical} twist, in which $\\n$ performs a uniform precession, making everywhere the angle $\\ca$ with a helix axis, $\\twist$, arbitrarily oriented in space.\nSuch a ground state should reflect the intrinsically less distorted molecular arrangement that results from minimizing the interaction energy of the achiral, bent molecules that comprise the medium.\n\nBy symmetry, there are indeed \\emph{two} such states, distinguished by the sense of precession (either clockwise or anticlockwise around $\\twist$). In general, borrowing a definition from Fluid Dynamics (see, for example, \\cite{moffatt:degree}), we call the pseudoscalar\n\\begin{equation}\\label{eq:helicity}\n\\eta:=\\n\\cdot\\curl\\n,\n\\end{equation}\nthe \\emph{helicity} of the director field $\\n$. We shall see now that it is precisely the sign of $\\eta$ that differentiates the two variants of the ground state of a $\\mathrm{N_{tb}}$.\n\nLetting $\\twist$ coincide with the unit vector $\\e_z$ of a Cartesian frame $(x,y,z)$, the fields $\\ground^\\pm$ representing the ground states can be written in the form\n\\begin{equation}\\label{eq:ground}\n\\begin{split}\n\\ground^\\pm=&\\sin\\ca\\cos(\\pm qz+\\az_0)\\,\\e_x\\\\+&\\sin\\ca\\sin(\\pm qz+\\az_0)\\,\\e_y+\\cos\\ca\\,\\e_z,\n\\end{split}\n\\end{equation}\nwhere $\\az_0$ is an arbitrary phase angle, $q$ is a prescribed \\emph{wave parameter}, taken to be non-negative, as characteristic of the condensed phase as the \\emph{cone angle} $\\ca$ (see Fig.~\\ref{fig:spherical}).\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\linewidth,angle=0]{Spherical.eps}\n\\caption{The angles $\\ca$ and $\\az=\\pm qz+\\az_0$ that in \\eqref{eq:ground} represent the fields $\\ground^\\pm$ are illustrated in a Cartesian frame $(x,y,z)$. Both $\\ca$ and $q$ are fixed parameters characteristic of the phase.}\n\\label{fig:spherical}\n\\end{figure}\nThe \\emph{pitch} $p$ corresponding to $q$ is given by\\footnote{No confusion should arise here between the pitch $p$ of the \\TBN phase and the polar vector $\\bm{p}$ mentioned in the Introduction. The former is macroscopic in nature, whereas the latter is microscopic.}\n\\begin{equation*}\\label{eq:p}\np:=\\frac{2\\pi}{q}.\n\\end{equation*}\nOn every two planes orthogonal to $\\e_z$ and $p$ apart, each field $\\ground^\\pm$ in \\eqref{eq:ground} delivers one and the same nematic director. A simple computation shows that\n\\begin{equation}\\label{eq:helicity_ground_state}\n\\eta^\\pm:=\\ground^\\pm\\cdot\\curl\\ground^\\pm=\\mp q\\sin^2\\ca.\n\\end{equation}\n\nWe shall call \\emph{helical nematic} each of the two phases for which either $\\ground^+$ or $\\ground^-$ is the nematic field representing the ground state. Here, for definiteness, we shall develop our theory as if only $\\ground^+$ represented the ground state. By \\eqref{eq:helicity_ground_state}, such a state has \\emph{negative} helicity. The corresponding case of positive helicity will be studied in Section~\\ref{sec:positive}. Until then we shall drop the superscript $^+$ from $\\ground^+$ to avoid clatter.\n\n\\subsection{Negative helicity}\\label{sec:negative}\nIt readily follows from \\eqref{eq:ground} that\\footnote{Here and below $\\times$ and $\\otimes$ denote the vector and tensor products of vectors. In particular, for any two vectors, $\\bm{a}$ and $\\bm{b}$, $\\bm{a}\\otimes\\bm{b}$ is the second-rank tensor that acts as follows on a generic vector $\\bm{v}$, $(\\bm{a}\\otimes\\bm{b})\\bm{v}:=(\\bm{b}\\cdot\\bm{v})\\bm{a}$, where $\\cdot$ denotes the inner product of vectors. An alternative way of denoting the dyadic product $\\bm{a}\\otimes\\bm{b}$ would be simply $\\bm{a}\\bm{b}$. In Cartesian components, they are both represented as $a_ib_j$.}\n\\begin{equation}\\label{eq:grad_ground}\n\\nabla\\ground=q\\left(\\e_z\\times\\ground\\right)\\otimes\\e_z.\n\\end{equation}\nMore generally, for $\\n$ prescribed at a point in space, the tensor\n\\begin{equation}\\label{eq:natural_distortion}\n\\Twist:=q(\\twist\\times\\n)\\otimes\\twist\n\\end{equation}\nexpresses the \\emph{natural} distortion\\footnote{A natural distortion is a distortion present in the ground state, when the latter fails to be the uniform nematic field $\\n$ for which $\\N\\equiv\\zero$.} that would be associated at that point with the preferred helical configuration agreeing with the prescribed nematic director $\\n$ and having $\\twist$ as helix axis. We imagine that in the absence of any frustrating cause, given $\\n$ at a point, the director field would attain in its vicinity a spatial arrangement such that\n\\begin{equation}\\label{eq:local_and_natural_distortions}\n\\nabla\\n=\\Twist,\n\\end{equation}\nwhere $\\Twist$ is as in \\eqref{eq:natural_distortion} and $\\twist$ is any unit vector such that\n\\begin{equation}\\label{eq:cone_constraint}\n\\n\\cdot\\twist=\\cos\\ca.\n\\end{equation}\nThis would make \\eqref{eq:grad_ground} locally satisfied, even though $\\n$ does not coincide with $\\ground$ in the large. Correspondingly, the elastic energy that would locally measure the distortional cost should be measured relative to the whole class of natural distortions, vanishing whenever any of the latter is attained. With this in mind, we write the elastic energy density $f^-_e$ in the form\\footnote{The superscript $^-$ will remind us that the ground state of the helical nematic phase we are considering here has a negative helicity $\\eta$,}\n\\begin{equation}\\label{eq:elastic_energy_density_definition}\nf^-_e(\\twist,\\n,\\N)=\\frac12(\\N-\\Twist)\\cdot\\elan[\\N-\\Twist],\n\\end{equation}\nwhere $\\elan$ is the most general positive-definite, symmetric fourth-order tensor invariant under rotations about $\\n$. Clearly, if for given $\\n$ and $\\N$, $\\twist$ can be chosen in \\eqref{eq:elastic_energy_density_definition} so that \\eqref{eq:local_and_natural_distortions} is satisfied, $f^-_e$ vanishes, attaining its absolute minimum. On the contrary, if  there is no such $\\twist$, then minimizing $f^-_e$ in $\\twist$ would identify the natural, undistorted state closest to the nematic distortion represented by $\\N$ in the metric induced by $\\elan$. For this reason, here both $\\n$ and $\\twist$ are to be considered as unknown fields linked by \\eqref{eq:cone_constraint}: at equilibrium, the free-energy functional that we shall construct is to be minimized in both these fields.\n\nCombining the general representation formula for $\\elan$ and the identities\\footnote{The superscript $\\T$ means tensor transposition.}\n\\begin{equation*}\\label{eq:tensor_identities}\n(\\N)\\T\\n=\\zero,\\qquad\\Twist\\T\\n=\\zero,\n\\end{equation*}\nwe can reduce $\\ela$ in \\eqref{eq:elastic_energy_density_definition} to the following equivalent form\n\\begin{equation}\\label{eq:K_reduced_representation}\n\\ela_{ijhk}=k_1\\delta_{ih}\\delta_{jk}+k_2\\delta_{ij}\\delta_{hk}+k_3\\delta_{ih}n_jn_k\n+k_4\\delta_{ik}\\delta_{jh},\n\\end{equation}\nwhere $k_1$, $k_2$, $k_3$, and $k_4$ are elastic moduli (as in \\cite[p.\\,114]{virga:variational}). By use of \\eqref{eq:K_reduced_representation}, of \\eqref{eq:cone_constraint}, and\n\\begin{equation*}\\label{eq:scalar_identities}\n\\tr\\Twist=0,\\qquad \\N\\cdot\\Twist=q(\\twist\\times\\n)\\cdot(\\N)\\twist,\n\\end{equation*}\nwe transform \\eqref{eq:elastic_energy_density_definition} into\n\\begin{equation}\\label{eq:elastic_energy_density_transformed}\n\\begin{split}\nf^-_e(\\twist,\\n,\\N)\n&=\\frac12\\Bigl\\{K_{11}(\\diver\\n)^2+K_{22}(\\n\\cdot\\curl\\n+q|\\twist\\times\\n|^2)^2\\\\\n&+K_{33}|\\n\\times\\curl\\n+q(\\n\\cdot\\twist)\\,\\twist\\times\\n|^2\\\\\n&+K_{24}[\\tr(\\N)^2-(\\diver\\n)^2]\\Bigr\\}\n-K_{24}q\\,\\twist\\times\\n\\cdot(\\N)\\T\\twist,\n\\end{split}\n\\end{equation}\nwhere $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$, which are analogous to the classical Frank's constants \\cite{frank:theory}, are related to the moduli $k_1$, $k_2$, $k_3$, and $k_4$ through the equations\n\\begin{equation*}\\label{eq:K_and_k}\nK_{11}=k_1+k_2+k_4,\\quad K_{22}=k_1,\\quad\nK_{33}=k_1+k_3,\\quad K_{24}=k_1+k_4.\n\\end{equation*}\nFor $f^-_e$ in \\eqref{eq:elastic_energy_density_transformed} to be positive definite, the elastic constants $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$ must obey the inequalities,\n\\begin{equation*}\\label{eq:Ericksen_inequalities}\n2K_{11}\\geqq K_{24},\\quad 2K_{22}\\geqq K_{24},\\quad\nK_{33}\\geqq0,\\quad K_{24}\\geqq0,\n\\end{equation*}\nwhich coincide with the classical Ericksen's inequalities for ordinary nematics \\cite{ericksen:inequalities}.\n\nThe total elastic energy $\\free^-_e$ is represented by the functional\n\\begin{equation*}\\label{eq:elastic_energy_functional}\n\\free^-_e[\\twist,\\n]:=\\int_\\body f^-_e(\\twist,\\n,\\N)dV,\n\\end{equation*}\nwhere both $\\twist$ and $\\n$ are subject to the pointwise constraint \\eqref{eq:cone_constraint} and either of them is prescribed on the boundary $\\boundary$ of the region in space occupied by the medium.\nIt is worth recalling that both $\\twist$ and $\\n$ are fields that need to be determined so as to obey \\eqref{eq:cone_constraint} and to minimize $\\free^-_e$. In this theory, the helix axis of the preferred conical state and the nematic director of the actual molecular organization equally participate in the energy minimization with the objective of reducing the discrepancy between natural and actual nematic distortions.\nPhysically, $\\twist$ represents the optic axis of the medium, likely to be the only optic observable when the pitch $p$ ranges in the the nanometric domain.\n\nSuch an abundance of state variables is a direct consequence of the degeneracy in the ground state admitted for helical nematics. The \\emph{vacuum manifold}, as is often called the set of distortions that minimize $f^-_e$, is indeed a three-dimensional orbit of congruent cones, identifiable with $\\sphere\\times\\Circle$, where $\\sphere$ is the unit sphere and $\\Circle$ the unit circle in three space dimensions. By \\eqref{eq:natural_distortion}, all natural distortions $\\Twist$ are represented by $\\twist\\in\\sphere$ and any $\\n$ in the cone of semi-amplitude $\\ca$ around $\\twist$. This shows, yet in another language, how rich in states is the single well where $f^-_e$ vanishes.\n\nA number of remarks are suggested by formula \\eqref{eq:elastic_energy_density_transformed}. First, it reduces to the elastic free-energy density of ordinary nematics, which features only $\\n$, when either the wave parameter $q$ or the cone angle $\\ca$ vanish, thus indicating two possible mechanisms to induce helicity in an ordinary nematic. Second, for $\\ca=\\frac\\pi2$, it delivers an alternative energy density for chiral nematics, which is positive-definite for all $K_{24}\\geqq0$ (whereas, to ensure energy positive-definiteness, the classical theory requires that $K_{24}=0$, see \\cite[p.\\,127]{virga:variational}). Finally, for arbitrary $q>0$ and $0<\\ca<\\frac{\\pi}{2}$, $f^-_e$ in \\eqref{eq:elastic_energy_density_transformed} is invariant under the reversal of either $\\twist$ or $\\n$, showing that both fields enjoy the nematic symmetry.\n\n\\section{Helix unwinding}\\label{sec:helix}\nIn the presence of an external field, say an electric field $\\field$, the free-energy density acquires an extra term, which we write as\n\\begin{equation*}\\label{eq:electric_energy density}\nf_E(\\n)=-\\frac12\\ez\\ea E^2(\\n\\cdot\\e)^2,\n\\end{equation*}\nwhere $\\ez$ is the vacuum permittivity, $\\ea$ is the (relative) dielectric anisotropy, and we have set $\\field=E\\e$, with $E>0$ and $\\e$ a unit vector. Accordingly, the total free-energy functional $\\free^-$ is defined as\n\\begin{equation}\\label{eq:free_energy_total}\n\\free^-[\\twist,\\n]:=\\int_\\body\\{f^-_e(\\twist,\\n,\\N)+f_E(\\n)\\}dV.\n\\end{equation}\n\nTo minimize $\\free^-$ when $\\body$ is the whole space, we shall assume that $\\twist$ is uniform and $\\n$ is spatially periodic with period $L$ and we shall compute the average $F^-$ of $\\free^-$ over an infinite slab of thickness $L$ orthogonal to $\\twist$. Letting $\\twist$ coincide with the unit vector $\\e_z$ of a given Cartesian frame $(x,y,z)$, we represent $\\n$ in precisely the same form adopted in \\eqref{eq:ground} for $\\ground$, but with $qz+\\az_0$ replaced by a function $\\az=\\az(z)$ such that\n\\begin{equation*}\\label{eq:periodicity_azimut}\n\\az(0)=0\\quad\\text{and}\\quad\\az(L)=2\\pi.\n\\end{equation*}\nWith no loss of generality, we may choose $\\e$ in the $(y,z)$ plane and represent it as\n\\begin{equation*}\\label{eq:a_representation}\n\\e=\\cos\\psi\\,\\e_z+\\sin\\psi\\,\\e_y.\n\\end{equation*}\nComputing $F^-$ on the ground state $\\az=2\\pi z\/p$ (where $L=p$), one easily sees that the average energy is smaller for $\\psi=\\frac\\pi2$ than for $\\psi=0$, whenever either\n\\begin{enumerate}[(a)]\n\\item\\label{case:1} $\\ea<0$ and $\\ca<\\cac:=\\arctan\\sqrt{2}\\doteq54.7\\degree$, or\n\\item\\label{case:2} $\\ea>0$ and $\\ca>\\cac$,\n\\end{enumerate}\nwhich are the only cases we shall consider here. In case \\eqref{case:1} (and for $\\psi=\\frac\\pi2$), $F^-$ reduces to\n\\begin{equation*}\\label{eq:F_unwinding}\n\\begin{split}\nF^-[L,\\az]=K\\sin^2\\ca\\left\\{\\frac{1}{2L}\\int_0^L\\left[\\az'^2+\\frac{1}{\\cohe^2}\\sin^2\\az\\right]dz -\\frac{2\\pi q}{L} \\right\\},\n\\end{split}\n\\end{equation*}\nwhere\n\\begin{equation*}\\label{eq:K_effective}\nK:=K_{22}\\sin^2\\ca+K_{33}\\cos^2\\ca\n\\end{equation*}\nis an effective \\emph{twist-bend} elastic constant and\n\\begin{equation}\\label{eq:coherence_length}\n\\cohe:=\\frac{1}{E}\\sqrt{\\frac{K}{\\ez|\\ea|}}\n\\end{equation}\nis a field \\emph{coherence length}.\n\nMinimizing $F^-$ in both $L$\\ and $\\az$ is a problem formally akin to the classical problem of unwinding the cholesteric helix \\cite{degennes:calul,meyer:effects,meyer:distortion}. The minimizing $\\az$ is determined implicitly by\n\\begin{equation*}\\label{eq:phi_unwinding}\n\\frac{z}{L}=\\frac14\\left(\\frac{\\ElF(\\az+\\frac\\pi2,k)}{\\ElK(k)}-1\\right),\n\\end{equation*}\nwhere $\\ElF$ and $\\ElK$ are the elliptic and complete elliptic integrals of the first kind, respectively, and $k$ is the root in the interval $[0,1]$ of the equation\n\\begin{equation}\\label{eq:root_k}\n\\frac{\\ElE(k)}{k}=\\pi^2\\frac{\\cohe}{p},\n\\end{equation}\nwhere $\\ElE$ is the complete elliptic integral of the second kind \\cite[p.\\,486]{olver:NHMF}. Equation \\eqref{eq:root_k}\nhas a (unique) root only for\n\\begin{equation*}\n\\cohe\\geqq\\cohec:=\\frac{p}{\\pi^2},\n\\end{equation*}\nfor which $L$ is correspondingly delivered by\n\\begin{equation*}\\label{eq:root_L}\nL=4\\cohe k\\ElK(k).\n\\end{equation*}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{LLL_Graph.eps}\n\\caption{\nThe graph of the spatial period $L$ of the distorted helix under field (scaled to the undistorted pitch $p$) against the field coherence length $\\cohe$ in \\eqref{eq:coherence_length}\n(equally scaled to $p$). The complete unwinding takes place for $\\cohe\/p=1\/\\pi^2\\doteq0.101$, where the graph diverges.}\n\\label{fig:L_graph}\n\\end{figure}\nFigure~\\ref{fig:L_graph} shows the graph of $L$ against $\\cohe$, which diverges as $\\cohe$ approaches $\\cohec$. As for ordinary chiral nematics, a field with coherence length $\\cohec$ unwinds completely the helix of a $\\mathrm{N_{tb}}$, but its actual strength now depends on the elastic constants $K_{22}$ and $K_{33}$, and the cone angle $\\ca$.\nThe measured values of $p$ range in the domain of tens of nanometers. Thus, the steepness of the graph in Fig.~\\ref{fig:L_graph} along with \\eqref{eq:coherence_length} suggest that in actual terms the field should be very strong for any unwinding to be noticed.\\footnote{A simple estimate based on \\eqref{eq:root_k} would show that a field strength larger than $100\\,\\mathrm{V}\/\\mu\\mathrm{m}$ is needed to unwind the \\TBN helix even if we assume $\\ea$ as large as $10$ and $K$ as small as $1\\,\\mathrm{pN}$.}\n\n\\section{Freedericks transition}\\label{sec:freedericks}\nIn ordinary nematics, the Freedericks transition is an instability that enables one to measure the classical Frank's elastic constants. For a $\\mathrm{N_{tb}}$, the setting is complicated by the role played by the additional field $\\twist$.\n\nWe consider a $\\mathrm{N_{tb}}$ liquid crystal confined between two parallel plates, placed in a Cartesian frame $(x,y,z)$ at $z=0$ and $z=d$, respectively. On both plates, we subject $\\n$ to a \\emph{conical} anchoring with respect to the plates' normal $\\e_z$ at precisely the cone angle $\\ca$, so that, by the constraint \\eqref{eq:cone_constraint}, $\\twist$ is there prescribed to coincide with $\\e_z$. Within the infinite cell bounded by these plates we allow $\\twist$ to vary in the $(y,z)$ plane, so that\n\\begin{equation*}\\label{eq:FrederikS_e_representation}\n\\twist=\\cos\\psi\\,\\e_z+\\sin\\psi\\,\\e_y,\n\\end{equation*}\nwhere $\\psi=\\psi(z)$. Letting\n\\begin{equation*}\n\\twist_\\perp:=\\sin\\psi\\,\\e_z-\\cos\\psi\\,\\e_y,\n\\end{equation*}\nwe represent the nematic director as\n\\begin{equation}\\label{eq:Freedericks_n_representation}\n\\n=\\cos\\ca\\,\\twist+\\sin\\ca\\cos\\az\\,\\twist_\\perp+\\sin\\ca\\sin\\az\\,\\e_x,\n\\end{equation}\nwhere $\\az=\\az(z)$ is the precession angle. The function $\\psi$ is subject to the conditions\n\\begin{equation}\\label{eq:Freedericks_psi_boundary_conditions}\n\\psi(0)=\\psi(d)=0,\n\\end{equation}\nwhile $\\az$ is free in the whole of $[0,d]$. The system is further subjected to an external field, $\\field=E\\e_z$. In the following analysis, we shall assume that either of the aforementioned  cases \\eqref{case:1} or \\eqref{case:2} occurs here.\n\nTaking $p\/d\\ll1$ and treating $\\psi$ as a perturbation, of which we only retain quadratic terms in the energy, when the precession angle represents the undistorted helix, $\\az=2\\pi z\/p$, we obtain the average $F^-$ of the free-energy functional $\\free^-$ in \\eqref{eq:free_energy_total}\n(per unit area of the plates) as\n\\begin{equation}\\label{eq:Freedericks_average_free_energy}\nF^-[\\psi]=\\frac12 K_{33}\\int_0^d\\left\\{A\\psi'^2+Bq^2\\psi^2\\right\\}dz,\n\\end{equation}\nwhere\n\\begin{gather*}\nA:=\\frac12\\Big[\\sin^2\\ca(k_{11}+k_{22}\\cos^2\\ca)+\\cos^2\\ca(1+\\cos^2\\ca)\\Big]\n\\\\\nB:=\\frac12\\Big[\\sin^2\\ca(k_{11}+k_{22}\\cos^2\\ca+\\sin^2\\ca\n-\\frac{1}{(\\cohe q)^2}|2\\cos^2\\ca-\\sin^2\\ca|\\Big]\n\\\\\nk_{11}:=\\frac{K_{11}}{K_{33}},\\quad k_{22}:=\\frac{K_{22}}{K_{33}},\\quad\\cohe:=\\frac{1}{E}\\sqrt{\\frac{K_{33}}{\\ez|\\ea|}}.\n\\end{gather*}\nIt is now easily seen that $\\psi\\equiv0$ is a locally stable extremum of the functional $F^-$\nin \\eqref{eq:Freedericks_average_free_energy} subject to \\eqref{eq:Freedericks_psi_boundary_conditions}\nfor $\\cohe>\\cohec$, whereas it is locally unstable for $\\cohe<\\cohec$, where\n\\begin{equation}\\label{eq:Freedericks_critical_xi_E}\n\\frac{\\cohec}{p}:=\\frac{\\sqrt{|2-t^2|(1+t^2)}}{\\pi\\sqrt{4[(1+k_{11})t^2+k_{11}+k_{22}]t^2+ \\frac{p^2}{d^2}[k_{11}t^4+(1+k_{11}+k_{22})t^2+2]}}\n\\end{equation}\nand $t:=\\tan\\ca$. Both this and the helix unwinding treated in Section~\\ref{sec:helix} are continuous transitions.\n\nSince in this theory $p\/d$ is small, $\\cohec$ is only weakly dependent on the cell thickness $d$. It should be noted however that for $\\ca=0$, \\eqref{eq:Freedericks_critical_xi_E} reduces to $\\cohec=d\/\\pi$, which reproduces the classical Freedericks threshold \\cite[p.\\,179]{virga:variational}.\n\nIn the special, hypothetical case of equal elastic constants (for which $k_{11}=k_{22}=1$), $\\cohec$ acquires a simpler form, which retains the qualitative features of \\eqref{eq:Freedericks_critical_xi_E}:\n\\begin{equation}\\label{eq:Freedericks_critical_xi_E_equal_constants}\n\\frac{\\cohec}{p}=\\frac{\\sqrt{|2-t^2|}}{\\pi\\sqrt{8t^2+\\frac{p^2}{d^2}\\frac{t^4+3t^2+2}{1+t^2}}}.\n\\end{equation}\n\\begin{figure}\n\\centering\n\\includegraphics[width=.9\\linewidth]{F_Graph.eps}\n\\caption{\nThe graphs of $\\cohec$ (scaled to the pitch $p$ of the undistorted helix) against the cone angle $\\ca$ (expressed in degrees), as delivered by \\eqref{eq:Freedericks_critical_xi_E_equal_constants} under the assumption of equal elastic constants, for $p\/d=0$ (solid line) and $p\/d=1$ (dotted line). For $\\ca=0$, the former graph diverges, while the latter reaches the value $\\cohec\/p=1\/\\pi\\doteq0.318$. For $0<p\/d<1$, the graphs of $\\cohec$, none of which diverges, fill the region bounded by the graphs shown here.}\n\\label{fig:F_Graph}\n\\end{figure}\nThe graph of $\\cohec$\nas delivered by \\eqref{eq:Freedericks_critical_xi_E_equal_constants}\nis plotted in Fig.~\\ref{fig:F_Graph} for $p\/d=0$ and $p\/d=1$, the graphs for all intermediate values of $p\/d$ being sandwiched between them. These graphs show that for $\\ca$ sufficiently large, in particular for $\\ca>\\cac$ (and $\\ea>0$), $\\cohec$ is virtually independent of $d$, whereas it is not so for $\\ca$ small. Moreover, for moderate values of $\\ca$ and $d$ much larger than $p$, $\\cohec$ can easily be made equal to several times the pitch of the undistorted helix.\nIf, as contemplated by \\eqref{eq:Freedericks_critical_xi_E}, the actual field required to ignite the Freedericks transition is weakly dependent on the cell thickness $d$, the corresponding critical potential $U_\\mathrm{c}$ would scale almost linearly with $d$.\\footnote{In the experiments performed in \\cite{borshch:nematic}, it was found that $U_\\mathrm{c}\\varpropto\\sqrt{d}$, but the boundary conditions imposed there seem to differ from the conical boundary conditions considered here. Moreover, in \\cite{borshch:nematic} the transition nucleated locally from the inside of the cell instead of happening uniformly, as presumed here. This might suggest that in the real experiment both helical variants present in a \\TBN are participating in the transition. It would then be advisable taking with a grain of salt any direct comparison of our theory for helical nematics with experiments available for the whole \\TBN phase.}\n\n\\section{Double-well energy}\\label{sec:Double-Well}\nA \\TBN phase can be regarded as a mixture of two helical nematic phases with opposite helicities. The ground  states of these phases corresponding to all admissible natural distortions are the members of two symmteric energy wells. In a way, this is reminiscent of the mixture of martensite twins in some solid crystals, which are equi-energetic variants with symmetrically sheared lattices (see, for example, \\cite[p.\\,129]{muller:entropy}). A thorough mathematical theory of these solid phases is based on a non-convex energy functional in the elastic deformation, featuring a multiplicity of energy wells \\cite{ball:fine,ball:proposed}. Below, adapting these ideas to the new context envisaged here, in which the energy depends on the local distortion of the molecular arrangement, and no deformation from a reference configuration is involved, we show how to construct a double-well elastic energy density $\\dens$ for a \\TBN phase starting from the energy densities for helical nematic phases of opposite helicities. To this end we need first supplement $\\dens^-$ in \\eqref{eq:elastic_energy_density_transformed} with the appropriate energy density $\\dens^+$ for a helical nematic phase of positive helicity.\n\n\\subsection{Positive helicity}\\label{sec:positive}\nPrescribing the helicity of the ground state of a helical nematic phase to be positive, instead of negative as above, would amount to replacing \\eqref{eq:elastic_energy_density_definition} by\n\\begin{equation}\\label{eq:elastic_energy_density_definition_positive}\nf^+_e(\\twist,\\n,\\N)=\\frac12(\\N+\\Twist)\\cdot\\elan[\\N+\\Twist],\n\\end{equation}\nstill with $\\Twist$ as in \\eqref{eq:natural_distortion} and $q>0$. Our development following \\eqref{eq:elastic_energy_density_definition} could be repeated verbatim here and it would lead us to the same conclusion obtained by subjecting \\eqref{eq:elastic_energy_density_transformed} to the \\emph{formal} change of $q$ into $-q$:\n\\begin{equation}\\label{eq:elastic_energy_density_transformed_positive}\n\\begin{split}\nf^+_e(\\twist,\\n,\\N)\n&=\\frac12\\Bigl\\{K_{11}(\\diver\\n)^2+K_{22}(\\n\\cdot\\curl\\n-q|\\twist\\times\\n|^2)^2\\\\\n&+K_{33}|\\n\\times\\curl\\n-q(\\n\\cdot\\twist)\\,\\twist\\times\\n|^2\\\\\n&+K_{24}[\\tr(\\N)^2-(\\diver\\n)^2]\\Bigr\\}\n+K_{24}q\\,\\twist\\times\\n\\cdot(\\N)\\T\\twist,\n\\end{split}\n\\end{equation}\nwhere $q$ remains a \\emph{positive} parameter.\n\nClearly, the energy well of \\eqref{eq:elastic_energy_density_definition_positive} and \\eqref{eq:elastic_energy_density_transformed_positive} is formally obtained from the corresponding well of \\eqref{eq:elastic_energy_density_definition} and \\eqref{eq:elastic_energy_density_transformed} by a sign inversion. \n\n\\subsection{\\TBN free energy density}\\label{sec:TBN}\nFollowing \\cite{truskinovsky:ericksen}, which studied systematically how to extend the non-convex energy first proposed by Ericksen~\\cite{ericksen:equilibrium} for a one-dimensional elastic bar, we consider two possible choices for the elastic energy density $\\dens$ of a \\TBN phase:\n\\begin{subequations}\\label{eq:f_e_TBN}\n\\begin{equation}\\label{eq:f_e_TBN_quadratic}\nf_e(\\twist,\\n,\\N)=\\min\\{f_e^+(\\twist,\\n,\\N),f_e^-(\\twist,\\n,\\N)\\},\n\\end{equation}\n\\begin{equation}\\label{eq:f_e_TBN_quartic}\nf_e(\\twist,\\n,\\N)=\\frac{1}{f_0}f_e^+(\\twist,\\n,\\N)f_e^-(\\twist,\\n,\\N),\n\\end{equation}\nwhere\n\\begin{equation*}\nf_0:=\\frac12\\sin^2\\ca(K_{22}\\sin^2\\ca+ K_{33}\\cos^2\\ca)\n\\end{equation*}\nis a normalization constant chosen so as to ensure that $\\dens(\\twist,\\n,\\zero)=\\dens^\\pm(\\twist,\\n,\\zero)$.\n\\end{subequations}\nWhile $\\dens$ in \\eqref{eq:f_e_TBN_quadratic} is quadratic around each well, it fails to be smooth for $\\N=\\zero$. On the other hand, $\\dens$ in \\eqref{eq:f_e_TBN_quartic} is everywhere smooth, but it is quartic. The differences between these energy densities are illustrated pictorially in Fig.~\\ref{fig:Free_Plots}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.7\\linewidth,angle=0]{Free_plots.eps}\n  \\caption{One-dimensional pictures for $\\dens$ in \\eqref{eq:f_e_TBN_quadratic} (solid line) and \\eqref{eq:f_e_TBN_quartic} (dashed line). Here $\\az=\\az(z)$ would represent the precession angle in a molecular arrangement such as that described by \\eqref{eq:Freedericks_n_representation} and $\\az'$ is its spatial derivative. Each minimum is representative for a three-dimensional well described by \\eqref{eq:natural_distortion} and its mirror image (with $q$ replaced by $-q$).}\n\\label{fig:Free_Plots}\n\\end{figure}\n\nWe shall not explore other possible forms $\\dens$. We only heed that both \\eqref{eq:f_e_TBN_quadratic} and \\eqref{eq:f_e_TBN_quartic} inherit the simple structure of the elastic energy density of a helical nematic phase, which features only four positive elastic constants, as in the classical Frank's theory of ordinary nematics. General considerations on how to match ground states extracted from two different wells of $\\dens$ (at zero energy cost) are independent of the peculiar form assumed for this function, as they are only consequences of the structure of each well. A study of the geometric compatibility conditions that arise in this case will be presented elsewhere \\cite{virga:double-well}.\n\n\\section{Conclusion}\\label{sec:conclusion}\nThe elastic energy density proposed in \\eqref{eq:elastic_energy_density_transformed} and \\eqref{eq:elastic_energy_density_transformed_positive} to describe the equilibrium distortions of each helical variant of a \\TBN phase featured just the classical four elastic constants and introduced the helix axis $\\twist$ in addition to the nematic director $\\n$. The instabilities studied above illustrated two second-order transitions that differ also qualitatively from their classical analogues. Only experiments may decide at this stage whether a quadratic elastic theory based on either \\eqref{eq:f_e_TBN_quadratic} or \\eqref{eq:f_e_TBN_quartic} above\\footnote{Even if globally quartic, the energy density in \\eqref{eq:f_e_TBN_quartic} is quadratic about each energy well. Moreover, it features only the four elastic constants required by the quadratic energy density of a helical phase with prescribed helicity.} is better suited to describe the novel \\TBN phases than the quartic theory proposed in \\cite{dozov:spontaneous}. The instabilities described in this paper for each \\TBN variant  just provide a theoretical means to set the quadratic theory to the test.\n\nTwo director fields, namely, $\\n$ and $\\twist$, were deemed necessary here to describe the local distortion of a \\TBN phase.\nThis poses the question as to which defects both fields may exhibit and how they are interwoven, in view of the constraint \\eqref{eq:cone_constraint}.\nAn extra field also requires extra boundary conditions. The question is how to set general boundary conditions for both fields to grant existence to the energy minimizers.\n\nFinally, no hydrodynamic considerations have entered our study, but the question should be asked as to whether the relaxation in time of $\\twist$ represents a further source of dissipation.\n\n\n\n\\section*{Acknowledgements}\nI wish to thank Oleg D. Lavrentovich for his encouragement to pursue this study and for his kindness in providing me with some of his results prior to their publication. I am also greatful to Mikhail A. Osipov for having suggested studying the work of Lorman and Mettout \\cite{lorman:theory} on the symmetry of helical nematics. I am finally indebted to the kindness of two anonymous Reviewers whose critical remarks and constructive suggestions improved considerably this manuscript.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Background and motivation}\n\nThe notion of Maslov indices arised from the study of semi-classical\napproximations in quantum mechanics \\cite{maslov1972theorie} in the\nphase space $(\\mathbb{R}^{2n},dx_{i}\\wedge dy_{i}$). It plays an\nimportant role in the problem of quantization as well as gives insights\ninto the geometric aspect of classical dynamics \\cite{dullin2005maslov}.\nIn \\cite{arnold1972classe} Arnold proved that the Maslov index of\na closed path $\\tilde{\\gamma}$ of Lagrangian planes in $\\mathbb{R}^{2n}$\ncan be characterized either as the (signed) number of intersections\nof $\\tilde{\\gamma}$ with the Maslov cycle $\\mathbb{\\mathfrak{M}}_{0}$\ninduced by the Lagrangian subspace $\\mathbb{E}_{0}=\\mathbb{R}^{n}\\times\\{0\\}$,\nor, equivalently, as the degree of the Maslov-Arnold map for $\\tilde{\\gamma}$:\n\\begin{equation}\n\\mathfrak{m}_{\\tilde{\\gamma}}:S^{1}\\xrightarrow{\\tilde{\\gamma}}\\Lambda(n)\\cong\\mathbb{U}(n)\\big\/\\mathbb{O}(n)\\xrightarrow{det_{\\mathbb{C}}^{2}}S^{1},\\label{eq:def_Maslov-Arnold}\n\\end{equation}\nwhere $\\Lambda(n)$ is the Lagrangian Grassmannian of $\\mathbb{R}^{2n}$.\nLet $\\frac{\\partial}{\\partial\\theta}=\\frac{d}{d\\theta}e^{2\\pi i\\cdot\\theta}$\nbe the standard vector field on $S^{1}$ and $d\\theta$ be its dual\n$1$-form. When $\\tilde{\\gamma}$ is smooth, $\\mathfrak{m}_{\\tilde{\\gamma}}$\ncan be calculated as \n\\begin{equation}\n\\mathfrak{m}_{\\tilde{\\gamma}}=\\int_{\\tilde{\\gamma}}\\tilde{\\eta}=\\int_{\\gamma}d\\theta,\\label{eq:maslov index by integration}\n\\end{equation}\nwhere $\\tilde{\\eta}$ is the pullback of the canonical $1$-form $d\\theta$\non $S^{1}$ via the Maslov-Arnold map\\cite{contreras2003asymptotic},\nand $\\gamma=det_{\\mathbb{C}}^{2}(\\tilde{\\gamma})$ is a loop in $S^{1}$. \n\nIn the general case where $(M,\\omega)$ is a symplectic manifold,\nthe Lagrangian Grassmannian will be replaced by the bundle of Lagrangian\nplanes $\\Lambda_{pl}$, which consists of all the Lagrangian subspaces\nof the tangent spaces of $M$ \\cite{contreras2003asymptotic,butler2009maslov}.\nNamely:\n\\begin{equation}\n\\Lambda_{pl}=\\bigcup_{p\\in M}\\{Lagrangian\\ subspaces\\ of\\ T_{p}M\\}.\\label{eq:Lambda_pl_over_M}\n\\end{equation}\nA Maslov cycle $\\mathbb{\\mathfrak{M}}$ consists of all the Lagrangian\nspaces that have nontrivial intersection with a Lagrangian subbundle\n$\\mathfrak{L}$, and the Maslov index of a loop in $\\Lambda_{pl}$\nis then defined in the same spirit as the number of intersections\nwith $\\mathbb{\\mathfrak{M}}$.\n\nAt the same time, $\\Lambda_{pl}$ can also be viewed as a structure\nassociated to an almost complex structure $J$ compatible with $\\omega$.\nIn the case $(M,\\omega)=(\\mathbb{R}^{2n},dx_{i}\\wedge dy_{i}$) the\nbundle of Lagrangian planes is a trivial bundle\n\\[\n\\mathbb{R}^{2n}\\times\\Lambda(n)=\\mathbb{R}^{2n}\\times\\mathbb{U}(n)\\big\/\\mathbb{O}(n),\n\\]\nwhere the last space can be interpreted as the quotient space of $\\mathbb{R}^{2n}\\times\\mathbb{U}(n)$\nby the $\\mathbb{O}(n)$ action. $\\mathbb{R}^{2n}\\times\\mathbb{U}(n)$\nis the unitary frame bundle consisting of all the unitary frames of\nthe tangent spaces of $\\mathbb{R}^{2n}$ with respect to the canonical\nalmost complex structure\n\\[\nJ_{0}=\\left[\\begin{array}{cc}\n0 & -I_{n}\\\\\nI_{n} & 0\n\\end{array}\\right]\n\\]\non the tangent bundle $T\\mathbb{R}^{n}$. The $S^{1}$ appearing at\nthe end of the sequence (\\ref{eq:def_Maslov-Arnold}) as the codomain\nof $\\mathfrak{m}_{\\tilde{\\gamma}}$ should be thought of as the $S^{1}$\nprincipal bundle \n\\[\n\\mathbb{R}^{2n}\\times\\mathbb{U}(n)\\big\/\\mathbb{EU}(n)\\cong\\mathbb{R}^{2n}\\times S^{1},\n\\]\nwhere $\\mathbb{EU}(n)$ is the subgroup of $\\mathbb{U}(n)$ consisting\nof all the matrices with complex determinants equal to $1$ or $-1$.\nFor an arbitrary symplectic manifold $(M,\\omega)$, an almost complex\nstructure $J$ compatible with the symplectic structure $\\omega$\ncan always be chosen, and the corresponding unitary frame bundle (associated\nto $J$)\n\\[\n\\pi_{Fr_{J}^{u}}:Fr_{J}^{u}\\rightarrow M\n\\]\nis a $\\mathbb{U}(n)-$principal bundle (with $\\mathbb{U}(n)$ acting\nfrom the right in our setting). Therefore we can define the associated\nfiber bundle \n\\[\n\\pi_{\\Lambda_{J}}:\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)\\rightarrow M\n\\]\nas well as the $S^{1}$-principal bundle\n\\[\n\\pi_{\\Gamma_{J}^{2}}:\\Gamma_{J}^{2}=Fr_{J}^{u}\\big\/\\mathbb{EU}(n)\\rightarrow M,\n\\]\nwhich are the counterparts of $\\mathbb{R}^{2n}\\times\\Lambda(n)$ and\n$\\mathbb{R}^{2n}\\times\\mathbb{U}(n)\\big\/\\mathbb{EU}(n)$, respectively,\nin the general setting. \n\nThe space $\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)$ is indeed in\none-one correspondence to $\\Lambda_{pl}$. This is because each fiber\n$\\pi_{\\Lambda_{J}}^{-1}(p)$ of $\\Lambda_{J}$ with $p\\in M$ is exactly\nthe Lagrangian Grassmannian of the symplectic vector space $(T_{p}M,\\omega_{p})$\nwith respect to the compatible linear complex structure $J\\big|_{T_{p}M}$.\nIn the text we will use interchangably the spaces $\\Lambda_{J}$ and\n$\\Lambda_{pl}$. From the set-theoretic point of view, the advantage\nof thinking about $\\Lambda_{pl}$ is that for any symplectic flow\n$\\varphi$ on $M$, the tangent maps $\\varphi_{*}^{t}$ induces a\nflow on $\\Lambda_{pl}$ in a natural way. However, the advantage of\nconsidering the space $\\Lambda_{J}$ is that its relation with $\\Gamma_{J}^{2}$\nis straigthforward. \n\nSince $\\mathbb{O}(n)\\subset\\mathbb{EU}(n)$ and they are both closed\nsubgroups of $\\mathbb{U}(n)$, each $\\mathbb{O}(n)$ orbit in $Fr_{J}^{u}$\nlies in a single $\\mathbb{EU}(n)$ orbit, and there is a natural quotient\nmap $det_{J}^{2}$ from $Fr_{J}^{u}\\big\/\\mathbb{O}(n)$ to $Fr_{J}^{u}\\big\/\\mathbb{EU}(n)$,\nsending each $\\mathbb{O}(n)$ orbit to its containing $\\mathbb{EU}(n)$\norbit. Viewed in proper local trivializations of the bundles $\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)\\rightarrow M$\nand $\\Gamma_{J}^{2}=Fr_{J}^{u}\\big\/\\mathbb{EU}(n)\\rightarrow M$,\nsuch a quotient map takes the form\n\\[\nU\\times\\mathbb{U}(n)\\big\/\\mathbb{O}(n)\\ni(p,[A])\\mapsto(p,det_{\\mathbb{C}}^{2}A)\\in U\\times S^{1}\n\\]\nwith respect to some local charts over some open set $U$ of $M$.\nDue to this reason, we denote this quotient map by\n\\[\ndet_{J}^{2}:\\Lambda_{J}\\rightarrow\\Gamma_{J}^{2}.\n\\]\nand the following composition\n\\[\ndet_{J}^{2}\\circ\\tilde{\\gamma}:S^{1}\\rightarrow\\Gamma_{J}^{2}\n\\]\nis then the Malsov-Arnold map for a closed path $\\tilde{\\gamma}$\nin $\\Lambda_{J}$. When the bundle admits a trivialization $\\Gamma_{J}^{2}\\cong M\\times S^{1}$\nwith respect to some global section $\\mathfrak{s}$, such a trivialization\nleads to the definition of an index. To be precise, with the isomorphism\nfrom $\\Gamma_{J}^{2}$ to $M\\times S^{1}$ being denoted by $tr_{\\mathfrak{s}}$,\nan index $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma})$ for a closed\npath $\\tilde{\\gamma}$ of Lagrangian planes can be defined as the\ndegree of the map defined in the following composition:\n\\begin{equation}\nS^{1}\\xrightarrow{\\tilde{\\gamma}}\\Lambda_{J}\\xrightarrow{det_{J}^{2}}\\Gamma_{J}^{2}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}\\xrightarrow{pr_{S^{1}}}S^{1}.\\label{eq:ind_=00007B=00005Cpsi=00007D}\n\\end{equation}\nNote that a globally defined Lagrangian vector subbundle gives a global\nsection $\\sigma$ to $\\Lambda_{J}$(or $\\Lambda_{pl}$), and then\nthe composition $\\mathfrak{s}=det_{J}^{2}\\circ\\sigma$ is a global\nsection to $\\pi_{\\Gamma_{J}^{2}}$, which gives a trivialization $\\Gamma_{J}^{2}\\cong M\\times S^{1}$.\nAlso note that this is always the case where $M$ is the cotangent\nbundle of some manifold, and the Lagrangian subbundle can be chosen\nas the vertical distribution. In this case, the index $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma})$\ndefined above is exactly the Maslov index (with respect to $\\mathfrak{s}$)\nin the usual sense. \n\nThe discussion above shows why we become interested in the principal\n$S^{1}$ bundle $\\Gamma_{J}^{2}$ and the other related structures,\n$Fr_{J}^{u}$ and $\\Lambda_{J}$, over a symplectic manifold $(M,\\omega)$.\nIn some cases it would be more convenient to consider the principal\n$S^{1}$ bundle $\\Gamma_{J}\\cong Fr_{J}^{u}\\big\/\\mathbb{SU}(n)$,\nwhich is related to $\\Gamma_{J}^{2}$ by the relation $\\Gamma_{J}^{2}\\cong\\Gamma_{J}\\big\/\\{\\pm1\\}$.\nIn the following text, we call $\\pi_{\\Gamma_{J}}:\\Gamma_{J}\\rightarrow M$\n(or simply $\\Gamma_{J}$) and $\\pi_{\\Gamma_{J}^{2}}:\\Gamma_{J}^{2}\\rightarrow M$\nMaslov $S^{1}$ bundles. Note that although from the set-theoretic\npoint of view, the definitions of $Fr_{J}^{u}$, $\\Lambda_{J}$ and\n$\\Gamma_{J}$ depend on the choice of $J$, their bundle structures\nare independent of the choice of the compatible almost complex structure. \n\n\\subsection{Maslov data: a nonintegrable version of Maslov indices}\n\nGenerally speaking, the principal $S^{1}$ bundles $\\Gamma_{J}$ and\n$\\Gamma_{J}^{2}$ are not necesarrily trivial, and the mapping \\ref{eq:ind_=00007B=00005Cpsi=00007D}\nis not available (this is the case for $S^{2}$, as we will explain\nlater). However, the integration \\ref{eq:maslov index by integration}\ncan still be defined by treating $\\tilde{\\eta}$ as the pullback of\nany connection $1$-form of $\\Gamma_{J}^{2}$ (or alternatively $\\Gamma_{J}$)\non $\\Lambda_{J}$. Following such a perspective, we note that a global\nsection of $\\Gamma_{J}^{2}$ is just an integral manifold of a flat\nconnection (which is an integrable horizontal distribution) on $\\Gamma_{J}^{2}$.\nTo generalize the notion of Maslov indices for the case where $\\Gamma_{J}^{2}$\nis a nontrivial bundle, we can replace the ``integrable horizontal\ndistribution'' with simply ``a connection''. We introduce a nonintegrable\nversion of Maslov indices for smooth loops in $\\Lambda_{pl}$ (or\n$\\Lambda_{J}$) with respect to an arbitrary connection $1$-form\n$\\beta=\\eta\\cdot\\frac{\\partial}{\\partial\\theta}$ on $\\Gamma_{J}^{2}$. \n\\begin{defn}\n\\label{def:Nonintegrable-Maslov-Quantity for =00005CLambda} For any\nsmooth loop $\\tilde{\\gamma}:[0,1]\\rightarrow\\Lambda_{J}$, define\nits Maslov data with respect to $\\beta$ to be\n\\[\n\\tilde{\\mathfrak{m}}_{\\beta}(\\tilde{\\gamma})=\\int_{\\tilde{\\gamma}}\\tilde{\\eta}\n\\]\nwith $\\tilde{\\eta}$ being the pullback of $\\eta$ on $\\Lambda_{J}$\nvia $det_{J}^{2}$. \n\\end{defn}\n\nNote that whenever $\\tilde{\\gamma}$ is a smooth loop in $\\Lambda_{J}$,\n\\[\n\\gamma=det_{J}^{2}\\circ\\tilde{\\gamma}\n\\]\nis a smooth loop in $\\Gamma_{J}^{2}$. Moreover, it holds\n\n\\begin{equation}\n\\int_{\\gamma}\\eta=\\int_{\\tilde{\\gamma}}\\tilde{\\eta}=\\tilde{\\mathfrak{m}}_{\\beta}(\\tilde{\\gamma}),\\label{eq:Maslov-quantity for a path\/loop}\n\\end{equation}\nand hence it will be convenient to consider $\\gamma$ instead of $\\tilde{\\gamma}$\nand think of $\\tilde{\\mathfrak{m}}_{\\beta}(\\tilde{\\gamma})$ as a\nquantity of $\\gamma$. We will use the symbol $\\mathfrak{m}_{\\beta}(\\gamma)$\nfor the integration $\\int_{\\gamma}\\eta$ and call it the Maslov data\nof $\\gamma$ with respect to $\\beta$. \n\nWhen the horizontal distribution $\\ker\\beta$ has an integral manifold\n$\\mathcal{S}_{0}$ being a one-sheet covering over $M$, Definition\n\\ref{def:Nonintegrable-Maslov-Quantity for =00005CLambda} gives the\nMaslov indices in the usual sense with respect to the Malsov cycle\n$\\mathfrak{M}_{0}=(det_{J}^{2})^{-1}(\\mathcal{S}_{0})$ (or to the\nsecion $\\mathfrak{s}$). Note that this is the case in \\cite{contreras2003asymptotic}\nwhere a dynamical system is considered in a vicinity of a connected\nsubmanifold $\\Sigma$ in $M$ on which (the restriction of) the bundle\nof Lagrangian planes $\\Lambda_{pl}$ can be trivialized as $\\Sigma\\times\\Lambda(n)$.\nAlso note that in general, $\\ker\\beta$ being flat does not necessary\nmean that the integral manifolds are single-sheeted coverings. A discussion\ntaking care of a nontrivial bundle $\\Gamma_{J}^{2}$ with flat connections\nwould however be beyond the scope this work, and therefore we will\nnot go into details for this issue. \n\\begin{rem}\nWe avoid using the term $index$ just for emphasizing that the Maslov\ndata is not necessarily a topological quantity, even when $\\tilde{\\gamma}$\nis a loop. \n\\end{rem}\n\n\n\\subsection{Purposes, Main Results and Layout}\n\nThis work is intended to be an exploration of the interplay between\ndynamics on $M$ and the geometry of $\\Gamma_{J}$(and\/or $\\Gamma_{J}^{2}$),\nas well as an investigation into the notion of Maslov data proposed\nin Definition \\ref{def:Nonintegrable-Maslov-Quantity for =00005CLambda}\nas a generalized\/nonintegrable version of ordinary Maslov indices. \n\nThe dynamics that we are interested in on the manifolds will be symplectic\ngroup actions. Note that given the flow of an arbitrary symplectic\nvector field $X$, $\\varphi:\\mathbb{R}\\times M\\rightarrow M$, the\ntangent map $\\varphi_{*}^{t}$ maps each Lagrangian space to another.\nNamely, $\\varphi_{*}$ defines a flow on $\\Lambda_{pl}$ which covers\n$\\varphi$. Since each trajectory $\\tilde{\\gamma}$ of $\\varphi_{*}$\nis mapped to a path $\\gamma$ in $\\Gamma_{J}^{2}$ by $\\gamma=det_{J}^{2}\\circ\\tilde{\\gamma}$,\nit is natural to expect that, under some conditions, the composition\n\\[\ndet_{J}^{2}\\circ\\varphi_{*}:\\mathbb{R}\\times\\Lambda_{pl}\\rightarrow\\Gamma_{J}^{2}\n\\]\nfactors through the space $\\mathbb{R}\\times\\Gamma_{J}^{2}$ and induces\na flow $\\varphi_{\\Gamma^{2}}$ on $\\Gamma_{J}^{2}$. Such a factorization\nindeed exists when the vector field $X$ is an infinitesimal generator\nof a symplectic group action by a compact Lie group $G$. Based on\nthis reason, we focus our study on compact and connected Lie group\nactions. \n\nIn the following we introduce the main results of this work. The simpliest\ncase where the Maslov bundles are non-trivial is given by $M=S^{2}$.\nNote that $S^{2}$ is a symplectic homogeneous space with a transitive\nsymplectic $\\mathbb{SO}(3)$ action. Also note that $H_{dR}^{1}(S^{2})=0$\nand hence any symplectic vector field on $S^{2}$ is Hamiltonian.\nA discussion of the $\\mathbb{SO}(3)$ action on $S^{2}$ (Section\n\\ref{sec:compact-group-actions}) shows that the lifted $\\mathbb{SO}(3)$\nactions on the Maslov $S^{1}$ bundles $\\Gamma_{S^{2}}$ and $\\Gamma_{S^{2}}^{2}$\nare transitive (Propersition \\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}),\nand the Hamiltonians for the infinitesimal generators can be expressed\nin terms of an invariant connection $1$-form of the bundle $\\Gamma_{S^{2}}\\rightarrow S^{2}$\n(Proposition \\ref{prop:symplectic=00003DHamiltonian on S^2}). These\nproperties are then extended to more general settings. To be specific,\nwe obtain the following result for symplectic homogeneous spaces:\n\\begin{thm}\n\\label{thm:=00005C=00007BGamma=00007D_J-homogeneous}Let $G$ be a\ncompact Lie group acting transitively and symplectically on $M$.\nIf the first real Chern class of the Maslov $S^{1}$ bundle $\\Gamma_{J}$\nis nonvanishing, then $\\Gamma_{J}$ is also a homogeneous $G$-space.\n\\end{thm}\n\n\\noindent We moreover get the following theorem which exitends the\nresult obtained in \\cite{lupton1995cohomologically,kaoru1992obstruction,cho2012chern}\nabout symplectic circle actions being Hamiltonian on monotone manifolds:\n\\begin{thm}\n\\label{thm:=00005C=00007Bomega=00007D=00003D=00003Dr=00005Bc=00005D:Symplectic--Hamiltonian}Let\n$(M,\\omega)$ be a symplectic manifold, and let $\\pi_{\\Gamma}:\\Gamma_{J}\\rightarrow M$\nbe its Maslov $S^{1}$ bundle with the first real Chern class $c_{\\Gamma}\\in H_{dR}^{2}(M)$.\nIf it satisfies $[\\omega]=r\\cdot c_{\\Gamma}$ for some real number\n$r$, then any symplectic action $\\Phi$ on $M$ by a compact Lie\ngroup $G$ is Hamiltonian. More specifically, for any $v\\in\\mathfrak{g}$,\nthere exists a Hamiltonian $H_{v}$ of $X_{v}$ such that $\\pi_{\\Gamma}^{*}(H_{v})=r\\cdot f_{\\beta}(\\mathcal{X}_{v})$\nfor some real number $r\\neq0$, with $\\beta=f_{\\beta}\\cdot\\frac{\\partial}{\\partial\\theta}$\nbeing a connection $1$-form invariant under the lifted $G$-action\n$\\Phi_{\\Gamma}$.\n\\end{thm}\n\n\\begin{rem}\n$X_{v}$ and $\\mathcal{X}_{v}$ in the statement of Theorem \\ref{thm:=00005C=00007Bomega=00007D=00003D=00003Dr=00005Bc=00005D:Symplectic--Hamiltonian}\nabove stand for the infinitesimal generators associated to $v$ of\nthe action $\\Phi$ on $M$ and the lifted action $\\Phi_{\\Gamma}$\non $\\Gamma_{J}$, respectively.\n\\end{rem}\n\nWe also discuss the Maslov data of symplectic circle actions. When\n$G=S^{1}$, the orbits of $\\Phi_{\\Gamma}$ in $\\Gamma_{J}$ covering\nthe same orbit of $\\Phi$ in $M$ have the same Maslov data. In other\nwords, for $w,w'\\in\\pi_{\\text{\\ensuremath{\\Gamma}}}^{-1}(p)$ with\n$\\gamma_{w}(z)=\\Phi_{\\Gamma}^{z}(w)$ and $\\gamma_{w'}(z)=\\Phi_{\\Gamma}^{z}(w')$,\n$\\mathfrak{m}_{\\beta}(\\gamma_{w})=\\mathfrak{m}_{\\beta}(\\gamma_{w'})$\nfor any connection $1$-form $\\beta$ on $\\Gamma_{J}$. Therefore,\nit defines a smooth function $Q_{\\theta}$ (which we call the $\\beta$-Maslov\ndata of $\\Phi$) on $M$ with $Q_{\\theta}(p)=\\mathfrak{m}_{\\beta}(\\gamma_{w})$.\nAlthough with a different connection $\\beta$ the function $Q_{\\theta}$\nmay be different, its values at the fixed points of the action $\\Phi$\nturn out to be independent of $\\beta$. When $[\\omega]=r\\cdot c_{\\Gamma}$,\n$Q_{\\theta}$ (after scaling) is a Hamiltonian of the action $\\Phi$.\n\nThis paper is organized as follows.\n\nIn Section \\ref{sec:Basics of Maslov-Index}, we discuss some basic\nproperties of Maslov indices from the perspective from the geometry\nof $\\Gamma_{J}^{2}$. In Section \\ref{sec:Basics-of-Principal S^1 bundles},\nwe recall some basics about principal $S^{1}$ bundles which will\nbe used in the later discussion. In Section \\ref{sec:compact-group-actions},\nwe discuss the lift of the $G$-action $\\Phi$ and study the case\nof $M=S^{2}$, which serves as an example in which the Maslov $S^{1}$\nbundle $\\Gamma_{S^{2}}$ is non-trivial. We explain that the lifted\n$\\mathbb{SO}(3)$ action acts transitively on $\\Gamma_{S^{2}}$ (Proposition\n\\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}) and\nshow that the Hamiltonians of its infinitesimal generators can be\nwritten down in terms of a connection $1$-form which is invariant\nunder the lifted $G$-action on $\\mathbb{SO}(3)\\cong\\Gamma_{S^{2}}$\n(Proposition \\ref{prop:symplectic=00003DHamiltonian on S^2}). In\nSection \\ref{sec:dynamics-on Maslov S^1 bundles}, we extend Proposition\n\\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D} and\n\\ref{prop:symplectic=00003DHamiltonian on S^2} to Theorem \\ref{thm:=00005C=00007BGamma=00007D_J-homogeneous}\nand \\ref{thm:=00005C=00007Bomega=00007D=00003D=00003Dr=00005Bc=00005D:Symplectic--Hamiltonian}.\nIn Section \\ref{sec:Beta-Maslov-Quantity for S^1-action}, we define\nthe $\\beta$-Maslov data of a symplectic $S^{1}$ action and discuss\nits properties, especially its values at the fixed points of the action.\n\n\\subsection{Notations}\n\nIn this paper we always assume $G$ to be a compact Lie group acting\nsympletically on $(M,\\omega)$. Denote the Lie algebra of $G$ by\n$\\mathfrak{g}$. The action of $G$ on $M$ is denoted by $\\Phi$.\nThe lifted actions on $\\Gamma_{J}$ and $\\Gamma_{J}^{2}$ are denoted\nby $\\Phi_{\\Gamma}$ and $\\Phi_{\\Gamma^{2}}$, respectively. For $v\\in\\mathfrak{g}$,\n$X_{v}$ is the corresponding infinitesimal generator of $\\Phi$ on\n$M$ , $\\mathcal{X}_{v}$ stands for the generator on $\\Gamma_{J}$(or\n$\\Gamma_{J}^{2}$), and $\\mathbb{X}_{v}$ is the generator of the\naction $\\Phi_{*}$ on $\\Lambda_{pl}$ which is induced naturally by\nthe tangent maps.\n\nWe denote by $\\frac{\\partial}{\\partial\\theta}$ the vector field on\n$S^{1}$ defined by $\\frac{d}{d\\theta}e^{2\\pi i\\cdot\\theta}$. Note\nthat, on a principal $S^{1}$ bundle $P$, there is a\/an structural\/inherent\n$S^{1}$ action. We also use the symbol $\\frac{\\partial}{\\partial\\theta}$\nto denote the corresponding infinitesimal generator of this $S^{1}$\naction.\n\n\\section{\\label{sec:Basics of Maslov-Index}Basic properties of Maslov indices}\n\n\\subsection{\\label{subsec:Lagrangian-subbundles and Maslov-indices}Lagrangian\nsubbundles and Maslov indices}\n\nLet $\\mathcal{N}$ be a submanifold in $M$ and $\\mathfrak{L}_{\\mathcal{N}}\\subset TM$\nbe a Lagrangian subbundle of $TM$ over $\\mathcal{N}$. That is, $\\mathfrak{L}_{\\mathcal{N}}\\big|_{b}$\nis a Lagrangian subspace for any $b\\in\\mathcal{N}$. It is straightward\nto see that $\\mathfrak{L}_{\\mathcal{N}}$ always induces a smooth\nsection $\\sigma_{\\mathcal{N}}$ from $\\mathcal{N}$ to $\\Lambda_{pl}\\cong\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)$\nwhich assigns to each $b\\in\\mathcal{N}$ the Lagrangian plane $\\mathfrak{L}_{\\mathcal{N}}\\big|_{b}$.\nMoreover, when $\\mathfrak{L}_{\\mathcal{N}}$ is orientable, it also\ninduces a section $\\tilde{\\sigma}_{\\mathcal{N}}$ from $\\mathcal{N}$\nto $Fr_{J}^{u}\\big\/\\mathbb{SO}(n)$. We give a brief account of the\nlatter case.\n\nWhen $\\mathfrak{L}_{\\mathcal{N}}$ is orientable, we can first fix\nan orientation. For any $b\\in\\mathcal{N}$, there is a neighbourhood\n$U_{b}$ in $\\mathcal{N}$ on which it admits an ordered local frame\n$f_{U_{b}}^{*}=(e_{1},...,e_{n})$ of $\\mathfrak{L}_{\\mathcal{N}}$\nwhich fits the orientation. With some modification, it can be extended\nto a(n) (ordered) unitary frame\n\\[\nf_{U_{b}}=(e_{1},...,e_{n},s_{1},...,s_{n})\n\\]\non $U_{b}$ for $TM$ with $s_{i}=Je_{i}$. Then \n\\[\n\\sigma_{b}:U_{b}\\ni x\\mapsto f_{U_{b}}(x)\\in Fr_{J}^{u}\n\\]\nis a local section for $Fr_{J}^{u}$. By composing with the quotient\nmap $q_{\\mathbb{SO}(n)}$ from $Fr_{J}^{u}$ to $Fr_{J}^{u}\\big\/\\mathbb{SO}(n)$\nwe get a local section $\\tilde{\\sigma}_{\\mathcal{S},b}$ from $U_{b}$\nto $Fr_{J}^{u}\\big\/\\mathbb{SO}(n)$. Namely,\n\\[\n\\tilde{\\sigma}_{\\mathcal{S},b}=q_{\\mathbb{SO}(n)}\\circ\\sigma_{b}.\n\\]\nThe collection $\\{U_{b}\\big|b\\in\\mathcal{N}\\}$ then constitutes an\nopen cover of $\\mathcal{N}$ over each element of which there is a\n(smooth) section $\\sigma_{\\mathcal{N},b}$. It remains to check that\nwhen $U_{b}\\cap U_{b'}\\neq\\emptyset$, $\\tilde{\\sigma}_{\\mathcal{S},b}$\nagrees with $\\tilde{\\sigma}_{\\mathcal{S},b'}$ on $U_{b}\\cap U_{b'}$.\nSince both $f_{U_{b}}^{*}$ and $f_{U_{b'}}^{*}$($=(e'_{1},...,e'_{n})$)\ncan be taken as orthonormal frames with respect to the Euclidean metric\n$g_{J}$ that fit the same orientation, at each $x\\in U_{b}\\cap U_{b'}$,\nthey are related by a matrix $C$ from $\\mathbb{SO}(n)$ via\n\\[\n(e'_{1},...,e'_{n})=(e_{1},...,e_{n})\\cdot C.\n\\]\nFor $f_{U_{b'}}=(e'_{1},...,e'_{n},s'_{1},...,s'_{n})$, it holds\n\\[\ns'_{i}=J\\cdot e'_{i},\n\\]\nand then it is straightforward to check that\n\\[\n(e'_{1},...,e'_{n},s'_{1},...,s'_{n})=(e_{1},...,e_{n},s_{1},...,s_{n})\\cdot C.\n\\]\nIt implies $\\tilde{\\sigma}_{\\mathcal{N},b'}(x)=\\tilde{\\sigma}_{\\mathcal{N},b}(x)$.\nThen we get $\\tilde{\\sigma}_{\\mathcal{N}}$ by simply piecing the\nlocal sections $\\tilde{\\sigma}_{\\mathcal{N},b}$ together.\n\nNow we consider the case $\\mathcal{N}=M$. Suppose that $\\mathcal{S}\\subset M$\nis a Lagrangian submanifold, and that $\\Gamma_{J}^{2}$ admits a global\nsection $\\mathfrak{s}^{2}$ which induces a trivialization\n\\[\ntr_{\\mathfrak{s}^{2}}:\\Gamma_{J}^{2}\\rightarrow M\\times S^{1}.\n\\]\nLet $\\gamma$ be a loop on $\\mathcal{S}$. Then $\\sigma_{\\mathcal{S}}\\circ\\gamma:t\\mapsto T_{\\gamma(t)}\\mathcal{S}$\nis a loop in $\\Lambda_{J}$. Composed with the map $det_{J}^{2}$\nit becomes a loop in $\\Gamma_{J}^{2}$, and the Maslov index $\\mathfrak{m}_{\\mathfrak{s}^{2}}(\\gamma)$\nis the degree of the following map\n\\[\nS^{1}\\stackrel{\\sigma_{\\mathcal{S}}\\circ\\gamma}{\\longrightarrow}\\Lambda_{J}\\stackrel{det_{J}^{2}}{\\longrightarrow}\\Gamma_{J}^{2}\\stackrel{tr_{\\mathfrak{s}^{2}}}{\\longrightarrow}M\\times S^{1}\\stackrel{pr_{S^{1}}}{\\longrightarrow}S^{1}.\n\\]\nNow suppose that $\\mathcal{S}$ is an orientable Lagrangian submanifold\nand that $\\Gamma_{J}$ admits a global section $\\mathfrak{s}$. Then\nwe have a global section of $\\Gamma_{J}^{2}$, which is given by $\\mathfrak{s}^{2}=q_{\\pm1}(\\mathfrak{s})$.\nHere $q_{\\pm1}:\\Gamma_{J}\\rightarrow\\Gamma_{J}^{2}\\big\/\\{\\pm1\\}$\nis the quotient map. As a consequence, we have the following commutative\ndiagram, which suggests that the Maslov index $\\mathfrak{m}_{\\mathfrak{s^{2}}}(\\gamma)$\nshould be an even number:\n\\begin{equation}\n\\text{\\ensuremath{\\begin{CD}S^{1}@>\\tilde{\\sigma}_{\\mathcal{S}}\\circ\\gamma>>Fr_{J}^{u}\\big\/\\mathbb{SO}(n)@>det_{J}>>\\Gamma_{J}@>tr_{\\mathfrak{s}}>>M\\times S^{1}@>pr_{S^{1}}>>S^{1}\\\\\n@|@Vq_{\\pm1}VV@Vq_{\\pm1}VV@Vq_{\\pm1}VV@V{\\scriptscriptstyle square}VV\\\\\nS^{1}@>\\sigma_{\\mathcal{S}}\\circ\\gamma>>\\Lambda_{J}@>det_{J}^{2}>>\\Gamma_{J}^{2}@>tr_{\\mathfrak{s}^{2}}>>M\\times S^{1}@>pr_{S^{1}}>>S^{1}\n\\end{CD}}}.\\label{Diagram:oriented Maslov indices and Maslov indices}\n\\end{equation}\n\n\\begin{rem}\nIf $\\mathfrak{L}$ is a Lagrangian subbundle of $TM$ over $M$, it\ninduces a global section $\\sigma:M\\rightarrow\\Lambda_{J}$ by sending\neach $b\\in M$ to the fiber $\\mathfrak{L}_{b}$ of $\\mathfrak{L}$\nabove it. Then \n\\[\n\\mathfrak{s}^{2}:=det_{J}^{2}\\circ\\sigma\n\\]\nis a global section to the bundle $\\Gamma_{J}^{2}$. Furthermore,\nif $\\mathfrak{L}$ is orientable, then it also induces a trivialization\nfor $\\Gamma_{J}$.\n\\end{rem}\n\n\n\\subsection{\\label{subsec:Simply connectedness and Local Maslov indices}Maslov\nindices on simply connected spaces}\n\nGenerally speaking, a loop $\\tilde{\\gamma}$ of Lagrangian planes\n(or its image $\\gamma$ in $\\Gamma_{J}^{2}$) will have different\nMaslov indices with respect to different global sections of $\\Gamma_{J}^{2}$.\nHowever, when the manifold $M$ is simply connected, Malsov indices\nof a loop with respect to different sections\/Maslov cycles will be\nidentical. We give a brief explanation of this fact from the perspective\nof the geometry of the bundle $\\Gamma_{J}^{2}$. \n\nSuppose that $\\mathfrak{s}$ and $\\mathfrak{s}'$ are sections of\n$\\text{\\ensuremath{\\Gamma^{2}}}$. In terms of the trivialization\n$\\text{\\ensuremath{\\Gamma^{2}}}\\xrightarrow{tr_{\\mathfrak{s}'}}M\\times S^{1}$\nwith respect to $\\mathfrak{s}'$, the section $\\mathfrak{s}:M\\rightarrow M\\times S^{1}$\ntakes the form $\\mathfrak{s}(p)=(p,\\theta_{p})$ with $\\theta:p\\mapsto\\theta_{p}$\nbeing a smooth map from $M$ to $S^{1}$. For a loop $\\gamma\\subset\\text{\\ensuremath{\\Gamma^{2}}}$,\nit takes the form $S^{1}\\ni z\\mapsto\\big(\\lambda(z),\\tau'(z)\\big)\\in M\\times S^{1}$\nin the trivialization $\\text{\\ensuremath{\\Gamma^{2}}}\\xrightarrow{tr_{\\mathfrak{s}'}}M\\times S^{1}$,\nand the form $S^{1}\\ni z\\mapsto\\big(\\lambda(z),\\tau(z)\\big)\\in M\\times S^{1}$\nin $\\text{\\ensuremath{\\Gamma^{2}}}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}$,\nwith the maps $\\lambda:S^{1}\\rightarrow M$, $\\tau',\\tau:S^{1}\\rightarrow S^{1}$.\nThe corresponding Maslov indices $\\mathfrak{m}_{\\mathfrak{s}'}(\\gamma)$\nand $\\mathfrak{m}_{\\mathfrak{s}}(\\gamma)$ are then the degrees of\nthe maps $\\tau'$ and $\\tau$, respectively. It holds that $\\tau'(z)=\\tau(z)\\cdot\\theta_{\\lambda(z)}$.\nSince $M$ is simply connected, the mapping $z\\rightarrow\\theta_{\\lambda(z)}$\nwhich is subject to the following factorization\n\\[\nS^{1}\\xrightarrow{\\lambda}M\\xrightarrow{\\theta}S^{1}\n\\]\nhas degree $0$, and hence $\\tau$ and $\\tau'$ have the same degree.\n\n\\section{\\label{sec:Basics-of-Principal S^1 bundles}Basics on Principal $S^{1}$-Bundles}\n\nFor later purposes, we recall in the section basic properties of a\nprincipal $S^{1}$ bundle. A principal $S^{1}$-bundle consists of\na total space $P$ on which there is a free $S^{1}$ action, its orbifold\n$B=P\\big\/S^{1}$, and the quotient map $\\pi_{P}:P\\rightarrow B$.\nTo distinguish it from other group actions, in the rest of the paper\nwe will call this $S^{1}$ action the inherent\/structural $S^{1}$\naction. For simplicity, we will also call $P$ the principal bundle\nwhen it is clear from the context.\n\nA connection $\\mathcal{H}$ on the bundle $\\pi_{P}:P\\rightarrow B$\nis a horizontal distribution invariant under the inherent $S^{1}$\naction, and the tangent bundle $TP$ is split as a direct sum $\\mathcal{H}\\oplus\\mathcal{V}$\nwith $\\mathcal{V}=\\ker\\pi_{P,*}$ being the vertical distribution.\nThe Lie algebra of $S^{1}$ is $T_{1}S^{1}=\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta}$\nwith $\\frac{\\partial}{\\partial\\theta}=\\frac{d}{d\\theta}\\bigg|_{\\theta=0}e^{i\\cdot2\\pi\\theta}.$\nThe following map\n\\[\n\\eta:P\\times\\bigg(\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta}\\bigg)\\rightarrow\\mathcal{V}\n\\]\nwith \n\\[\n\\bigg(p,r\\frac{\\partial}{\\partial\\theta}\\bigg)\\mapsto\\frac{d}{d\\theta}\\bigg|_{\\theta=0}(p\\cdot e^{i\\cdot2\\pi r\\cdot\\theta})\n\\]\nis a vector bundle isomorphism. For simplicity we also denote the\nvector $\\frac{d}{d\\theta}\\bigg|_{\\theta=0}(p\\cdot e^{i\\cdot2\\pi r\\cdot\\theta})$\nby $r\\frac{\\partial}{\\partial\\theta}$. The connection $1$-form associated\nto $\\mathcal{H}$ is\n\\[\n\\alpha:TP\\xrightarrow{pr_{\\mathcal{V}}}\\mathcal{V}\\cong P\\times(\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta})\\rightarrow\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta}\n\\]\nwith $pr_{\\mathcal{V}}$ being the projection associated to the splitting\n$TP=\\mathcal{H}\\oplus\\mathcal{V}$ onto $\\mathcal{V}$. $\\alpha$\nis $S^{1}$ invariant, and there is an $S^{1}$-invariant $1$-form\n$f_{\\alpha}$ such that $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$.\nBy abusing the terminology we also call $f_{\\alpha}$ a connection\n$1$-form on $P$.\n\nNote that $f_{\\alpha}(\\frac{\\partial}{\\partial\\theta})\\equiv1$. The\n$S^{1}$-invariance of $f_{\\alpha}$ implies $\\mathcal{L}_{\\frac{\\partial}{\\partial\\theta}}f_{\\alpha}=0$.\nThen by Cartan's formula, we get\n\\[\n0=\\mathcal{L}_{\\frac{\\partial}{\\partial\\theta}}f_{\\alpha}=\\iota_{\\frac{\\partial}{\\partial\\theta}}df_{\\alpha}+d\\bigg(f_{\\alpha}\\big(\\frac{\\partial}{\\partial\\theta}\\big)\\bigg)=\\iota_{\\frac{\\partial}{\\partial\\theta}}df_{\\alpha}.\n\\]\nWith $pr_{\\mathcal{H}}$ being the projection of $\\mathcal{H}\\oplus\\mathcal{V}$\nonto $\\mathcal{H}$, the equation above implies that \n\\[\ndf_{\\alpha}(pr_{\\mathcal{H}}\\cdot,pr_{\\mathcal{H}}\\cdot)=df_{\\alpha}.\n\\]\nSince $df_{\\alpha}$, $\\mathcal{H}$ and $\\mathcal{V}$ are all $S^{1}$-invariant,\nthis implies that there is a closed $2-$form $\\Omega_{\\alpha}$ on\n$B$ such that $\\pi_{P}^{*}(\\Omega_{\\alpha})=df_{\\alpha}$. $[\\Omega_{\\alpha}]\\in H_{dR}^{2}(B)$\nis independent of $\\alpha$ and is called the characteristic class\nof the bundle $P\\rightarrow B$.\n\nBy the definition of $f_{\\alpha}$, $\\ker f_{\\alpha}=\\mathcal{H}$.\nAccording to the Frobenius integrability theorem, $\\mathcal{H}$ is\nan integrable distribution if and only if $f_{\\alpha}\\wedge df_{\\alpha}=0$.\nIn this case, such an identity actually requires $df_{\\alpha}$ to\nbe $0$. This is because if $df_{\\alpha}\\neq0$, then there exist\n$u,v\\in\\mathcal{H}_{p}=\\ker f_{\\alpha}\\big|_{p}$ such that $df_{\\alpha}(u,v)\\neq0$,\nand then \n\\[\nf_{\\alpha}\\wedge df_{\\alpha}(\\frac{\\partial}{\\partial\\theta},u,v)=f_{\\alpha}(\\frac{\\partial}{\\partial\\theta})\\cdot df_{\\alpha}(u,v)=df_{\\alpha}(u,v)\\neq0,\n\\]\nyielding a contradiction. As a result, the corresponding $2$-form\n$\\Omega_{\\alpha}=0$, and hence $[\\Omega_{\\alpha}]=0$.\n\nConversely, if $[\\Omega_{\\alpha}]=0$, then there exists a $1$-form\n$\\tau$ on $B$ such that $d\\tau=\\Omega_{\\alpha}$. Let $\\alpha'=\\alpha-\\pi_{\\Gamma_{J}}^{*}(\\tau)\\cdot\\frac{\\partial}{\\partial\\theta}$.\nThen $\\alpha'$ is also $S^{1}$-invariant with\n\\[\n\\alpha'(r\\frac{\\partial}{\\partial\\theta})=\\alpha(r\\frac{\\partial}{\\partial\\theta})=r\\frac{\\partial}{\\partial\\theta},\n\\]\nnamely, it is also a connection $1$-form, and $\\mathcal{H}'=\\ker\\alpha'$\ndefines another connection on $P\\rightarrow B$. With $f_{\\alpha'}\\cdot\\frac{\\partial}{\\partial\\theta}=\\alpha'$,\nit holds\n\\[\nf_{\\alpha'}=f_{\\alpha}-\\pi_{\\Gamma_{J}}^{*}(\\tau).\n\\]\nThen \n\\[\ndf_{\\alpha'}=\\pi_{\\Gamma_{J}}^{*}(\\Omega_{\\alpha})-\\pi_{\\Gamma_{J}}^{*}(d\\tau)=0,\n\\]\nimplying $\\mathcal{H}'$ to be integrable. Therefore, the characterstic\nclass of a principal $S^{1}$-bundle is zero if and only if the bundle\nhas an integrable connection.\n\n\\section{\\label{sec:compact-group-actions}Compact group actions on Maslov\n$S^{1}$ bundles}\n\nLet\n\\[\n\\Phi:G\\times M\\rightarrow M\n\\]\nbe a symplectic left action on $M$ by a compact Lie group $G$ .\nNamely, $\\Phi^{h'h}=\\Phi^{h'}\\circ\\Phi^{h}$ and $\\Phi^{h}{}^{*}(\\omega)=\\omega$\nfor $h,h'\\in G$. In this secion, we first show that $\\Phi$ can be\nlifted to a $G$-action on $\\Gamma_{J}$. Namely, there is a $G$-action\n$\\Phi_{\\Gamma}$ on $\\Gamma_{J}$ that covers $\\Phi$. Moreover, $\\Phi_{\\Gamma}$\ncommutes with the inherent $S^{1}$ action on $\\Gamma_{J}$. Then\nwe study such a lifted $G$-action for the case where $M$ is a homogeneous\n$G$-space.\n\\begin{defn}\n\\label{def:group action on the bundle}Let $P\\rightarrow B$ be a\nprincipal $S^{1}$-bundle. A group action on the bundle $P\\rightarrow B$\n(or simply saying the bundle $P$) by $G$ is a $G$-action on the\nmanifold $P$ such that it commutes with the inherent $S^{1}$-action.\n\\end{defn}\n\n\n\\subsection{$G$-actions on $Fr_{J}^{u}$ and $\\Gamma_{J}$}\n\nThe $G$-action $\\Phi$ on $M$ can be lifted to a $G$-action on\nthe bundle $Fr_{J}^{u}\\rightarrow M$ by resorting to a $G$-equivariant\nalmost complex structure $J$. In the following we briefly explain\nhow this is done.\n\nLet $g$ be an arbitrary Riemannian metric on $M$, and let $dh$\nbe a probability measure on $G$ which is invariant under the right\ntranslations. Define a new Riemannian metric $\\bar{g}$ with\n\\[\n\\bar{g}(u,v)=\\int_{G}\\Phi_{h}{}^{*}(g)(u,v)dh\n\\]\nwith $u,v\\in T_{x}M$ and $x\\in M$. Then $\\bar{g}$ is invariant\nunder the (left) $G$-action $\\Phi$. Let $\\mathcal{A}$ be the vector\nbundle isomorphism on $TM$ defined by \n\\begin{equation}\n\\omega(u,\\cdot)=\\bar{g}(\\mathcal{A}u,\\cdot)\\label{eq:def-=00005Cmathcal=00007BA=00007D}\n\\end{equation}\nand let \n\\begin{equation}\n\\bar{J}=\\mathcal{A}^{-1}\\sqrt{-\\mathcal{A}^{2}}.\\label{eq:def-=00005Cbar=00007BJ=00007D}\n\\end{equation}\nThen $\\text{\\ensuremath{\\bar{J}}}$ is an almost complex structure\ncompatible with $\\omega$, and $g_{\\bar{J}}(\\cdot,\\cdot)=\\omega(\\bar{J}\\cdot,\\cdot)$\ndefines a Riemannian metric. For convenience we denote by $h_{*}$\nthe pushforward $\\Phi_{h,*}$. It is a standard result that $\\bar{J}$\ncommutes with $h_{*}$ for all $h\\in G$, and $g_{\\bar{J}}$ is then\ninvariant under the $G$-action. For completeness we give an argument\nin the following.\n\\begin{lem}\n\\label{lem:G-invariant-unitary-structure}Suppose that $\\bar{g}$\nis a Riemannian metric invariant under the $G$-action $\\Phi$. Let\n$\\mathcal{A}$ and $\\bar{J}$ be endomorphisms on $TM$ defined respectively\nby (\\ref{eq:def-=00005Cmathcal=00007BA=00007D}) and (\\ref{eq:def-=00005Cbar=00007BJ=00007D})\nabove. Then for any $h\\in G$, it holds\n\\begin{equation}\nh_{*}\\circ\\bar{J}=\\bar{J}\\circ h_{*}.\\label{eq:h^*J=00003DJh^*}\n\\end{equation}\nand\n\\begin{equation}\nh^{*}g_{\\bar{J}}(\\cdot,\\cdot)=g_{\\bar{J}}(\\cdot,\\cdot).\\label{eq:h^*g=00003Dg}\n\\end{equation}\nAs a consequence, $h_{*}$ maps the unitary frame bundle $Fr_{\\bar{J}}^{u}$\nto itself. \n\\end{lem}\n\n\\begin{proof}\nSuppose that (\\ref{eq:h^*J=00003DJh^*}) holds, then \n\\[\nh^{*}g_{\\bar{J}}(\\cdot,\\cdot)=\\omega(\\bar{J}\\circ h_{*}\\cdot,h_{*}\\cdot)=\\omega(h_{*}\\circ\\bar{J}\\cdot,h_{*}\\cdot)=g_{\\bar{J}}(\\cdot,\\cdot)\n\\]\ngives (\\ref{eq:h^*g=00003Dg}).\n\nWe start to prove (\\ref{eq:h^*J=00003DJh^*}) by looking at the bundle\nisomorphism $\\mathcal{A}$. Since $G$ acts symplectically on $M$,\nit holds\n\\[\n\\begin{aligned}\\bar{g}(\\mathcal{A}h_{*}u,h_{*}\\cdot) &  & = & \\omega(h_{*}u,h_{*}\\cdot)\\\\\n &  & = & \\omega(u,\\cdot)\\\\\n &  & = & \\bar{g}(\\mathcal{A}u,\\cdot)\\\\\n &  & = & \\bar{g}(h_{*}\\mathcal{A}u,h_{*}\\cdot)\n\\end{aligned}\n.\n\\]\nNow that $\\mathcal{A}$ is bijective and $u$ runs over $TM$, this\nimplies \n\\[\n\\mathcal{A}h_{*}=h_{*}\\mathcal{A}.\n\\]\nConsequently, $h_{*}$ also commutes with $\\mathcal{A}^{-1}$ and\n$-\\mathcal{A}^{2}$. For proving (\\ref{eq:h^*J=00003DJh^*}), it remains\nto show that \n\\[\n\\mathcal{B}h_{*}=h_{*}\\mathcal{B}\n\\]\nwith $\\mathcal{B}=\\sqrt{-\\mathcal{A}^{2}}$, i.e. $\\mathcal{B}^{2}=-\\mathcal{A}^{2}$. \n\nNote that, at each $x$ in $M$, $\\mathcal{B}_{x}^{2}=\\mathcal{B}^{2}\\big|_{T_{x}M}$\nand $\\mathcal{B}_{x}$ are both self-adjoint positive operators with\nrespect to $\\bar{g}$, and a vector $v\\in T_{x}M$ is an eigenvector\nof $\\mathcal{B}_{x}^{2}$ with eigenvalue $\\lambda^{2}$ if and only\nif it is an eigenvector of $\\mathcal{B}_{x}$ with eigenvalue $\\lambda>0$,\ni.e.\n\\[\n\\mathcal{B}_{x}^{2}(v)=\\lambda^{2}v\\iff\\mathcal{B}_{x}(v)=\\lambda v.\n\\]\nResorting to the commutativity, we have\n\\[\n\\mathcal{B}_{h\\cdot x}^{2}h_{*}(v)=h_{*}\\mathcal{B}_{x}^{2}(v)=\\lambda^{2}h_{*}(v),\n\\]\nand then \n\\[\n\\mathcal{B}_{h\\cdot x}h_{*}(v)=\\lambda h_{*}(v).\n\\]\nThe discussion above amounts to\n\\[\n\\mathcal{B}_{x}(v)=\\lambda v\\iff\\mathcal{B}_{h\\cdot x}h_{*}(v)=\\lambda h_{*}(v).\n\\]\nNow let $\\{e_{1},...,e_{2n}\\}$ be a basis of $T_{x}M$ with each\n$e_{i}$ being an eigenvector of $\\mathcal{B}_{x}$ with eigenvalue\n$\\lambda_{i}$. Then\n\\[\nh_{*}\\mathcal{B}_{x}(e_{i})=\\lambda_{i}h_{*}(e_{i})=\\mathcal{B}_{h\\cdot x}h_{*}(e_{i}).\n\\]\nSince $\\{e_{1},...,e_{2n}\\}$ is a basis, this concludes the proof.\n\\end{proof}\nThe tangent maps of the actions by the elements in $G$ gives the\nfollowing $G$-action on $Fr_{\\bar{J}}^{u}$:\n\n\\[\n\\Phi_{\\#}:G\\times Fr_{\\bar{J}}^{u}\\rightarrow Fr_{\\bar{J}}^{u}\n\\]\n\\[\n\\Phi_{\\#}^{h}(u_{i},v_{i})=\\big(h_{*}(u_{i}),h_{*}(v_{i})\\big).\n\\]\nFor simplicity, we write $\\Phi_{\\#}^{h}(u_{i},v_{i})$ as $h_{\\#}(u_{i},v_{i})$.\nFor any $C\\in\\mathbb{U}(n)$, it can also be checked that\n\\begin{equation}\n\\big(h_{*}(u_{1}),...,h_{*}(v_{n})\\big)\\cdot C=h_{\\#}\\big((u_{1},...,v_{n})\\cdot C\\big),\\label{eq:h=000023_is_equivariant}\n\\end{equation}\nand hence $\\Phi_{\\#}$ is a $G$-action on $Fr_{\\bar{J}}^{u}\\rightarrow M$.\nDue to (\\ref{eq:h=000023_is_equivariant}), $\\Phi_{\\#}$ induces a\nsmooth $G$-action $\\Phi_{\\Gamma}$ on $\\Gamma_{\\bar{J}}=Fr_{\\bar{J}}^{u}\\big\/\\mathbb{SU}(n)$.\nMoreover, $\\Phi_{\\Gamma}$ on $\\Gamma_{\\bar{J}}$ commutes with the\n$S^{1}$ action. To see this, note that \n\\[\n\\mathbb{U}(n)\\big\/\\mathbb{SU}(n)\\ni[C]\\mapsto det_{\\mathbb{C}}(C)\\in S^{1}\n\\]\nis an isomorphism, and the $S^{1}$ action on $\\Gamma_{J}$ is given\nby\n\\[\n[u_{1},...,v_{n}]\\cdot e^{i\\theta}:=[(u_{1},...,v_{n})]\\cdot[C]=[(u_{1},...,v_{n})\\cdot C]\n\\]\nwith $det_{\\mathbb{C}}(C)=e^{i\\theta}$. Then \n\\[\n\\begin{aligned}\\Phi_{\\Gamma}^{h}\\big([u_{1},...,v_{n}]\\cdot e^{i\\theta}\\big) &  & = & \\Phi_{\\Gamma}^{h}[(u_{1},...,v_{n})\\cdot C]\\\\\n &  & = & [(h_{*}u_{1},...,h_{*}v_{n})\\cdot C]\\\\\n &  & = & \\Phi_{\\Gamma}^{h}\\big([u_{1},...,v_{n}]\\big)\\cdot e^{i\\theta}\n\\end{aligned}\n.\n\\]\nSince the unitary structure on $M$ is unique up to isomorphism, the\ndiscussion in this subsection amounts to the following proposition:\n\\begin{prop}\n\\label{prop:Existence of =00005CPhi=000023}Suppose that $\\Phi$ is\na symplectic group action on $(M,\\omega)$ by a compact Lie group\n$G$. Then there exists a $G$-action $\\Phi_{\\#}$ on $Fr_{J}^{u}\\rightarrow M$\nthat covers $\\Phi$. As a consequence, it induces $G$-actions $\\Phi_{\\Gamma}$\nand $\\Phi_{\\Gamma^{2}}$ on the bundles $\\Gamma_{J}$ and $\\Gamma_{J}^{2}$,\nrespectively. These actions covers $\\Phi$ and are covered by $\\Phi_{\\#}$.\n\\end{prop}\n\n\n\\subsection{\\label{subsec:An-example: S^2}An example: $S^{2}$}\n\nWe can get the idea of the main result in this section by looking\ninto the case where $M=S^{2}$. This might be the simplest example\nin which the Maslov $S^{1}$ bundles are non-trivial.\n\nConsider $S^{2}$ as an embedded submanifold in $\\mathbb{R}^{3}$.\nAt each point $p\\in S^{2}$, the tangent space $T_{p}S^{2}$ is a\nsubspace of $T_{p}\\mathbb{R}^{3}$. Let $\\bar{n}$ be the restriction\nof the vector field $x\\frac{\\partial}{\\partial x}+y\\frac{\\partial}{\\partial y}+z\\frac{\\partial}{\\partial z}$\non $S^{2}$. With $T_{p}\\mathbb{R}^{3}$ being identified with $\\mathbb{R}^{3}$,\nit holds $\\bar{n}_{p}=p$. Then\n\\[\n\\omega_{S^{2}}=\\iota_{\\bar{n}}dx\\wedge dy\\wedge dz.\n\\]\nis a symplectic structure on $S^{2}$. Viewing $u,v\\in T_{p}S^{2}$\nand as vectors in $\\mathbb{R}^{3}$, it holds\n\\[\n\\omega_{S^{2}}(u,v)=\\bar{n}\\cdot(u\\times v).\n\\]\nWith respect to $\\omega_{S^{2}}$, \n\\[\nJ_{S^{2}}(u):=-\\bar{n}\\times u\n\\]\ndefines a compatible almost complex structure, and \n\\[\ng_{S^{2}}(\\cdot,\\cdot):=\\omega_{S^{2}}(J_{S^{2}}\\cdot,\\cdot)\n\\]\nis the restriction of the standard Riemannian metric on $\\mathbb{R}^{3}$\nto $S^{2}$. \n\nDenote by $\\Gamma_{S^{2}}$ and $Fr_{S^{2}}^{u}$, respectively, the\nMaslov $S^{1}$ bundle ($\\Gamma_{J}$) and the unitary frame bundle\n($Fr_{J}^{u}$) of $(S^{2},\\omega_{S^{2}})$. Over each $p\\in S^{2}$,\neach unitary frame can be denoted uniquely and distinctly as\n\\[\n\\big(u,J_{S^{2}}(u);p\\big)=(u,\\bar{n}_{p}\\times u;p)=(u,p\\times u;p)\n\\]\nwith $u\\in\\mathbb{R}^{3}$ tangent to $S^{2}$ at $p$. When written\nas a matrix $[u,p\\times u,p]$ with $u$ , $p\\times u$ and $p$ being\nthe columns, this is an element in $\\mathbb{SO}(3)$. More precisely,\nthe following map\n\\[\n\\mathfrak{mat}:Fr_{S^{2}}^{u}\\ni\\big(u,-J_{S^{2}}(u);p\\big)\\mapsto[u,p\\times u,p]\\in\\mathbb{SO}(3)\n\\]\nis a diffeomorphism from $Fr_{S^{2}}^{u}$ to $\\mathbb{SO}(3)$. Since\n$\\mathbb{SU}(1)=\\{1\\}$, $\\Gamma_{S^{2}}=Fr_{S^{2}}^{u}$. Since $\\mathbb{SO}(3)$\nacts symplectically on $(S^{2},\\omega_{S^{2}})$, there is an $\\mathbb{SO}(3)$-action\non the bundle $\\pi_{\\Gamma_{S^{2}}}:\\Gamma_{S^{2}}\\rightarrow S^{2}$\nwhich covers the action on $S^{2}$. In fact, this action is exactly\ngiven by \n\\[\n\\mathbb{SO}(3)\\times\\Gamma_{S^{2}}\\xrightarrow{id\\times\\mathfrak{mat}}\\mathbb{SO}(3)\\times\\mathbb{SO}(3)\\xrightarrow{multiply}\\mathbb{SO}(3)\\xrightarrow{\\mathfrak{mat}^{-1}}\\Gamma_{S^{2}}\n\\]\nwith\n\\[\n\\big(A,[u,p\\times u,p]\\big)\\mapsto[Au,Ap\\times Au,Ap].\n\\]\n\nHere $\\mathbb{SO}(3)\\times\\mathbb{SO}(3)\\xrightarrow{multiply}\\mathbb{SO}(3)$\nis simply the group multiplication $(A,B)\\mapsto A\\cdot B$, and hence\nthe lifted group action is recognized as the left action of $\\mathbb{SO}(3)$\non itself (identified as $\\Gamma_{S^{2}}$). As a result, we have\nthe following proposition.\n\\begin{prop}\n\\label{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}The\nlifted $\\mathbb{SO}(3)$ action is transitive on $\\Gamma_{S^{2}}$.\n\\end{prop}\n\n\\begin{proof}\nThis is because the left action of $\\mathbb{SO}(3)$ on itself is\ntransitive.\n\\end{proof}\nLet $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$ be a\nconnection $1$-form on $\\Gamma_{S^{2}}$. Since $S^{2}$ is simply\nconnected and the principal bundle $\\pi_{\\Gamma_{S^{2}}}:\\Gamma_{S^{2}}\\rightarrow S^{2}$\nis not trivial, there is no integrable connection on $\\Gamma_{S^{2}}$\nand hence $df_{\\alpha}\\neq0$. Note that $df_{\\alpha}$ descends to\na nondegenerate closed $2-$form $\\Omega_{\\alpha}$ on $S^{2}$, and\nsince $H_{dR}^{2}(S^{2})=\\mathbb{R}$ it satisfies $[\\Omega_{\\alpha}]=r\\cdot[\\omega]$\nfor some real number $r\\neq0$. Moreover, we have the following proposition.\n\\begin{lem}\n\\label{lem:symp-potential on S^2}There exists an $\\mathbb{SO}(3)$-invariant\nconnection $1$-form $\\bar{\\alpha}$ on $\\Gamma_{S^{2}}$ such that\n$df_{\\bar{\\alpha}}=r\\cdot\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)$ with $r$\nbeing some non-zero real number. \n\\end{lem}\n\n\\begin{proof}\nStarting with an arbitrary connection $1$-form $\\alpha$, for each\nelement $v$ of the tangent bundle $T\\Gamma_{S^{2}}$, define\n\\[\n\\bar{f}(v):=\\int_{\\mathbb{SO}(3)}f_{\\alpha}(h_{*}v)dh\n\\]\nwith $dh$ being a right invariant probability measure on $\\mathbb{SO}(3)$.\nThen $\\bar{f}$ is both $S^{1}$-invariant and $\\mathbb{SO}(3)$ invariant,\nand $\\bar{f}(\\frac{\\partial}{\\partial\\theta})\\equiv1$. Therefore,\n$\\bar{\\alpha}:=\\bar{f}\\cdot\\frac{\\partial}{\\partial\\theta}$ defines\na connection $1$-form which is invariant under the $\\mathbb{SO}(3)$-action.\nThen $d\\bar{f}$ is also $\\mathbb{SO}(3)$ invariant, and so is $\\Omega_{\\bar{\\alpha}}$\nwith $d\\bar{f}=\\pi_{\\Gamma_{S^{2}}}^{*}(\\Omega_{\\bar{\\alpha}})$.\nSince $\\Omega_{\\bar{\\alpha}}\\neq0$ and $S^{2}$ is $2$ dimensional,\nthere exists a non-zero real number $r$ such that $\\Omega_{\\bar{\\alpha}}=r\\cdot\\omega$.\nNote that $\\bar{f}=f_{\\bar{\\alpha}}$ and this concludes the proof.\n\\end{proof}\n\\begin{prop}\n\\label{prop:symplectic=00003DHamiltonian on S^2}There exists a connection\n$1$-form $\\bar{\\beta}:=f_{\\bar{\\beta}}\\cdot\\frac{\\partial}{\\partial\\theta}$\nsuch that, for each $v\\in\\mathfrak{so}(3)$, the corresponding infinitesimal\ngenerator $X_{v}$ has a Hamiltonian function $H_{v}$ on $S^{2}$\nsatisfying $H_{v}\\circ\\pi_{\\Gamma_{S^{2}}}=-\\frac{1}{r}\\cdot f_{\\bar{\\beta}}(\\mathcal{X}_{v})$\nwith some non-zero constant $r$.\n\\end{prop}\n\n\\begin{proof}\nLet $\\bar{\\alpha}$ be the connection $1$-form in Lemma \\ref{lem:symp-potential on S^2}.\nFor simplicity, we write the symplectic form $\\omega_{S^{2}}$ as\n$\\omega$ in the following discussion. Applying the averaging method\nagain to $\\bar{\\alpha}$ with the $G$-action on $\\Gamma_{S^{2}}$:\n\\[\n\\bar{\\beta}(v):=\\int_{G}\\bar{\\alpha}\\circ h_{*}(v)dh\n\\]\n with $dh$ being a right invariant probability measure on $G$. $f_{\\bar{\\beta}}$\nis then invariant under the $G$-action, and\n\\[\n\\begin{aligned}df_{\\bar{\\beta}}(u,v) &  & = & \\int_{G}h^{*}(df_{\\bar{\\alpha}})(u,v)dh\\\\\n &  & = & r\\cdot\\int_{G}h^{*}\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)(u,v)dh\\\\\n &  & = & r\\cdot\\int_{G}\\pi_{\\Gamma_{S^{2}}}^{*}h^{*}(\\omega)(u,v)dh\\\\\n &  & = & r\\cdot\\int_{G}\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)(u,v)dh\\\\\n &  & = & r\\cdot\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)(u,v)\n\\end{aligned}\n.\n\\]\nBy Cartan's formula,\n\\[\n\\begin{aligned}d\\big(f_{\\bar{\\beta}}(\\mathcal{X}_{v})\\big)(\\cdot) &  & = & \\mathcal{L}_{\\mathcal{X}_{v}}f_{\\bar{\\beta}}-\\iota_{\\mathcal{X}_{v}}df_{\\bar{\\beta}}\\\\\n &  & = & -r\\cdot\\omega\\big(\\pi_{\\Gamma_{S^{2}},*}(\\mathcal{X}_{v}),\\pi_{\\Gamma_{S^{2}},*}\\cdot\\big)\\\\\n &  & = & -r\\cdot\\omega(X_{v},\\pi_{\\Gamma_{S^{2}},*}\\cdot)\n\\end{aligned}\n.\n\\]\nNote that the $G$-action commutes with the $S^{1}$-action on $\\Gamma_{S^{2}}$,\nand hence $\\mathcal{X}_{v}$ is $S^{1}$-invariant. As a result, $-\\frac{1}{r}\\cdot f_{\\bar{\\beta}}(\\mathcal{X}_{v})$\nis also $S^{1}$-invariant, and there is a fucntion $H_{v}$ on $M$\nsuch that\n\\[\n-\\frac{1}{r}\\cdot f_{\\bar{\\beta}}(\\mathcal{X}_{v})=H_{v}\\circ\\pi_{\\Gamma_{S^{2}}}.\n\\]\nWith the identity deduced above it yields \n\\[\ndH_{v}\\circ\\pi_{\\Gamma_{S^{2}},*}=\\omega\\big(X_{v},\\pi_{\\Gamma_{S^{2}},*}\\cdot\\big)\n\\]\nand then \n\\[\ndH_{v}=\\omega(X_{v},\\cdot).\n\\]\n\\end{proof}\n\n\\section{\\label{sec:dynamics-on Maslov S^1 bundles}Dynamics on the Maslov\n$S^{1}$ Bundles}\n\nIn this section we extend the properities obtained for $S^{2}$, Proposition\n\\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D} and\n\\ref{prop:symplectic=00003DHamiltonian on S^2}, to more general settings.\n\n\\subsection{Extension for Proposition \\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}\nto symplectic homogeneous $G$-spaces}\n\nIn this subsections we extend Proposition \\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}\nto the case where the group action $\\Phi$ by $G$ is transitive on\n$M$ . Namely, $M$ is a symplectic homogeneous $G$-space. For each\n$p\\in M$ and $w\\in\\Gamma_{J}$, denote by $G_{p}$ the isotropy group\nof the action $\\Phi$ at $p$, and by $G_{w}$ the isotropy group\nof $\\Phi_{\\Gamma}$ at $w$.\n\nBy the transitivity of the $G$-action, $M$ is diffeomorphic to $G\\big\/G_{p_{0}}$\nfor any fixed point $p_{0}$ through the map\n\\[\n\\mathcal{F}_{p_{0}}:G\\big\/G_{p_{0}}\\ni[h]\\mapsto h\\cdot p_{0}\\in M.\n\\]\nLet $w_{0}$ be a point in $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$. Then $G_{w_{0}}\\subset G_{p_{0}}$.\nSince $\\Phi_{\\Gamma}$ on $\\Gamma_{J}$ covers $\\Phi$ on $M$, $G_{p_{0}}$\nacts on the fiber $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$. That is, $h\\cdot w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$\nfor any $h\\in G_{p_{0}}$ and $w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$.\nIt turns out that $G_{w}=G_{w_{0}}$ for any $w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$,\nand that the $G_{p_{0}}$-action on $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$\ninduces a homomorphism to $S^{1}$ with $G_{w_{0}}$ being the kernel. \n\\begin{prop}\n\\label{prop:=00005Cphi_p}There exists a Lie group homomorphism $\\phi_{p}$\nfrom $G_{p}$ to $S^{1}$ with $G_{w}$ being the kernel such that,\nfor $w\\in\\pi_{\\Gamma_{J}}^{-1}(p)$ and $h\\in G_{p}$, it holds\n\\[\nh\\cdot w=w\\cdot\\phi_{p}(h).\n\\]\nMoreover, the family $\\{\\mathcal{\\phi}_{p}\\}$ is $G$-related in\nthe sense that\n\\begin{equation}\n\\phi_{h'\\cdot p}\\circ Ad_{h'}=\\phi_{p}\\label{eq:adjoint action on =00007B=00005Cphi=00007D}\n\\end{equation}\nwith $p\\in M$ and $h'\\in G$. \n\\end{prop}\n\n\\begin{proof}\nWe start by arguing with the fixed points $p_{0}$ and $w_{0}$. Since\n$S^{1}$ acts transitively and freely on $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$,\nfor each $h\\in G_{p_{0}}$ there exists a unique $z_{h}\\in S^{1}$\nsuch that $h\\cdot w_{0}=w_{0}\\cdot z_{h}$. Then \n\\[\nw\\cdot z_{h'h}=(h'h)\\cdot w_{0}=w_{0}\\cdot z_{h}\\cdot z_{h'}=w_{0}\\cdot(z_{h'}z_{h}),\n\\]\nand hence $z_{h'h}=z_{h'}z_{h}.$ Define $\\phi_{p_{0}}$with $\\phi_{p_{0}}(h)=z_{h}.$\nThen this is a homomorphism, and $h\\cdot w_{0}=w_{0}$ if and only\nif $\\phi_{p_{0}}(h)=1_{G}$. Hence $\\ker\\phi_{p_{0}}=G_{w_{0}}$.\nNote that $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$ is diffeomorphic to $S^{1}$\nvia $\\bar{\\theta}_{w_{0}}:w_{0}\\cdot z\\mapsto z$. Since $\\phi_{p_{0}}(h)=\\bar{\\theta}_{w_{0}}\\circ\\Phi_{\\Gamma}^{h}(w_{0})$,\n$\\phi_{p_{0}}$ is a smooth map, and therefore it is a Lie group homomorphism.\n\nThe assignment of $z_{h}$ to $h$ is actually independent of the\nchoice of $w_{0}$. Namely, the identity $h\\cdot w=w\\cdot z_{h}$\nholds for all $w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$. To see this, first\nnote that there existis $z\\in S^{1}$ such that $w_{0}\\cdot z=w$,\nand then\n\\[\nh\\cdot w=h\\cdot(w_{0}\\cdot z)=(h\\cdot w_{0})\\cdot z=w_{0}\\cdot z_{h}\\cdot z=w\\cdot z_{h}.\n\\]\nMoreover, the argument above holds for any point $p$, and then the\nhomomorphism $\\mathcal{\\phi}_{p}$ can be constructed in the same\nway. \n\nIt remains to show that $G$ acts on the family $\\{\\mathcal{\\phi}_{p}\\}$\naccording to Eq.\\ref{eq:adjoint action on =00007B=00005Cphi=00007D}.\nLet $w$ be a point on the fiber $\\pi_{\\Gamma}^{-1}(p)$. Then $w'=h'\\cdot w\\in\\pi_{\\Gamma}^{-1}(h'\\cdot p)$.\nFor any $h\\in G_{p}$,\n\\[\n\\begin{aligned}h'hh'^{-1}\\cdot w' &  & = & h'hw\\\\\n &  & = & h'\\cdot\\big(w\\cdot\\phi_{p}(h)\\big)\\\\\n &  & = & (h'\\cdot w)\\cdot\\phi_{p}(h)\\\\\n &  & = & w'\\cdot\\phi_{p}(h),\n\\end{aligned}\n\\]\nwhich implies $\\phi_{h'\\cdot p}(h'hh'^{-1})=\\phi_{p}(h)$ and concludes\nthe proof.\n\\end{proof}\nDue to (\\ref{eq:adjoint action on =00007B=00005Cphi=00007D}) in Proposition\n\\ref{prop:=00005Cphi_p} above, for any $p,p'\\in M$, $\\text{Im}\\phi_{p}=\\text{Im}\\phi_{p'}$,\nand we denote this subgroup by $S_{\\phi}^{1}$. Since $G_{p}$ is\ncompact, $S_{\\phi}^{1}=\\phi_{p}(G_{p})$ is a compact subgroup of\n$S^{1}$. Therefore it is either a finite cyclic group $\\{e^{i\\frac{2\\pi}{k}\\cdot l}\\big|l=0,....,k-1\\}$,\nor $S^{1}$ itself.\n\nFor each element $v$ of the Lie algebra $\\mathfrak{g}$ of $G$,\ndenote by $\\mathcal{X}_{v}$ the infinitesimal generator of $\\Phi_{\\Gamma}$\nin the direction $v$. That is, $\\mathcal{X}_{v}(p)=\\frac{d}{dt}exp(tv)\\cdot p$\nwith $t\\mapsto exp(tv)$ being the one parameter subgroup of $G$\ngenerated by $v$. We claim:\n\\begin{prop}\n\\label{prop:If:S_=00005Cphi-finite}$S_{\\phi}^{1}$ is either $S^{1}$\nor a finite subgroup of $S^{1}$. If $S_{\\phi}$ is finite, then the\ninifinitesimal generators $\\mathcal{X}_{v}$ span an integrable connection\n$\\mathcal{D}$ on the bundle $\\Gamma_{J}$ with the orbits $G\\cdot w$\nbeing the maximal connected integral manifolds.\n\\end{prop}\n\n\\begin{proof}\nAs an orbit of a compact Lie group action, $G\\cdot w$ is an embedded\nsubmanifold in $\\Gamma_{J}$, and then its dimension is no larger\nthan $2n+1$. If $\\dim G\\cdot w=2n+1$, $G\\cdot w$ is an open set\nin $\\Gamma_{J}$ and then $G\\cdot w\\cap\\pi_{\\Gamma_{J}}^{-1}(p)$\nis open in $\\pi_{\\Gamma_{J}}^{-1}(p)$ (and then it is the whole $\\pi_{\\Gamma_{J}}^{-1}(p)$\nsince it is open and compact for any $p\\in M$). Now suppose that\n$\\dim G\\cdot w\\leq2n+1$. Since $G$ acts transitively on $M$ and\n$\\pi_{\\Gamma}\\circ\\Phi_{\\#}=\\Phi\\circ\\pi_{\\Gamma}$, it holds\n\\[\n\\pi_{\\Gamma,*}\\big(T_{w}(G\\cdot w)\\big)=T_{\\pi_{\\Gamma}(w)}M,\n\\]\nand hence $\\dim G\\cdot w\\geq2n$. Together this yields $\\dim G\\cdot w=2n$.\nDue to the commutativity of $\\Phi_{\\Gamma}$ and the $S^{1}$action\non $\\Gamma_{J}$, $(G\\cdot w)\\cdot z=G\\cdot(w\\cdot z)$ with $z\\in S^{1}$.\nThis means $(G\\cdot w)\\cdot z$ is exactly the $G$-orbit through\n$w\\cdot z$, and the $S^{1}$ action maps orbits to orbits. Since\n\\[\nT_{w}(G\\cdot w)=span\\{\\mathcal{X}_{v}\\big|v\\in\\mathfrak{g}\\}=\\mathcal{D}_{w},\n\\]\n$\\mathcal{D}$ is a $2n$ dimensional distribution invariant under\nthe $S^{1}$ action, and $\\pi_{\\Gamma,*}(\\mathcal{D}_{w})=T_{\\pi_{\\Gamma}(w)}M$.\n\\end{proof}\nTherefore, from Proposition \\ref{prop:If:S_=00005Cphi-finite} we\ncan deduce:\n\\begin{thm}\nLet $G$ be a compact Lie group acting transitively and symplectically\non $M$. If the characteristic class of the Maslov $S^{1}$ bundle\n$\\Gamma_{J}$ is non-zero, then $\\Gamma_{J}$ is also a homogeneous\n$G$-space.\n\\end{thm}\n\n\\begin{proof}\nIf $\\Phi_{\\Gamma}$ does not act transitively on $\\Gamma_{J}$, then\n$S_{\\phi}^{1}$ should be a finite subgroup of $S^{1}$. According\nto Proposition \\ref{prop:If:S_=00005Cphi-finite}, $\\mathcal{D}$\nwould be an integrable connection, and then the characteristic class\nwould be zero, which violates the condition.\n\\end{proof}\n\n\\subsection{Extension for Proposition \\ref{prop:symplectic=00003DHamiltonian on S^2}\nto the case $[\\omega]=r\\cdot c_{\\Gamma}$}\n\nThe argument for Proposition \\ref{lem:symp-potential on S^2} can\nactually be applied to any symplectic action on a symplectic manifold\n$(M,\\omega)$ satisfying $[\\omega]=r\\cdot c_{\\Gamma}$ for $r\\in\\mathbb{R}$.\nInstead of directly giving a proof, we introduce some notions and\nformalize the demonstration in such a way that the relevant structures\nare better illustrated.\n\nRecall that a symplectic $G$-action on $M$ is Hamiltonian if and\nonly if there is a smooth map (called a momentum map)\n\\[\nF:M\\rightarrow\\mathfrak{g}^{*}\n\\]\nsuch that for each $h\\in G$ and $v\\in\\mathfrak{g}$, $F$ is $G$-coadjoint\nequivariant, i.e.\n\\[\nF_{h\\cdot x}=Ad_{h^{-1}}^{*}F_{x},\n\\]\nand the mapping \n\n\\[\nF^{v}:M\\ni x\\mapsto F_{x}(v)\\in\\mathbb{R}\n\\]\ndefines a Hamiltonian for $\\mathcal{X}_{v}$ with $v\\in\\mathfrak{g}$.\n\\begin{rem}\nIn the rest of this subsection the symbol ``$\\Gamma$'' stands for\nboth the spaces $\\Gamma_{J}$ and $\\Gamma_{J}^{2}$, since the same\nargment works for both of them.\n\\end{rem}\n\n\\begin{defn}\nAn $S^{1}$-invariant $1$ form $\\eta$ on the bundle $\\pi_{\\Gamma}:\\Gamma\\rightarrow M$\nis called a symplectic potential if $d\\eta=-\\pi_{P}^{*}(\\omega)$.\n\\end{defn}\n\nIt is straighforward to check that averging a symplectic potential\n$\\eta$ with the (lifted) $G$ action $\\Phi_{\\Gamma}$ gives a $G$-invariant\nsymplectic potential $\\bar{\\eta}$. We have the following lemma.\n\\begin{lem}\n\\label{lem:coadjoint-equivariant map from a symplectic potential}Let\n$\\bar{\\eta}$ be a $G$-invariant symplectic potential. For each $p\\in\\Gamma$,\nlet $\\mu_{p}$ be the element in $\\mathfrak{g}^{*}$ with $\\mu_{p}(v)=\\bar{\\eta}(\\mathcal{X}_{v})(p)$\nfor $v\\in\\mathfrak{g}$, where $\\mathcal{X}_{v}$ is the infinitesimal\ngenerator of $\\Phi_{\\Gamma}$ corresponding to $v$. Then the map\n$\\mu$ from $\\Gamma$ to $\\mathfrak{g}^{*}$ defined by \n\\[\n\\mu:p\\mapsto\\mu_{p}\n\\]\nis a momentum map for $G$.\n\\end{lem}\n\n\\begin{proof}\nFor the smoothness of $\\mu$, it suffices to show that the map $\\bar{\\mu}:\\Gamma\\times\\mathfrak{g}\\rightarrow\\mathbb{R}$\ndefined by $\\bar{\\mu}(p,v)=\\mu_{p}(v)$ is smooth. Observe that $\\bar{\\mu}$\nfactors as\n\\[\n\\bar{\\mu}:\\Gamma\\times\\mathfrak{g}\\stackrel{\\mathcal{X}}{\\longrightarrow}T\\Gamma\\stackrel{\\bar{\\eta}}{\\longrightarrow}\\mathbb{R}\n\\]\nwith $\\mathcal{X}$ being the map from $P\\times\\mathfrak{g}$ to $TP$\nsending $(p,v)$ to $\\mathcal{X}_{v}(p)$. The factorization below\nshows smoothness of $\\mathcal{X}$:\n\n\\[\n\\mathcal{X}:\\Gamma\\times\\mathfrak{g}\\stackrel{\\sigma}{\\longrightarrow}T\\Gamma\\times TG\\cong T(\\Gamma\\times G)\\stackrel{D\\Phi_{\\Gamma}}{\\longrightarrow}T\\Gamma.\n\\]\nHere $\\sigma$ is the map sending $(p,v)\\in\\Gamma\\times\\mathfrak{g}$\nto $(\\bar{\\boldsymbol{0}}_{p},v)\\in T_{p}\\Gamma\\times T_{1}G$ with\n$\\bar{\\boldsymbol{0}}$ being the zero section from $\\Gamma$ to $T\\Gamma$,\nand $D\\Phi_{\\Gamma}$ is the tangent map of $\\Phi_{\\Gamma}$. Hence\n$\\mathcal{X}$ is smooth. As a consequence, $\\bar{\\mu}$ is also smooth.\n\nSince $\\bar{\\eta}$ and $\\mathcal{X}_{v}$ are both $S^{1}$-invariant\non $\\Gamma$, the function $\\bar{\\eta}(\\mathcal{X}_{v})$ is constant\nalong each $S^{1}$-fiber, and hence there exists a function $H_{v}$\non $M$ such that $H_{v}\\circ\\pi=\\bar{\\eta}(\\mathcal{X}_{v})$. Check\nthat $0=\\mathcal{L}_{\\mathcal{X}_{v}}(\\bar{\\eta})=d\\iota_{\\mathcal{X}_{v}}\\bar{\\eta}+\\iota_{\\mathcal{X}_{v}}d\\bar{\\eta}$\nand then it holds\n\\[\nd\\big(\\bar{\\eta}(\\mathcal{X}_{v})\\big)=-\\iota_{\\mathcal{X}_{v}}d\\bar{\\eta}=\\iota_{\\mathcal{X}_{v}}\\pi_{P}^{*}(\\omega),\n\\]\nimplying $\\iota_{X_{v}}\\omega=dH_{v}$, where $X_{v}=\\pi_{\\Gamma,*}(\\mathcal{X}_{v})$\nis the generator of $\\Phi$ corresponding to $v$.\n\nIt remains to show that $\\mu$ is $G$-coadjoint equivariant. Note\nthat for $h\\in G$ and $p\\in\\Gamma$,\n\\[\nh_{*}(\\mathcal{X}_{v}\\big|_{p})=\\mathcal{X}_{Ad_{h,*}(v)}(h\\cdot p).\n\\]\nThen \n\\[\n\\begin{aligned}\\mu_{h\\cdot p}(v) &  & = & \\bar{\\eta}(\\mathcal{X}_{v})(h\\cdot p)\\\\\n &  & = & \\bar{\\eta}\\big|_{h\\cdot p}\\big(\\mathcal{X}_{v}(h\\cdot p)\\big)\\\\\n &  & = & \\bar{\\eta}\\big|_{h\\cdot p}\\big(h_{*}\\mathcal{X}_{Ad_{h^{-1},*}(v)}(p)\\big)\\\\\n &  & = & h^{*}(\\bar{\\eta}\\big|_{h\\cdot p})\\big(\\mathcal{X}_{Ad_{h^{-1},*}(v)}(p)\\big)\\\\\n &  & = & \\bar{\\eta}\\big|_{p}\\big(\\mathcal{X}_{Ad_{h^{-1},*}(v)}(p)\\big)\\\\\n &  & = & \\mu_{p}\\circ Ad_{h^{-1},*}(v)\n\\end{aligned}\n\\]\nand this concludes the proof.\n\\end{proof}\nWe give the following remark for later reference.\n\\begin{thm}\n\\label{thm:Symplectic=00003D=00003DHamiltonian}If $[\\omega]=r\\cdot c_{\\Gamma}$\nwith $c_{\\Gamma}$ being the first real Chern class of $\\Gamma$ and\n$r\\in\\mathbb{R}$, then any symplectic action $\\Phi$ on $M$ by a\ncompact Lie group $G$ is Hamiltonian. Moreover, if $r\\neq0$, then\nthere exists a connection $1$-form $\\beta=f_{\\beta}\\cdot\\frac{\\partial}{\\partial\\theta}$\nwith some nonzero real number $r$ such that $\\frac{1}{r}f_{\\beta}$\nis a $G$-invariant symplectic potential, and then $\\frac{1}{r}f_{\\beta}(\\mathcal{X}_{v})$\nis an Hamiltonian for $X_{v}$.\n\\end{thm}\n\n\\begin{proof}\nIf the homology class $[\\omega]=0$, then there exists a $1$-form\non $M$ with $d\\tau=\\omega$. It is straightforward to check that\n$\\pi_{P}^{*}(\\tau)$ is an $S^{1}$-invariant $1$-form on $\\Gamma$\nand is a symplectic potential. If $[\\omega]\\neq0$, then there exists\na non-zero constant $r$ such that $[r\\cdot\\omega]=[\\Omega_{\\alpha}]$\nwith $\\pi_{\\Gamma}^{*}(\\Omega_{\\alpha})=df_{\\alpha}$ for a connection\n$1$-form $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$.\nThen it holds $r\\cdot\\omega=\\Omega_{\\alpha}+d\\tau$ for some $1$-form\n$\\tau$ on $M$. Let $f_{\\beta}=f_{\\alpha}+\\pi_{\\Gamma_{J}}^{*}(\\tau)$\nand then $df_{\\beta}=r\\cdot\\pi_{\\Gamma}^{*}(\\omega)$. Hence $\\frac{1}{r}f_{\\beta}$\nis a symplectic potential on the bundle $\\Gamma\\rightarrow M$. \n\nThe conclusion then follows from Lemma \\ref{lem:coadjoint-equivariant map from a symplectic potential}.\n\\end{proof}\n\n\\subsection{Conservation laws}\n\nWhen $\\eta$ (or $\\beta$) is invariant under the action $\\Phi_{\\Gamma^{2}}$,\nit is always true that $\\eta(\\mathcal{X}_{v})$ is constant along\nthe flow of $\\mathcal{X}_{v}$, i.e. $\\mathcal{L}_{\\mathcal{X}_{v}}\\eta(\\mathcal{X}_{v})=0$.\nThis is because $\\mathcal{X}_{v}$ and $\\eta$ are invariant under\nthe pushforward and pullback of the flow of $\\mathcal{X}_{v}$, respectively.\nMoreover, since $\\mathcal{X}_{v}$ and $\\eta$ are also invariant\nunder the inherent $S^{1}$ action of the bundle $\\Gamma^{2}$, there\nalways exists a function $Q_{v}$ on $M$ such that $\\eta(\\mathcal{X}_{v})=Q_{v}\\circ\\pi_{\\Gamma^{2}}$.\nSince $\\pi_{\\Gamma^{2}}$ is a submersion, the smoothness of $\\eta(\\mathcal{X}_{v})$\nimplies the smoothness of $Q_{v}$. Check that $d\\eta(\\mathcal{X}_{v})=dQ_{v}\\circ\\pi_{\\Gamma^{2},*}$\nand then $0=dQ_{v}\\circ\\pi_{\\Gamma^{2},*}(\\mathcal{X}_{v})=dQ_{v}(X_{v})$,\nmeaning $Q_{v}$ is invariant under the flow $\\varphi_{v}$ of $X_{v}$.\nThis implies in particular that, if the trajectory of the one-parameter\nsubgroup $\\exp(tv)$ is dense in $G$, then $Q_{v}$ is invariant\nunder the $G$-action.\n\n\\section{\\label{sec:Beta-Maslov-Quantity for S^1-action}$\\beta$-Maslov data\nfor Symplectic $S^{1}$ Actions}\n\nIn this section we consider the case $G=S^{1}$. For a connection\n$1$-form $\\beta=\\eta\\cdot\\frac{\\partial}{\\partial\\theta}$ on $\\Gamma^{2}$,\nthe $\\beta$-Maslov data $Q_{\\beta}$ of the $S^{1}$ action $\\Phi$\nis the function on $M$ defined by\n\\begin{equation}\nQ_{\\beta}(p)=\\mathfrak{m}_{\\beta}(\\gamma_{w})=\\int_{\\gamma_{w}}\\eta\\label{eq:def: Maslov-Quantity Q_=00005Ctheta}\n\\end{equation}\nwith $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$ and $\\gamma_{w}(z)=\\Phi_{\\Gamma^{2}}^{z}(w)$\nfor $z\\in S^{1}$. Note that the integral in (\\ref{eq:def: Maslov-Quantity Q_=00005Ctheta})\nis independent of the choice of $w$ on $\\pi_{\\Gamma^{2}}^{-1}(p)$\nand hence $Q_{\\beta}$ is a well-defined function on $M$. To see\nthis, note that for $w,w'\\in\\pi_{\\Gamma^{2}}^{-1}(p)$, there exists\nan element $z\\in S^{1}$ such that $w'=w\\cdot z$ and then $\\gamma_{w'}=\\gamma_{w}\\cdot z$,\nand $\\int_{\\gamma_{w}}\\eta=\\int_{\\gamma_{w'}}\\eta$ follows from the\nfact that $\\eta$ is invariant under the inherent $S^{1}$ action\non $\\Gamma^{2}$ (since $\\beta$ is a connection $1$-form).\n\nFor simplicity we denote by $\\varphi$ the period-$1$ flow of the\ncircle action, namely, $\\frac{d}{dt}\\varphi^{t}=X_{\\theta}$. \n\n\\subsection{\\label{subsec:The-Maslov-index of the S^1 action}The Maslov index\nof the $S^{1}$ action}\n\nIn this subsection we look at the case when $\\Gamma^{2}$ is a trivial\nbundle. \n\nLet $\\mathfrak{s}$ be a global section of $\\Gamma^{2}$. For each\nLagrangian plane $w$, $\\tilde{\\gamma}_{w}(z)=\\Phi_{*}^{z}(w)$ with\n$z\\in S^{1}$ is a loop in $\\Lambda_{pl}$ and hence has a Maslov\nindex $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma}_{w})$ with respect\nto $\\mathfrak{s}$. Note that $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma}_{w})$\nequals to the degree of the map obtained by the following composition\n\n\\begin{equation}\nS^{1}\\cong[0,1]\\big\/\\{0,1\\}\\xrightarrow{\\Phi_{*}(w)}\\Lambda_{pl}\\xrightarrow{det_{J}^{2}}\\Gamma_{J}^{2}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}\\xrightarrow{pr_{S^{1}}}S^{1},\\label{eq:Maslov index for circle actions (Lambda)}\n\\end{equation}\nwhich, together with the connectedness of $\\Lambda_{pl}$, implies\nthat $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma}_{w})$ is actually\nindependent of $w$. Namely, all the orbits in $\\Lambda_{pl}$ of\nthe action $\\Phi_{*}$ have the same Maslov index. \n\\begin{rem}\nNote that the construction above is equivalent to looking at the degree\n$\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})$ of the following map\n\\begin{equation}\nS^{1}\\cong[0,1]\\big\/\\{0,1\\}\\xrightarrow{\\Phi_{\\Gamma^{2}}^{(\\cdot)}(w)}\\Gamma_{J}^{2}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}\\xrightarrow{pr_{S^{1}}}S^{1}\\label{eq:Maslov index for circle actions (Gamma)}\n\\end{equation}\nwith $\\gamma_{w}(z)=\\Phi_{\\Gamma^{2}}^{z}(w)$.\n\\end{rem}\n\nWe show further that when the action has a fixed point, then $\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})$\nis also independent of the choice of $\\mathfrak{s}$, and hence we\ncan simply talk about the Maslov index of the flow with respect to\n$\\mathfrak{s}$ and denote it by $\\mathfrak{m}_{\\mathfrak{s}}(\\Phi)$.\n\nLet $p$ be an arbitrary fixed point. Then on the fiber $\\pi_{\\Gamma^{2}}^{-1}(p)$,\nthe vector field takes the form as $\\mathcal{X}_{\\theta}(w)=a_{w}\\frac{\\partial}{\\partial\\theta}$\nfor all $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$. Since $\\mathcal{X}_{\\theta}$\nis invariant under the inherent $S^{1}$ action, it holds $a_{w}\\equiv k_{p}$\non $\\pi_{\\Gamma^{2}}^{-1}(p)$. Let $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$\nbe the connection $1$-form that has (the image of) $\\mathfrak{s}$\nas an integral manifold of its kernel. Now that $\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})$\nis independent of the choice of $w$, we particularly choose $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$.\nThen it yields\n\\begin{equation}\n\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})=\\int_{0}^{1}f_{\\alpha}(\\mathcal{X}_{\\theta})\\circ\\gamma_{w}(t)dt=k_{p}.\\label{eq:def_local-Maslov}\n\\end{equation}\nand our arugment is concluded by the fact that $k_{p}$ is independent\nof $\\mathfrak{s}$. \n\\begin{rem}\n\\label{rem:X_=00005Ctheta=00003Dkp}According to the discussion above\nwe have $\\mathcal{X}_{\\theta}=k_{p}\\frac{\\partial}{\\partial\\theta}$\non the fiber $\\pi_{\\Gamma^{2}}^{-1}(p)$.\n\\end{rem}\n\nWe call $k_{p}$ defined by \\ref{eq:def_local-Maslov} the local Maslov\nindex of the $S^{1}$ action at the fixed point $p$. Note that, on\nthe one hand this definition does not depend on the triviality of\n$\\Gamma_{J}^{2}$, and on the other hand it equals to the Maslov index\nof the restricted action on an invariant Daboux chart of $p$. The\ndiscussion above leads to the following proposition.\n\\begin{prop}\n\\label{prop:local maslov index=00003D=00003Dglobal maslov index}When\nthe bundle $\\Gamma_{J}^{2}$ is trivial and the $S^{1}$ action $\\Phi$\nhas fixed points, then for any fixed points $p,p'$ and sections $\\mathfrak{s},\\mathfrak{s}'$,\nit holds\n\\[\nk_{p}=k_{p'}=\\mathfrak{m}_{\\mathfrak{s}}(\\Phi)=\\mathfrak{m}_{\\mathfrak{s}'}(\\Phi).\n\\]\n\\end{prop}\n\n\n\\subsection{The $\\beta$-Maslov data $Q_{\\beta}$}\n\nNow we take a look that the function $Q_{\\beta}$ defined in \\ref{eq:def: Maslov-Quantity Q_=00005Ctheta}.\nAs is noted in Remark \\ref{rem:X_=00005Ctheta=00003Dkp}, on the fiber\nover a fixed point $p$, it holds $\\mathcal{X}_{\\theta}=k_{p}\\frac{\\partial}{\\partial\\theta}$\nand then $\\eta(\\mathcal{X}_{\\theta})\\equiv k_{p}$ on the loop the\nloop $\\gamma_{w}(z)=\\Phi_{\\Gamma^{2}}^{z}(w)$ with $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$.\nHence\n\\begin{equation}\nQ_{\\beta}(p):=\\int_{\\gamma_{w}}\\eta=\\int_{0}^{1}\\eta(\\mathcal{X}_{\\theta})_{\\gamma_{w}(t)}dt=k_{p}.\\label{eq:Q(p)=00003Dk_p}\n\\end{equation}\nIn particular, if $\\beta$ is invariant under $\\Phi_{\\Gamma_{J}^{2}}$,\nthen $\\eta(\\mathcal{X}_{\\theta})$ is constant along any orbit of\n$\\Phi_{\\Gamma_{J}^{2}}$ and it holds that $\\pi_{\\Gamma^{2}}^{*}(Q_{\\beta})=\\eta(\\mathcal{X}_{\\theta})$.\n\nThe following remark comes as an observation on Eq.(\\ref{eq:Q(p)=00003Dk_p}). \n\\begin{rem}\nAlthough the construction of $Q_{\\beta}$ depends on the choice of\n$\\beta$, its values at a fixed point of the $S^{1}$ action do not.\nIn particular, if the action has fixed points with different local\nMaslov indices, then $Q_{\\beta}$ is never constant.\n\\end{rem}\n\nNote that for some invariant neighbourhood a fixed point $p$ of the\nsymplectic $S^{1}$ action, a Daboux chart\n\\[\n(U,dq_{i}\\wedge dp_{i})\\xrightarrow{\\phi}(M,\\omega)\n\\]\nwith $\\phi(0)=p$ can be chosen such that viewed in the chart the\nrestricted $S^{1}$ action is linearized as $(e^{2m_{1}\\pi i\\cdot t},...,e^{2m_{n}\\pi i\\cdot t}).$\nWe denote this linearized $S^{1}$ action on $U$ by $L_{\\Phi}$.\nWe call $(m_{1},...,m_{n})\\in\\mathbb{Z}^{n}$ the resonance type of\nthe fixed point $p$. From the structure of $Q_{\\beta}$ we have the\nfollowing proposition. \n\\begin{prop}\n\\label{prop:Maslov data at the fixed points}At the fixed point $p$\nof resonant type $(m_{1},...,m_{n})$, the local Maslov index is\n\\begin{equation}\nk_{p}=Q_{\\beta}(p)=2\\cdot\\sum_{i=1}^{n}m_{i}.\\label{eq:Maslov data=000026resonant type}\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nSince $\\phi$ is a symplectomorphism, it induces a bundle isomorphism\n$\\phi_{*}$ from $\\Gamma_{U}^{2}$ to $\\Gamma_{J}^{2}\\big|_{\\tilde{U}}$\nwith $\\tilde{U}=\\phi(U)$. Here $\\Gamma_{U}^{2}$ is the Maslov $S^{1}$\nbundle for $(U,dq_{i}\\wedge dp_{i})$ and it is isomorphic to $U\\times S^{1}$.\nThe lifted $S^{1}$ action $L_{\\Phi}$ on $\\Gamma_{U}^{2}$ and the\naction $\\Phi_{\\Gamma^{2}}$ on $\\Gamma_{J}^{2}\\big|_{\\tilde{U}}$\nare topologically conjugate via $\\phi_{*}$. Then it is straigtforward\nto check that the local Maslov index of $L_{\\Phi}$ at $0\\in U$ is\nthe same as that of $\\Phi_{\\Gamma^{2}}$ at $p$ (both equal to $k_{p})$.\nDue to Proposion \\ref{prop:local maslov index=00003D=00003Dglobal maslov index},\nthis index equals the ordinary Maslov index of the linearized flow\n$(e^{2m_{1}\\pi i\\cdot t},...,e^{2m_{n}\\pi i\\cdot t})$ on $(U,dq_{i}\\wedge dp_{i})$,\nwhich can be computed directly via the Maslov-Arnold map and is equal\nto $2\\cdot\\sum_{i=1}^{n}m_{i}$.\n\\end{proof}\nThe following corollary is a consequence of Theorem \\ref{thm:Symplectic=00003D=00003DHamiltonian}\nand Proposition \\ref{prop:Maslov data at the fixed points}.\n\\begin{cor}\nIf $[\\omega]=r\\cdot c_{\\Gamma}$ and $\\beta$ is invariant under $\\Phi_{\\Gamma_{J}^{2}}$,\n$h_{\\theta}=r\\cdot Q_{\\beta}$ is an Hamiltonian for $\\Phi$. Then\n$h_{\\theta}$ takes the same value at all its critical points of the\nsame resonance type. Moreover, the critical values of $h_{\\theta}$\nlie in the lattice $r\\cdot\\mathbb{Z}$.\n\\end{cor}\n\nWhile we can tell directly from Eq.(\\ref{eq:Maslov data=000026resonant type})\nthat the local Maslov index at a fixed point is an even number, it\ncan also be observed from the bundle structures of $\\Gamma$ and $\\Gamma^{2}$.\nAt a fixed point $p$, since $\\dot{\\gamma}_{w}(t)=\\mathcal{X}_{\\theta}=k_{p}\\frac{\\partial}{\\partial\\theta}$\nand $\\gamma_{w}(0)=\\gamma_{w}(1)=w$, the number $k_{p}$ is an integer\nand it counts how many rounds the orbit $\\gamma_{w}$ winds around\nthe fiber $\\pi_{\\Gamma^{2}}^{-1}(p)\\cong S^{1}$. Note that $\\Phi$\nis also lifted to the action $\\Phi_{\\Gamma}$ on $\\Gamma_{J}$ and\nit satisfies\n\\begin{equation}\nq_{\\pm}\\circ\\Phi_{\\Gamma}=\\Phi_{\\Gamma^{2}}\\circ q_{\\pm},\\label{eq:=00005CPhi-q=00003Dq-=00005CPhi}\n\\end{equation}\nwhere $q_{\\pm}:\\Gamma_{J}\\rightarrow\\Gamma_{J}^{2}$ is the map in\nthe commutative diagram \\ref{Diagram:oriented Maslov indices and Maslov indices}\n(in Subsection \\ref{subsec:Lagrangian-subbundles and Maslov-indices}).\nFrom Eq.(\\ref{eq:=00005CPhi-q=00003Dq-=00005CPhi}) we can tell that\nthe closed orbit(s) of $\\Phi_{\\Gamma}$ has (have) nontrivial winding\nalong the fiber $\\pi_{\\Gamma}^{-1}(p)$ if and only if the closed\norbit(s) of $\\Phi_{\\Gamma^{2}}$ has (have) nontrivial winding along\nthe fiber $\\pi_{\\Gamma^{2}}^{-1}(p)$, and that when $\\Phi_{\\Gamma}$\ngoes one round, $\\Phi_{\\Gamma^{2}}$ goes twice. Therefore, $k_{p}$\nis always an even number.\n\nIt is then straightforward to deduce the following corollary from\nthe results and discussion above, as well as the theorem of Delzant's\npolytopes (for references about Delzant's polytopes, see \\cite{mcduff2017introduction}\nor \\cite{da2008lectures}).\n\\begin{cor}\n(A Delzant picture) Suppose that $G=\\mathbb{T}^{l}$. Then there is\na smooth map $\\mathcal{Q}_{\\beta}=(Q_{\\beta}^{(1)},...,Q_{\\beta}^{(l)})$\nfrom $M$ to $\\mathbb{R}^{l}$ such that $\\mathcal{Q}$ maps the fixed\npoints of the $\\mathbb{T}^{l}$ action to the lattice $2\\cdot\\mathbb{Z}^{l}\\subset\\mathbb{R}^{l}$.\nIf $M$ is compact and $[\\omega]=r\\cdot c_{\\Gamma}$, then the image\n$\\text{Im}\\mathcal{Q}_{\\beta}$ is a polytope with the vertices taking\nthe form $(k_{p}^{1},...,k_{p}^{l})$ with $k_{p}^{j}$ being local\nMaslov indices at $p$.\n\\end{cor}\n\n\\begin{prop}\nSuppose that $c_{\\Gamma}=0$ and $\\Phi$ is a symplectic $S^{1}$\naction on $M$. Then all the local Maslov indices at the fixed points\nare equal.\n\\end{prop}\n\n\\begin{proof}\nSince $c_{\\Gamma}=0$, there is a connection $1$-form $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$\non $\\Gamma_{J}^{2}$ such that $df_{\\alpha}=0$. For any two fixed\npoints $p_{0},p_{1}$ of $\\Phi$, choose $w_{0}\\in\\pi_{\\Gamma^{2}}^{-1}(p_{0})$\nand $w_{1}\\in\\pi_{\\Gamma^{2}}^{-1}(p_{1})$. Let $\\lambda:[0,1]\\rightarrow\\Gamma_{J}^{2}$\nbe a smooth path from $w_{0}$ to $w_{1}$. Define a map $h$ from\n$S^{1}\\times[0,1]$ to $\\Gamma_{J}^{2}$ by\n\\[\nh(z,t)=\\Phi_{\\Gamma^{2}}^{z}\\circ\\lambda(t).\n\\]\nLet $\\tilde{f}_{\\alpha}$ be the pulled-back $1$-form of $f_{\\alpha}$\non $S^{1}\\times[0,1]$, i.e. $\\tilde{f}_{\\alpha}=h^{*}(f_{\\alpha})$.\nThen it holds that $d\\tilde{f}_{\\alpha}=h^{*}(df_{\\alpha})=0$. By\nStokes' formula,\n\\begin{equation}\n\\int_{S^{1}\\times\\{0\\}}\\tilde{f}_{\\alpha}=\\int_{S^{1}\\times\\{1\\}}\\tilde{f}_{\\alpha}.\\label{eq:boundary integration}\n\\end{equation}\nNote that for $i\\in\\{0,1\\}$, it holds $h_{*}(\\frac{\\partial}{\\partial\\theta}\\bigg|_{(z,i)})=\\mathcal{X}_{\\theta}\\big(\\Phi_{\\Gamma^{2}}^{z}(w_{i})\\big)$.\nAs a consequence, we have\n\\[\n\\int_{S^{1}\\times\\{i\\}}\\tilde{f}_{\\alpha}=\\int_{0}^{1}\\tilde{f}_{\\alpha}(\\frac{\\partial}{\\partial\\theta})\\bigg|_{(e^{2\\pi i\\cdot t},i)}dt=\\int_{0}^{1}f_{\\alpha}(\\mathcal{X}_{\\theta})\\bigg|_{\\gamma_{w_{i}}(t)}dt=k_{p_{i}}\n\\]\nwith $\\gamma_{w_{i}}(t)=\\Phi_{\\Gamma^{2}}^{e^{2\\pi i\\cdot t}}(w_{i})$.\nTogether with Eq.(\\ref{eq:boundary integration}) it concludes the\nproof.\n\\end{proof}\nThe following two remarks can be obtained by averaging the Liouville\n$1$-form of a cotangent bundle, but they also appear as consequences\nof the discussion above about the local Maslov indices. Note that\nwhen $M$ is a cotangent bundle, it holds that $c_{\\Gamma}=[\\omega]=0$,\nand the Maslov $S^{1}$ bundles are trivial.\n\\begin{rem}\nSuppose that $M$ is a connected cotangent bundle and $h_{\\theta}$\nis a Hamiltonian of the $S^{1}$ action. Since $k_{p}=k_{p'}$ for\nany fixed points, $h_{\\theta}$ has at most $1$ critical value.\n\\end{rem}\n\n\\begin{rem}\nConsider the case where $M$ is a cotangent bundle and $\\dim M=4$.\nNote that in a neighbourhood of a fixed point $p$, the $S^{1}$ action\ncan be linearized in a Daboux chart as $t\\mapsto(e^{2\\pi i\\cdot mt},e^{-2\\pi i\\cdot nt})$.\nWhen $k_{p}=0$, we have $m-n=0$, and hence the fixed points are\nall $1:-1$ resonances.\n\\end{rem}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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The in-plane directions are bulk-like\nand can be separated within the 3D Schr\\\"odinger equation. The variation over the different layers can then be described by a 1D model, most commonly by a square quantum well, where the eigenenergies increase quadratically and scale with $1\/d^2$, $d$ being the well width. In many cases, however, a deviation from this standard\ncase is observed.\nFor epitaxially grown SnO$_2$, InSb, and Pb films with low charge densities, measurements with exponents about $1.5$ have been reported~\\cite{SciRep.5.17424, PhysStatSolB.33.425, PhysRevB.66.233408}.\nFurthermore, Kr\\\"oger et al.~\\cite{PhysRevB.97.045403} and Hirahara et al.~\\cite{PhysRevB.75.035422} investigated Bi films \n\\cite{PhysRevLett.115.106803, PhysRevLett.93.046403, PhysRevLett.117.236402, JElectronSpec.201.98, NanoLett.12.1776}\nand demonstrated that these exhibit equidistant energy levels with a $1\/d$ thickness dependence\n\\cite{PhysRevB.97.045403, PhysRevB.75.035422}.\nFor this, Kr\\\"oger et al.\\ prepared epitaxially grown Bi(111) films on Si(111) samples.\nThe thicknesses varied between 20 and 100 bilayers (i.e.\\ $8\\,\\text{nm}$ to $40\\,\\text{nm}$).\nThe conductance $G$ was measured as a function of temperature $T$.\nA general dependence was fitted to this data and an effective bandgap $E_\\text{g}$ was extracted by describing the temperature dependence of the conductance by thermal excitation, $G\\propto\\text{exp}(E_\\text{g}\/k_\\text{B}T)$, where $k_\\text{B}$ is the Boltzmann constant.\nHirahara et al.\\ prepared Bi(001) films grown on Si(111) samples.\nThe thicknesses varied from 7 to 40 bilayers (i.e.\\ $2.8\\,\\text{nm}$ to $16\\,\\text{nm}$).\nThe energy levels were measured with angle-resolved photoemission spectroscopy (ARPES) and were found to be equidistant.\nThis motivates a description by the well-known harmonic oscillator potential in contrast to the usually found square well potential, which was also stated by Kr\\\"oger et al.\nAn explanation given by Hirahara et al.\\ is the phase shift accumulation model, originating from the Bohr-Sommerfeld quantization condition, which yields the $1\/d$-dependence if the dispersion near the Fermi energy is linear~\\cite{PhysRevB.75.035422}.\nNevertheless, the question must be asked whether equidistant energy levels actually necessitate a harmonic potential or whether alternative potential shapes with equidistant levels exist, especially considering that only a limited energy range is examined and a finite measurement uncertainty is present.\nThe general answer is ``no''~\\cite{CommMathPhys.82.471}, which was demonstrated by previous analytic derivations resulting in different potentials by generalizing the creation\/annihilation operator.\nFor this purpose, the factorization method \\cite{JMathPhys.25.3387, TheorChemAcc.110.403} or the more general shift-operator approach~\\cite{Chaos.4.47, SovPhysJETP.75.446} was used.\n\nFirst principle studies of any material are expected to result in a potential which oscillates with the atomic positions of the real structure which is put in.\nAs the harmonic oscillator is a valid model, an expectation could be that the potential of thin Bi films would on average and at a scale larger than the atomic structure be quadratic with additional spatial variations at smaller scale.\nWithin this manuscript we focus on the model point of view and calculate model potentials satisfying the previous expectation.\nWe hope that this publication will serve to interpret experimental results more carefully and clarify that even when the energy levels are known, a wide range of potentials must still be considered.\nIn the following, we assume equidistant energy levels and investigate what follows for the potential.\nIn \\sec{Experiments} the experimental results are modeled by a truncated harmonic oscillator, yielding good agreement of the energy levels with the analytical harmonic oscillator case up to a certain energy and providing a more physical description including the specific layer thickness.\n\\Sec{Shift_operator} treats the 1D  Schr\\\"odinger equation as an inverse problem by calculating the potentials from the given equidistant energy spectrum.\nThis is done using the shift-operator approach.\nWe present some previous analytical results and some new numerical calculations and possible interpretations concerning real structures.\nIn \\sec{Perturbation} we show by perturbation theory that polynomial potentials apart from the quadratic harmonic oscillator potential cannot exhibit an infinite number of \\textbf{exactly} equidistant energy levels.\n\n\\section{Comparison with experiments}\\label{sec:Experiments}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{graphics\/experiment.pdf}\n\t\\caption{Experimental data of Kr\\\"oger et al.~\\cite{PhysRevB.97.045403} (3 different Bi films) and Hirahara et al.~\\cite{PhysRevB.75.035422} in comparison with a fit to the harmonic oscillator.}\n\t\\label{fig:HO:experiment}\n\\end{figure}\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/oscillator_truncated.pdf}\n\t\\caption{Left: energy levels. Center: energy level spacings. Right: thickness-dependent energy level. Colored lines denote the 1D harmonic oscillator. Black dots denote the truncated 1D harmonic oscillator of 16\\,nm thickness.}\n\t\\label{fig:HO:truncated}\n\\end{figure}\n\nIn this section, we concentrate on the interpretation of experimental results for Bismuth films from the literature.\nThe experimental results of Kr\\\"oger et al.~\\cite{PhysRevB.97.045403} and Hirahara et al.~\\cite{PhysRevB.75.035422}, i.e.\\ the energy level spacings $\\Delta E$ as a function of the Bi film thickness $d$, are depicted in \\fig{HO:experiment} (colored symbols).\n$d_\\text{BL}$ is the thickness of a Bi bilayer, which form a stable unit in (111) direction.\nFor the Kr\\\"oger data, the measured bandgap $E_\\text{g}$ is interpreted as energy spacing $\\Delta E$.\nThe inverse thickness dependence can be clearly seen. The solid line shows a fit with a $d^{-1}$ thickness dependence.\n\nTo describe the experimental data and to connect the eigenenergies to a physical situation, the model of a truncated quadratic potential\n\\begin{flalign}\n\tV(x) &= V_0\\begin{cases}\n\t\t1 & |x|\\geq\\frac{d}{2}\\\\\n\t\t\\left(\\frac{2x}{d}\\right)^2 & |x|\\leq\\frac{d}{2}\n\t\\end{cases} &\n\\end{flalign}\nis used, where $d$ is the thickness of the material and the potential at the well edge $x=\\pm\\frac{d}{2}$ equals the well depth $V_0$.\nExemplary results are shown in \\fig{HO:truncated} for $V_0\/m^\\ast=8\\,\\text{eV}\/m_\\text{e}$ and $d=16\\,\\text{nm}$, where $m^\\ast$ is the effective mass and $m_\\text{e}$ is the electron mass.\nThe energy eigenvalues $E_n$ and the energy level spacings $\\Delta E$ of the truncated potential are in very good agreement with the non-truncated potential (i.e.\\ the harmonic oscillator) for $V<0.9V_0$, yielding\n\\begin{align}\n\tE_n = \\Delta E\\left(n+\\dfrac{1}{2}\\right) \\quad,\\quad \\Delta E = \\frac{2\\hbar}{d}\\sqrt{\\frac{2V_0}{m^\\ast}} \\label{eqn:fit}\n\\end{align}\nwith equidistant energy spacing $\\Delta E$, which scales like $1\/d$.\n\nA regression according to~\\eqn{fit} for reproducing the thickness dependence of the experimental data in \\fig{HO:experiment} yields $\\Delta E = 3.34\\,\\text{eV}\\cdot d_\\text{BL}\/d$ and $V_0\/m^\\ast=2.93\\,\\text{eV}\/m_\\text{e}$.\nAll data are in good agreement with the regression. We conclude that the experimental data may be well fitted with a truncated harmonic oscillator potential which yields the desired equidistant energy states. Nevertheless, the question remains whether this is the only possible shape for the potential. The next section thus deals with finding alternative potential shapes with the same energy spectrum.\n\n\\section{Anharmonic oscillators with equidistant energy levels}\\label{sec:Shift_operator}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{graphics\/L2.pdf}\n\t\\caption{Potential and lowest states of the second-order shift operator for $A=1$. The offset of the states are the corresponding eigenenergies.}\n\t\\label{fig:L2}\n\\end{figure}\n\nIn this section we show how the creation\/annihilation operators of the harmonic oscillator can be generalized with the aim of finding further potentials with equidistant spectra.\nAn approach to model equidistant energy levels is using the shift operator to treat the Schr\\\"odinger equation as an inverse problem, explained in the following.\nWe first give a general introduction and show previous analytical results by Dubov et al.~\\cite{Chaos.4.47, SovPhysJETP.75.446}.\nAfterwards we present new numerical calculations and possible interpretations concerning real structures.\n\n\\subsection{Method}\n\nThe general idea is to define for an arbitrary potential the shift operator $\\mathcal{L}$, which shifts the eigenstates $\\psi$ in energy:\n\\begin{align}\n\t\\mathcal{L}\\psi(\\xi,\\epsilon) &= \\psi(\\xi,\\epsilon+1) \\quad.\n\\end{align}\nHere, $\\xi=\\sqrt{m\\omega\/\\hbar}x$ and $\\epsilon=E\/\\hbar\\omega$ are the dimensionless real space and energy.\nIf the states are equidistant, $\\mathcal{L}$ transforms state $\\psi_n$ into state $\\psi_{n+1}$, resulting in the operator equation\n\\begin{align}\n\t[\\mathcal{H},\\mathcal{L}] &= \\mathcal{L} \\quad.\n\\end{align}\nIt can be solved by applying\n\\begin{align}\n\t\\mathcal{L} = \\sum_{k=0}^K \\alpha_k(\\xi)(\\text{i}\\mathcal{P})^k \\quad \\text{with}\\quad K\\geq 1 \\quad.\n\\end{align}\n$\\mathcal{P}$ is the momentum operator.\n$\\alpha_k(\\xi)$ are free parameters.\n\nThe first-order case ($K=1$) leads to the harmonic potential $U_0=\\frac{1}{2}\\xi^2$ and $\\mathcal{L}$ equals the creation operator.\nThe derivations can be found in the Supplementary Material.\nThe solution of the second-order shift operator gives the isotonic oscillator~\\cite{PhysLettA.70.177, IntJQuantumChem.110.1317, JPhysAMathGen.20.4331, JPhysAMathTheo.41.085301}\n\\begin{align}\n\tU_1 = \\frac{1}{8}\\xi^2 + \\frac{A}{\\xi^2} \\label{eqn:U_2}\n\\end{align}\nwith either $\\xi>0$ or $\\xi<0$, depicted in \\fig{L2} (black curve).\nIt can be interpreted as a half harmonic potential with a smooth wall, where the smoothness can be adjusted with $A$.\nFor the half harmonic oscillator (with a hard wall) only the odd solutions remain, resulting in a doubled energy level difference.\nThe potential in \\eqn{U_2} yields very similar eigenstates and the same eigenenergies as the half harmonic oscillator, but slightly shifted in energy.\nThe eigenstates have a similar form to the harmonic potential:\n\\begin{align}\n\t\\epsilon_n &= \\frac{1}{2} + n + \\frac{1}{4}\\sqrt{1+8A} \\quad,\\\\\n\t\\psi_n(\\xi) &= C_n \\xi^{(1+\\sqrt{1+8A})\/2} \\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\sum_{k=0}^n a_k \\xi^{2k} \\quad.\n\\end{align}\nThe eigenstates are also shown in \\fig{L2}.\nIt can be concluded that with this potential it is possible to model asymmetric minima in contrast to the pure harmonic potential.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_U.pdf}\n\t\\caption{Potentials of the third-order shift operator $U_0 \\ldots U_{21}$.}\n\t\\label{fig:L3}\n\\end{figure}\n\nThe third-order shift operator results in the differential equation\n\\begin{gather}\n\t\\frac{3}{2}\\left(W^2\\right)'' - \\frac{1}{4}W'''' + \\xi^2W'' + 3\\xi W' = 0 \\label{eqn:L3:U:DGL}\n\\end{gather}\nwith $U = W + \\frac{1}{2}\\xi^2$.\nThe prime denotes the first derivative.\n The equation can be solved as an initial value problem or a boundary value problem with any given conditions, resulting in many different solutions for the potential.\nPossible solutions obtained by a Darboux transformation~\\cite{Chaos.4.47, OsakaJMath.36.949} are\n\\begin{align}\n\tU_m &= -\\frac{1}{2}\\xi^2 - \\frac{2}{3}(2m+1) + \\left( \\frac{P_m'(\\xi)}{P_m(\\xi)} + \\xi \\right)^2 \\quad,\\\\\n\tP_{2m} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k)!}\\xi^{2k} \\quad,\\\\\n\tP_{2m+1} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k+1)!}\\xi^{2k+1} \\quad.\n\\end{align}\nThe index $m$ numbers the solutions.\nThey are drawn in \\fig{L3}.\nA further shift of the solutions in $\\xi$ and $U$ is valid, but not a superposition due to the non-linearity of the constituting equations.\n$U_{2m}$ are quadratic-like potentials with a dip at the minimum.\n$U_0$ is the harmonic potential.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=1$.\nThe ground state of potential $U_{2m}$ is additionally lowered by $2m$.\n$U_{2m+1}$ are singular potentials at ${\\xi=0}$.\n$U_1$ is one solution of the second-order shift operator.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=2$ ($\\Delta\\epsilon=1$ can be achieved by rescaling $\\xi=\\tilde{\\xi}\/2$).\nThe eigenstates of these anharmonic potentials are also similar to the ones of the harmonic potential.\nEspecially the potentials $U_1$ and $U_{2n+1}$ allow to construct asymmetric 1D models which describe equidistant energy levels, providing a much more flexible tool than a restriction to the analytical harmonic oscillator offers.\n\nFurther examples and their derivations can be found in the Supplementary Material.\n\n\\subsection{Numerical results}\n\n\\Eqn{L3:U:DGL} or its first integrals\n\\begin{gather}\n\t\\frac{3}{2}\\xi\\left(W^2\\right)' - \\frac{3}{2}W^2 - \\frac{1}{4}\\xi W''' + \\frac{1}{4}W'' + \\xi^3W' = A \\quad,\\label{eqn:L3:U:DGL:2}\\\\\n\t-\\frac{1}{2}\\left( \\frac{A + \\frac{3}{2}W^2 - \\frac{1}{4}W''}{\\xi} \\right)^2 - \\frac{1}{2}W^3 + \\frac{1}{8}\\left(W'\\right)^2 = AW + B\\label{eqn:L3:U:DGL:3}\n\\end{gather}\ncan also be solved numerically to achieve further types of potentials.\nWe calculated the potential using the explicit Runge--Kutta method of 4th order or higher as implemented in the software package Mathematica~\\cite{Mathematica.12.1}.\nWe quantify the numerical error of the solution using the local relative residuum, i.e. the residuum normalized to the maximum absolute additive contribution to the differential equation, evaluated as a function of $\\xi$.\nWe get values below $10^{-4}$, which is sufficiently small to consider the solution correct.\nThe Schr\\\"odinger equation is solved by discretizing the differential operator and the potential on an equidistant grid ($\\Delta x<0.01\\,\\text{nm}$) and solving the resulting matrix eigenvalue problem as implemented in Mathematica.\nFor this, the calculated spatial dimension is much larger than presented in the following, i.e. large enough for the potential at the boundary to be higher than the highest considered energy level, to prevent numerical errors from limiting the spatial dimension.\n\nThree different potentials and corresponding solutions of the Schr\\\"odinger equation are depicted in Fig.~\\ref{fig:L3:num_1}, Fig.~\\ref{fig:L3:num_2}, and Fig.~\\ref{fig:L3:num_3}.\nThey represent three different types of solutions, named type-1, type-2, and type-3 potentials in the following.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_numerisch1.pdf}\n\t\\caption{Numerical solution of \\eqn{L3:U:DGL:2} with $A=-0.4$ and $W(1)=W'(1)=W''(1)=0$. Left: type-1 potential and lowest states. The offset of the states are the corresponding eigenenergies. Right: energy levels. The inset shows the energy level spacings.}\n\t\\label{fig:L3:num_1}\n\\end{figure}\n\n\\begin{figure}[!b]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_numerisch2.pdf}\n\t\\caption{Numerical solution of \\eqn{L3:U:DGL:2} with $A=-0.001$ and $W(1)=W'(1)=W''(1)=0$. Left: type-2 potential and selected states (quantum numbers 1, 11, 22, 32, 42, 52, 63, 73, 84, 95). The offset of the states are the corresponding eigenenergies. Two different classes of states are denoted by color (class 1: red, class 2: blue). Right: energy levels. Arrows mark the states shown in (a). The insets show the energy level spacings.}\n\t\\label{fig:L3:num_2}\n\\end{figure}\n\n\\fig{L3:num_1} shows a potential with two singularities and a minimum in between.\nThe singularities behave like $1\/x^2$, similar to the $U_{2n+1}$ potential.\nThe wave functions are also of similar shape as the ones of the former potentials.\nThe energy levels are roughly equidistant in the depicted range with $\\Delta E\\approx 1040\\,\\text{meV}$.\nThe higher states increase quadratically in energy instead of linearly.\nThis could be due to the finite discretization and the resulting bad energetic resolution or due to a badly converged potential near the singularities.\nHowever, the $1\/x^2$ dependence near the singularities acts like a square box, resulting in quadratically increasing energy levels.\n\nThe second type of potentials, see example in \\fig{L3:num_2}, also has a $1\/x^2$ singularity for positive values of $x$, similar to $U_{2n+1}$.\nYet for $x<0$, the potential oscillates and the strength of these oscillations increases with rising absolute value of $x$.\nThe states of this potential can be separated into two classes:\n(1) States localized in the oscillation minima (red states in \\fig{L3:num_2}).\n(2) Delocalized states oscillating between the smooth wall and the potential oscillation peaks of the corresponding energy range (blue states in \\fig{L3:num_2}).\nThe depicted example exhibits nearly linearly increasing energy levels.\nAdditional small irregular variations are due to the 2 classes of states.\nThe insets in \\fig{L3:num_2} show the energy level spacings of these distinct classes.\nThe class-1 states (upper inset) show linearly increasing energy levels with $\\Delta E\\approx 358\\,\\text{meV}$.\nThe class-2 states (lower inset) vary between $\\Delta E=300\\,\\text{meV}$ and $\\Delta E=450\\,\\text{meV}$.\nThe superposition of these two classes of states with different energy level spacings leads to the variations in the combined picture and although the class-2 states are not truly equidistant, the deviations from the overall linearity of all states are small.\nThe overall average energy level spacing is $\\Delta E\\approx 180\\,\\text{meV}$.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_numerisch3.pdf}\n\t\\caption{Numerical solution of \\eqn{L3:U:DGL:3} with $A=0$, $B=-1$, and $W(1)=W'(1)=0$. Left: type-3 potential and selected states (quantum numbers 5, 13, 24, 33, 44, 53, 62, 72, 82). The offset of the states are the corresponding eigenenergies. Two different classes of states are denoted by color (class 1: red and green, class 2: blue). Right: energy levels. Arrows mark the states shown in (a). The insets show the energy level spacings (red data points are behind green data points).}\n\t\\label{fig:L3:num_3}\n\\end{figure}\n\n\\Fig{L3:num_3} depicts the third type of potentials with no singularities, but oscillations in both directions and also displaying two different\nclasses of states.\nClass 1 are states mostly localized within the potential oscillations at the left or right (red and green in \\fig{L3:num_3}).\nThere are also states similar to the depicted ones, but located at the other side.\nClass 2 are delocalized states oscillating between the left and right  potential oscillation peaks of corresponding energy (blue in \\fig{L3:num_3}).\nEnergy levels increase almost linearly; the small periodic variations are due to the two classes of states.\nThe insets in \\fig{L3:num_3} show the energy level spacings of these distinct classes.\nIt can be clearly seen that for each class linearly increasing energy levels are present.\nClass 1 (upper inset in \\fig{L3:num_3}) has $\\Delta E=358\\,\\text{meV}$ for states located at the left (green) as well as for states located at the right (red).\nThe states are just shifted by a small energy due to the slightly asymmetry of the potential.\nClass 2 (lower inset) has $\\Delta E\\approx 267\\,\\text{meV}$.\nThe class-2 energy level spacings vary slightly (about 10\\%).\nBut as this is a differential value, deviations from linearity are sufficiently small and in good agreement with experimental results.\nThe superposition of these two classes of states with different energy level spacings leads to the irregular variations in the combined picture.\nIn total, the energy levels show a linear dependence with an overall average energy level spacing of $\\Delta E\\approx 108\\,\\text{meV}$.\nThe additional irregular variation on the large scale are small.\nWith regard to experimental results with some error bars these irregularities may not be visible.\n\nThe presented 1D potentials allow more interpretations concerning the comparison with real structures than a pure 1D harmonic oscillator model.\nThey can be related to different possible cases occurring in reality.\nFor example, a thin, free-standing slab, surrounded by vacuum, has states which are trapped within the slab in the direction perpendicular to the slab and exponentially decaying into vacuum.\nThis could be modeled by the type-1 potential of the numerical calculations, which diverges at the slab-vacuum interface.\nA thin layer on a substrate could be modeled by the type-2 potential.\nThis diverges at the layer-vacuum interface and oscillates into the bulk-substrate, representing the periodic structure.\nA thin sandwich-layer between two other substrates could be modeled by the type-3 potential, which oscillates into both directions to describe the penetration into the periodic bulk material.\nAnother interpretation could be to use the type-2 or type-3 potential to describe the oscillating atomic structure within the layer itself.\nThe experimental results from literature~\\cite{PhysRevB.97.045403, PhysRevB.75.035422, PhysRevLett.115.106803} have the structure silicon--bismuth--vacuum and fall into these categories.\nThe $\\Delta E=358\\,\\text{meV}$ achieved in \\fig{L3:num_2} is in the range of possible experimental $\\Delta E$ of \\fig{HO:experiment}.\nUsing the result of the fit in \\eqn{fit}, this corresponds to a thickness of $9.3$ bismuth bilayers, i.e. $3.7\\,\\text{nm}$.\nWithin the potential in \\fig{L3:num_2} this approximately matches the width of the non-oscillating part at the right near the singularity.\nThe oscillating part represents the transition into the bulk-like part with wave lengths between $0.2\\,\\text{nm}$ (for $-10\\,\\text{nm}<x<-8\\,\\text{nm}$) and $0.6\\,\\text{nm}$ (for $-4\\,\\text{nm}<x<-2\\,\\text{nm}$).\nThis nicely matches the periodicity of possible atomic structures (the bismuth bilayer thickness is approximately $0.4\\,\\text{nm}$, the silicon lattice constant is $0.54\\,\\text{nm}$).\n\nIn this section spectra have been studied which are equidistant in a limited energy range or contain a subset of equidistant states.\nIn the next section we show what can be learned from perturbation theory.\nThe difference to the former shift-operator approach is that, in the following, searching for equidistant eigenstates means an infinite number of exactly equidistant states.\nThe formerly shown results are less strict and do not belong to this for the following reasons:\nPotentials $U_{2m+1}$ and the numerical type-1 potential have non-polynomial shapes and quadratic non-integrable singularities.\nPotentials $U_{2m}$ consider parts of the spectrum being non-equidistant (i.e. the ground state).\nThe numerical type-2 and type-3 potentials lead to states which split into two classes with separate energy spacing each.\n\n\\section{Perturbed 1D harmonic oscillator}\\label{sec:Perturbation}\n\nIn this section, we use perturbation theory to show that an \\textbf{exactly} equidistant energy spectrum \\textbf{over the whole energy range} cannot be achieved by a polynomial potential other than the harmonic (quadratic) potential.\nTo this end, the 1D harmonic oscillator is perturbed with an arbitrary polynomial perturbation potential ${u(\\xi)=\\sum_{j=0}^Nu_j\\xi^j}$,\n\\begin{align}\n\t\\left[ -\\dfrac{\\partial^2}{\\partial\\xi^2} + \\xi^2 + \\lambda u(\\xi) \\right] \\psi_n(\\xi) &= \\epsilon_n^{(0)}\\psi_n(\\xi).\n\\end{align}\nAs the equations are linear, we consider each monomial $\\xi^j$ individually.\nThe perturbation potential ${u(\\xi)=\\xi^j}$ in the basis of the harmonic oscillator eigenstates gives the perturbation coefficients\n\\begin{align}\n\tu_{mn}^{(j)} &= \\frac{1}{\\sqrt{\\pi 2^{m+n}m!n!}} \\int\\limits_{-\\infty}^\\infty \\xi^j\\text{e}^{-\\xi^2}H_m(\\xi)H_n(\\xi) \\,\\text{d}\\xi \\quad,\n\\end{align}\nwhere $H_n(\\xi)$ are the Hermite polynomials.\n\nThe diagonal elements $u_{kk}^{(j)}$ equal the first-order energy correction $\\epsilon_k^{(1)}$.\nThey are equal to zero for odd values of $j$.\nFor even numbers $j$, $u_{kk}^{(j)}$ is a polynomial in $k$ of order $j\/2$ as shown in \\fig{u_kk_j} for different monomial perturbations ${u(\\xi)=\\xi^j}$.\nFor the smallest values of $j$ one gets\n\\begin{align}\n\tu_{kk}^{(0)} = 1 \\quad,\\quad\n\tu_{kk}^{(2)} = \\frac{1}{2} + k \\quad,\\quad\n\tu_{kk}^{(4)} = \\frac{3}{4} + \\frac{3}{2}k + \\frac{3}{2}k^2 \\quad.\n\\end{align}\nConsequently, if the harmonic oscillator is perturbed with a quadratic potential, $\\epsilon_k^{(1)}$ is linear in $k$, reproducing the equidistant energy levels of the harmonic oscillator.\nIf the harmonic oscillator is perturbed with a non-quadratic polynomial potential, $\\epsilon_k^{(1)}$ is a superposition of different powers of $k$, yielding a different spectrum.\nThus, only the quadratic polynomial gives equidistant first-order energy corrections.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{graphics\/u_kk.pdf}\n\t\\caption{$u_{kk}^{(j)}$ for different perturbation potentials $u(\\xi)=\\xi^j$.}\n\t\\label{fig:u_kk_j}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/u_kk-d.pdf}\n\t\\caption{$u_{k,k-l}^{(j)}$ for different $l$ and for different perturbation potentials $u(\\xi)=\\xi^j$.}\n\t\\label{fig:u_mn_j}\n\\end{figure}\n\nThe non-diagonal elements $u_{mn}^{(j)}$ are needed to calculate the higher order energy corrections $\\epsilon_k^{(\\geq 2)}$.\nFor $p$th order a $(p-1)$-fold sum over a $p$-fold product of $u_{mn}^{(j)}$ with different indices has to be calculated.\nIf $m+n+j$ is odd or if $|m-n|>j$, then ${u_{mn}^{(j)}=0}$.\nThis leads to finite summations over no more than $j+1$ elements $u_{k,k+l}^{(j)}$ with $-j\\leq l\\leq j$.\nThe $u_{k,k+l}^{(j)}$ are shown in \\fig{u_mn_j} for different $l$ and for different polynomials ${u(\\xi)=\\xi^j}$.\nThe non-zero elements for a specific $l$ can be written as\n\\begin{align}\n\tu_{k,k-l}^{(j)} = \\sqrt{\\frac{k!}{2^l(k-l)!}}p_{kl}^{(j)}\n\\end{align}\nwith a polynomial $p_{kl}^{(j)}$ in $k$ of order $(j-l)\/2$.\nThus, $(u_{k,k-l}^{(j)})^2$ is a polynomial in $k$ of order $j$ for all $l$.\nThe previous diagonal case $l=0$ is included.\nIn particular, for the smallest values of $j$ one gets\n\\begin{flalign}\n\tp_{k,1}^{(1)} &= 1 \\quad,\n\t& p_{k,1}^{(3)} &= \\frac{3}{2}k \\quad,\n\t& p_{k,1}^{(5)} &= \\frac{5}{4}(1+2k^2) \\quad,\\\\\n\tp_{k,2}^{(2)} &= 1 \\quad,\n\t& p_{k,2}^{(4)} &= -1+2k \\quad,\n\t& p_{k,2}^{(6)} &= \\frac{15}{4}(1-k+k^2) \\quad.&&\\hspace{15em}\n\\end{flalign}\n\nWith this, the second order energy correction\n\\begin{align}\n\t\\epsilon_k^{(2)} &= -\\sum_{n\\neq k}\\frac{u_{nk}u_{kn}}{\\epsilon_n^{(0)}-\\epsilon_k^{(0)}} = -\\frac{1}{2} \\sum_{\\substack{1\\leq l\\leq j\\\\l-j~\\text{even}}}\\frac{(u_{k,k+l}^{(j)})^2-(u_{k,k-l}^{(j)})^2}{l} = \\mathcal{O}(k^{j-1})\n\\end{align}\nis a polynomial in $k$ of order $j-1$, as the highest order in $k$ cancels out with the difference $(u_{k,k+l}^{(j)})^2-(u_{k,k-l}^{(j)})^2$.\nConsequently, the second order energy corrections scale linearly with $k$ only for the quadratic potential ($j=2$).\nTreating the higher-order perturbations similarly, in the $k^{3j\/2}$-scaling two hierarchic differences lead to a cancellation of the two highest orders of $k$.\nWith this, $\\epsilon_k^{(3)}$ is a polynomial in $k$ of order $3j\/2-2$ and also linear scaling only for the quadratic potential.\n\nAnother argumentation can be done using the $p$th order perturbation corrections, written as a recursion:\n\\begin{align}\n\t\\epsilon_k^{(p)} &= \\sum_nu_{kn}c_{kn}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{kk}^{(p-q)} \\\\\n\tc_{km}^{(p)} &= \\frac{1}{\\epsilon_k^{(0)}-\\epsilon_m^{(0)}}\\left( \\sum_nu_{mn}c_{kn}^{(p-1)} - u_{kk}c_{km}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{km}^{(p-q)} \\right) \\quad.\n\\end{align}\nThe general argumentation is as follows:\nFor the harmonic oscillator, $c_{km}^{(0)}$ is independent of $k$ and $\\epsilon_k^{(0)}$ is linear in $k$.\nIf $c_{km}^{(p-1)}$ is independent of $k$ and $\\epsilon_k^{(p-1)}$ is linear in $k$, which is fulfilled for $p=1$, then $c_{km}^{(p)}$ is also independent of $k$ and $\\epsilon_k^{(p)}$ is also linear in $k$.\nFor the case $j\\neq 2$ yields: if $c_{km}^{(p-1)}$ scales like $k^x$ and $\\epsilon_k^{(p-1)}$ scales like $k^y$, then $c_{km}^{(p)}$ scales like $k^{x+j\/2-1}+k^{x+y-1}$ and $\\epsilon_k^{(p)}$ scales like $k^{x+j\/2}+k^{x+y}$.\nConsequently, a linear dependence of $\\epsilon_k$ on $k$ can only be achieved if $y=1$, thus $x=0$, giving $j=2$, which is the harmonic oscillator.\nIn total, for a general perturbation potential ${u(\\xi)=\\sum_{j=0}^Nu_j\\xi^j}$, the perturbed energy levels are a sum of linear functions of $k$ for the quadratic case and a sum of functions of $k$ with different exponents for the non-quadratic cases, which does not sum up to a linear dependence.\n\nThough it may seem so, \\Sec{Shift_operator} and \\Sec{Perturbation} do not contradict each other. \nThe previous argumentation holds only for a polynomial potential which produces an equidistant spectrum for all eigenstates and with no tolerance in the level spacing.\nIt does not apply to power series and spectra which are only equidistant in a specific energy range or approximately equidistant spectra, as required to model experimental spectra.\n\nThe derivations of this section and some more results can be found in the Supplementary Material.\n\n\\section{Summary and conclusions}\n\nIn this publication, we discuss the implications that may be drawn from the experimental observation of an equidistant energy level spacing. \nThough a harmonic potential comes to mind immediately, this shape of potential is sufficient, but not necessary for the equidistant spacing, especially considering that experimental\nresults cover a finite energy range with a finite measurement uncertainty. The question is thus more involved than is visible at first sight.\nIn this publication, we discuss different aspects of the problem.\n \nIn the first part we compare a 1D harmonic oscillator model with experimental results, looking at the analytical harmonic potential as well as on\na truncated version. \nThe energy levels measured in thin Bi films of different thicknesses and the resulting thickness-dependence fit the theoretical model with comparable energy values and dimensions.\nThus, the 1D harmonic oscillator is a valid model to describe the quantum well states in thin films as shown for Bi films.\n\nHowever, this still leaves the question whether it is the only possible potential to explain the measurements.\nTherefore, in the second part of the paper, we discuss anharmonic potentials of non-polynomial shape and apply the shift-operator \napproach to explicitly calculate potentials with an equidistant spectrum.\nWe present a number of analytic and numeric examples representing different types of potentials including asymmetric minima, singularities, and oscillations.\nThese potentials seem well-suited 1D models to describe the potential of thin slabs, thin films on substrates or thin sandwich films between two substrates.\nThey also fit the expectation that first principle calculations of real materials should result in oscillating potentials which are in average quadratic to fit the harmonic oscillator model.\n\nIn the last part, we apply $n$th-order perturbation theory with a polynomial perturbation potential to the quantum-mechanical harmonic oscillator problem.\nWe calculate and discuss the perturbation integrals $u_{k,k-l}$ and the resulting $n$th order energy corrections as a function of $k$ for single monomials $\\xi^j$,\nshowing that $u_{k,k-l}^2$ is a polynomial in $k$ of order $j$ and arguing that every non-quadratic perturbation monomial leads to a non-quadratic $k$-dependence of the $n$th-order energy corrections, leaving only the harmonic term to produce an equidistant spectrum.\n\nThus we conclude that all possible anharmonic potentials producing exactly equidistant spectra must have a non-polynomial form.\nAnharmonic non-polynomial potentials exist, which could describe real structures and produce spectra which mirror the experimental results.\n\n\\section*{Acknowledgment}\n\nWe thank C. Tegenkamp and H.-R. Berger for helpful discussions.\n\n\n\\section{Shift-operator approach}\n\n\\subsection{Definition and basic operator equation}\n\n\\noindent The Schr\\\"odinger equation in dimensionless coordinates\n\\begin{align}\n\\begin{aligned}\n\t\\xi &= \\sqrt{\\frac{m\\omega}{\\hbar}}x \\quad,& \\epsilon_n &= \\frac{E_n}{\\hbar\\omega} \\quad,& \\psi(\\xi) &= \\sqrt[4]{\\frac{\\hbar}{m\\omega}}\\Psi(x)\\quad,& U(\\xi) &= \\frac{V(x)}{\\hbar\\omega}\n\\end{aligned}\n\\end{align}\nis\n\\begin{align}\n\t\\left[ -\\dfrac{1}{2}\\dfrac{\\partial^2}{\\partial\\xi^2} + U(\\xi) \\right] \\psi_n(\\xi) = \\epsilon_n\\psi_n(\\xi) \\quad .\n\\end{align}\nThe shift operator\n\\begin{align}\n\t\\mathcal{L}\\psi(\\xi,\\epsilon) = \\psi(\\xi,\\epsilon+1) \\quad,\\quad \\mathcal{L}^\\dagger\\psi(\\xi,\\epsilon) = \\psi(\\xi,\\epsilon-1)\n\\end{align}\nshifts each state in energy by $\\hbar\\omega$.\nApplying the Hamiltonian operator gives\n\\begin{align}\n\\begin{aligned}\n\t\\mathcal{H}\\mathcal{L}\\psi(\\xi,\\epsilon) &= (\\epsilon+1)\\psi(\\xi,\\epsilon+1) \\quad,& \\mathcal{H}\\mathcal{L}^\\dagger\\psi(\\xi,\\epsilon) &= (\\epsilon-1)\\psi(\\xi,\\epsilon-1) \\quad,\\\\\n\t\\mathcal{L}\\mathcal{H}\\psi(\\xi,\\epsilon) &= \\epsilon\\psi(\\xi,\\epsilon+1) \\quad,& \\mathcal{L}^\\dagger\\mathcal{H}\\psi(\\xi,\\epsilon) &= \\epsilon\\psi(\\xi,\\epsilon-1) \\quad.\n\\end{aligned}\n\\end{align}\nFor an equidistant spectrum $\\epsilon_{n+1}=\\epsilon_n+1$ this gets\n\\begin{align}\n\\begin{aligned}\n\t\\mathcal{H}\\mathcal{L}\\psi_n(\\xi) &= \\epsilon_{n+1}\\psi_{n+1}(\\xi) \\quad,& \\mathcal{H}\\mathcal{L}^\\dagger\\psi_n(\\xi) &= \\epsilon_{n-1}\\psi_{n-1}(\\xi) \\quad,\\\\\n\t\\mathcal{L}\\mathcal{H}\\psi_n(\\xi) &= \\epsilon_n\\psi_{n+1}(\\xi) \\quad,& \\mathcal{L}^\\dagger\\mathcal{H}\\psi_n(\\xi) &= \\epsilon_n\\psi_{n-1}(\\xi) \\quad.\n\\end{aligned}\n\\end{align}\nWith this, a resulting operator equation can be set up using commutators:\n\\begin{align}\n\t[\\mathcal{H},\\mathcal{L}] = \\mathcal{L} \\quad,\\quad [\\mathcal{H},\\mathcal{L}^\\dagger] &= -\\mathcal{L}^\\dagger \\quad.\\label{eqn:shift_operator_equation}\n\\end{align}\nIt follows $[\\mathcal{H},\\mathcal{L}\\mathcal{L}^\\dagger] = [\\mathcal{H},\\mathcal{L}^\\dagger\\mathcal{L}] = 0$.\nConsequently, $\\mathcal{L}\\mathcal{L}^\\dagger$, $\\mathcal{L}^\\dagger\\mathcal{L}$, and $[\\mathcal{L},\\mathcal{L}^\\dagger]$ are functions of $\\mathcal{H}$. Thus, $\\mathcal{L}$ can be written as a polynomial of the dimensionless momentum operator $\\mathcal{P}=-\\text{i}\\frac{\\partial}{\\partial\\xi}$:\n\\begin{align}\n\t\\mathcal{L} = \\sum_{k=0}^K \\alpha_k(\\xi)(\\text{i}\\mathcal{P})^k \\quad.\\label{eqn:L:general}\n\\end{align}\n$K$ is the order of the shift operator.\n$\\alpha_k(\\xi)$ are free parameters.\nInserting \\eqn{L:general} into \\eqn{shift_operator_equation} and comparing the coefficients yields $K+2$ differential equations for $K+1$ unknown functions $\\alpha_k(\\xi)$ and the additional unknown function $U(\\xi)$.\n\n\\subsection{First-order shift operator}\n\n\\noindent Inserting the first-order shift operator\n\\begin{align}\n\t\\mathcal{L} = \\alpha_0(\\xi) + \\alpha_1(\\xi)\\text{i}\\mathcal{P}\n\\end{align}\ninto \\eqn{shift_operator_equation} yields\n\\begin{alignat}{4}\n\\begin{aligned}\n\t0 &= \\alpha_0 + \\frac{1}{2}\\alpha_0'' + U'\\alpha_1 \\quad,\\\\\n\t0 &= \\alpha_0' + \\alpha_1 + \\frac{1}{2}\\alpha_1'' \\quad,\\\\\n\t0 &= \\alpha_1' \\quad.\n\\end{aligned}\n\\end{alignat}\nThe solution for the $\\alpha_k$ yields the condition\n\\begin{align}\n\tU' = \\xi \\quad.\n\\end{align}\nIntegrating this yields the harmonic oscillator potential\n\\begin{align}\n\tU_0 = \\frac{1}{2}\\xi^2 \\quad.\n\\end{align}\nThe corresponding shift operator\n\\begin{align}\n\t\\mathcal{L} = -\\xi + \\text{i}\\mathcal{P} \\quad,\\quad \\mathcal{L}^\\dagger = \\xi + \\text{i}\\mathcal{P}\n\\end{align}\nis the creation resp.~the annihilation operator.\nThe corresponding solution of the Schr\\\"odinger equation is\n\\begin{align}\n\t\\epsilon_n &= \\frac{1}{2} + n \\quad,\\quad \\psi_n(\\xi) = \\frac{1}{\\sqrt{2^nn!\\sqrt{\\pi}}} H_n(\\xi)\\exp\\left(-\\frac{1}{2}\\xi^2\\right) \\quad,\n\\end{align}\nwhere $H_n(\\xi)$ are the Hermite polynomials.\nThe potential and the states of the first-order shift operator are depicted in \\fig{L1}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{supplement_graphics\/L1.pdf}\n\t\\caption{Potential and lowest states of the first-order shift operator.}\n\t\\label{fig:L1}\n\\end{figure}\n\n\\subsection{Second-order shift operator}\n\n\\noindent Inserting the second-order shift operator\n\\begin{align}\n\t\\mathcal{L} = \\alpha_0(\\xi) + \\alpha_1(\\xi)\\text{i}\\mathcal{P} + \\alpha_2(\\xi)(\\text{i}\\mathcal{P})^2\n\\end{align}\ninto \\eqn{shift_operator_equation} yields\n\\begin{alignat}{5}\n\\begin{aligned}\n\t0 &= \\alpha_0 + \\frac{1}{2}\\alpha_0'' + U'\\alpha_1 + U''\\alpha_2 = 0 \\quad,\\\\\n\t0 &= \\alpha_0' + \\alpha_1 + \\frac{1}{2}\\alpha_1'' + 2U'\\alpha_2 \\quad,\\\\\n\t0 &= \\alpha_1' + \\alpha_2 + \\frac{1}{2}\\alpha_2'' \\quad,\\\\\n\t0 &= \\alpha_2' \\quad.\n\\end{aligned}\n\\end{alignat}\nThe solution for the $\\alpha_k$ yields the condition\n\\begin{align}\n\t\\xi U' + 2U = \\frac{1}{2}\\xi^2 \\quad.\n\\end{align}\nMultiplying with $\\xi$ and summarizing using the product rule yields\n\\begin{align}\n\t(\\xi^2U)' = \\frac{1}{2}\\xi^3 \\quad.\n\\end{align}\nIntegrating and dividing by $\\xi^2$ yields\n\\begin{align}\n\tU_1 = \\frac{1}{8}\\xi^2 + \\frac{A}{\\xi^2} \\quad.\n\\end{align}\nThe shift operator is\n\\begin{align}\n\\begin{aligned}\n\t\\mathcal{L} &= \\frac{1}{4}\\xi^2 - \\frac{2A}{\\xi^2} - \\frac{1}{2} - \\xi\\text{i}\\mathcal{P} - (\\text{i}\\mathcal{P})^2 \\quad,\\quad \\mathcal{L}^\\dagger = \\frac{1}{4}\\xi^2 - \\frac{2A}{\\xi^2} + \\frac{1}{2} + \\xi\\text{i}\\mathcal{P} - (\\text{i}\\mathcal{P})^2 \\quad.\n\\end{aligned}\n\\end{align}\nThe solution of the Schr\\\"odinger equation can be achieved with the asymptotic behaviour\n\\begin{align}\n\t\\xi &\\rightarrow\\infty: & \\psi_\\infty''-\\frac{1}{2}\\xi^2\\psi_\\infty &= 0 \\quad, & \\psi_\\infty &\\sim \\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\quad,\\hspace{20em}\\\\\n\t\\xi &\\rightarrow 0: & \\psi_0''-\\frac{4A}{\\xi^2}\\psi_0 &= 0 \\quad, & \\psi_0 &\\sim \\xi^{(1+\\sqrt{1+8A})\/2}\n\\end{align}\nand the consequential approach\n\\begin{align}\n\t\\psi_n(\\xi) &= \\xi^{(1+\\sqrt{1+8A})\/2}\\sum_{k=0}^na_k\\xi^k\\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\quad.\n\\end{align}\nIt follows\n\\begin{align}\n\t\\epsilon_n &= \\frac{1}{2} + n + \\frac{1}{4}\\sqrt{1+8A} \\quad,\\\\\n\t\\psi_n(\\xi) &= C_n J_n(\\xi) \\left(\\frac{\\xi^2}{2}\\right)^{(1+\\sqrt{1+8A})\/4} \\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\quad,\\\\\n\tJ_n(\\xi) &= \\sum_{k=0}^n (-1)^k \\begin{pmatrix}n\\\\k\\end{pmatrix} \\frac{\\Gamma\\left(\\frac{\\sqrt{1+8A}}{2}+1\\right)}{\\Gamma\\left(\\frac{\\sqrt{1+8A}}{2}+1+k\\right)} \\left(\\frac{\\xi^2}{2}\\right)^k \\quad,\\\\\n\tC_n &= \\sqrt{\\frac{\\sqrt{2}\\Gamma\\left(\\frac{\\sqrt{1+8A}}{2}+1+n\\right)}{n!\\Gamma^2\\left(\\frac{\\sqrt{1+8A}}{2}+1\\right)}} \\quad.\n\\end{align}\nThe potential and the states of the second-order shift operator are depicted in \\fig{L2}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{supplement_graphics\/L2.pdf}\n\t\\caption{Potential and lowest states of the second-order shift operator.}\n\t\\label{fig:L2}\n\\end{figure}\n\n\\subsection{Third-order shift operator}\n\n\\noindent Inserting the thirt-order shift operator\n\\begin{align}\n\t\\mathcal{L} = \\alpha_0(\\xi) + \\alpha_1(\\xi)\\text{i}\\mathcal{P} + \\alpha_2(\\xi)(\\text{i}\\mathcal{P})^2 + \\alpha_3(\\xi)(\\text{i}\\mathcal{P})^3\n\\end{align}\ninto \\eqn{shift_operator_equation} yields\n\\begin{alignat}{6}\n\t0 &= \\alpha_0 + \\frac{1}{2}\\alpha_0'' + U'\\alpha_1 + U''\\alpha_2 + U'''\\alpha_3 \\quad,\\\\\n\t0 &= \\alpha_0' + \\alpha_1 + \\frac{1}{2}\\alpha_1'' + 2U'\\alpha_2 + 3U''\\alpha_3 \\quad,\\\\\n\t0 &= \\alpha_1' + \\alpha_2 + \\frac{1}{2}\\alpha_2'' + 3U'\\alpha_3 \\quad,\\\\\n\t0 &= \\alpha_2' + \\alpha_3 + \\frac{1}{2}\\alpha_3'' \\quad,\\\\\n\t0 &= \\alpha_3' \\quad.\n\\end{alignat}\nThe solution for the $\\alpha_k$ yields the shift operator\n\\begin{align}\n\t\\mathcal{L} &= \\left( \\frac{3}{2}\\left(U^2\\right)' - \\frac{1}{4}U''' - \\frac{\\xi^2+3}{2}U' + \\frac{1}{2}\\xi \\right) + \\left( \\frac{1}{2}\\xi^2 - 3U - 1 \\right) \\text{i}\\mathcal{P} - \\xi(\\text{i}\\mathcal{P})^2 + (\\text{i}\\mathcal{P})^3 \\quad,\\\\\n\t\\mathcal{L}^\\dagger &= -\\left( \\frac{3}{2}\\left(U^2\\right)' - \\frac{1}{4}U''' - \\frac{\\xi^2+3}{2}U' - \\frac{1}{2}\\xi \\right) + \\left( \\frac{1}{2}\\xi^2 - 3U + 1 \\right) \\text{i}\\mathcal{P} + \\xi(\\text{i}\\mathcal{P})^2 + (\\text{i}\\mathcal{P})^3\n\\end{align}\nand the condition\n\\begin{align}\n\t\\frac{3}{2}\\left(W^2\\right)'' - \\frac{1}{4}W'''' + \\xi^2W'' + 3\\xi W' = 0\n\\end{align}\nwith $U = W + \\frac{1}{2}\\xi^2$.\nMultiplying with $\\xi$ and summarizing using the product rule yields the first integral\n\\begin{align}\n\t\\frac{3}{2}\\xi\\left(W^2\\right)' - \\frac{3}{2}W^2 - \\frac{1}{4}\\xi W''' + \\frac{1}{4}W'' + \\xi^3W' = A \\quad.\\label{eqn:DGL:W}\n\\end{align}\nMultiplying with $W'$, using the identities of \\eqn{DGL:W}\n\\begin{gather}\n\t\\frac{3}{2}\\xi\\left(W^2\\right)' - \\frac{1}{4}\\xi W''' + \\xi^3W' = A + \\frac{3}{2}W^2 - \\frac{1}{4}W'' \\quad,\\\\\n\t \\xi^3W' = -\\xi^2\\left( \\frac{A + \\frac{3}{2}W^2 - \\frac{1}{4}W''}{\\xi} \\right)' \\quad,\n\\end{gather}\nand summarizing using the chain rule yields another first integral\n\\begin{align}\n\t&-\\frac{1}{2}\\left( \\frac{A + \\frac{3}{2}W^2 - \\frac{1}{4}W''}{\\xi} \\right)^2 - \\frac{1}{2}W^3 + \\frac{1}{8}\\left(W'\\right)^2 = AW + B \\quad.\n\\end{align}\nUsing a Darboux transformation, a possible solution can be written in the form\n\\begin{align}\n\tW &= -\\xi^2 + \\frac{4}{3}\\mu + \\left( \\frac{\\chi_\\mu'}{\\chi_\\mu} \\right)^2\n\\end{align}\nwith\n\\begin{align}\n\t-\\frac{1}{2}\\chi_\\mu'' + \\frac{1}{2}\\xi^2\\chi_\\mu = \\mu\\chi_\\mu \\quad.\n\\end{align}\nWith $\\mu=-\\left(m+\\dfrac{1}{2}\\right)$ two possible classes of solutions are\n\\begin{align}\n\tU_m &= -\\frac{1}{2}\\xi^2 - \\frac{2}{3}(2m+1) + \\left( \\frac{P_m'(\\xi)}{P_m(\\xi)} + \\xi \\right)^2 \\quad,\\\\\n\tP_{2m} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k)!}\\xi^{2k} \\quad,\\\\\n\tP_{2m+1} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k+1)!}\\xi^{2k+1} \\quad.\n\\end{align}\n$U_{2n}$ are quadratic potentials with a dip at the minimum.\n$U_0$ is the harmonic potential.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=1$.\nThe ground state of potential $U_{2n}$ is additionally lowered by $2n$.\n$U_{2n+1}$ are singular potentials at $\\xi=0$.\n$U_1$ is the one of the second-order shift operator.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=2$ ($\\Delta\\epsilon=1$ can be achieved by rescaling $\\xi=\\xi'\/2$).\nThe potentials of the third-order shift operator are depicted in \\fig{L3:U}. The potentials $U_2$ and $U_4$ and corresponding lowest states are shown in \\fig{L3:U2} and \\fig{L3:U4}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\TwoColumnWidth]{supplement_graphics\/L3_U.pdf}\n\t\\caption{Potentials of the third-order shift operator obtained by a Darboux transformation.}\n\t\\label{fig:L3:U}\n\\end{figure}\n\n\\begin{figure}[!b]\n\t\\centering\n\t\\begin{minipage}{\\OneColumnWidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{supplement_graphics\/L3_U2.pdf}\n\t\t\\caption{Potential $U_2$ and corresponding lowest states of the third-order shift operator.}\n\t\t\\label{fig:L3:U2}\n\t\\end{minipage}\\hfill\n\t\\begin{minipage}{\\OneColumnWidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{supplement_graphics\/L3_U4.pdf}\n\t\t\\caption{Potential $U_4$ and corresponding lowest states of the third-order shift operator.}\n\t\t\\label{fig:L3:U4}\n\t\\end{minipage}\n\\end{figure}\n\n\\section{Perturbed 1D harmonic oscillator}\n\n\\noindent The Schr\\\"odinger equation for the 1D harmonic oscillator is\n\\begin{align}\n\t\\left[ -\\frac{\\hbar^2}{2m}\\dfrac{\\partial^2}{\\partial x^2} + \\frac{m\\omega^2}{2}x^2 \\right] \\Psi_n(x) &= E_n^{(0)}\\Psi_n(x) \\quad.\n\\end{align}\nSubstituting with the dimensionless variables\n\\begin{align}\n\tx &= \\sqrt{\\frac{\\hbar}{m\\omega}}\\xi \\quad,\\quad E_n^{(0)} = \\hbar\\omega\\epsilon_n^{(0)} \\quad,\\quad \\Psi(x) = \\sqrt[4]{\\frac{m\\omega}{\\hbar}}\\psi(\\xi)\n\\end{align}\nyields the dimensionless formula\n\\begin{align}\n\t\\left[ -\\frac{1}{2}\\dfrac{\\partial^2}{\\partial\\xi^2} + \\frac{1}{2}\\xi^2 \\right] \\psi_n(\\xi) &= \\epsilon_n^{(0)}\\psi_n(\\xi) \\quad.\n\\end{align}\nIts solution is\n\\begin{align}\n\t\\epsilon_n^{(0)} &= n+\\frac{1}{2} \\quad,\\quad \\psi_n(\\xi) = \\frac{1}{\\sqrt{2^nn!\\sqrt{\\pi}}} \\exp\\left(-\\frac{1}{2}\\xi^2\\right) H_n\\left(\\xi\\right) \\quad,\n\\end{align}\nwhere $H_n$ are the Hermite polynomials. They can be expressed the following:\n\\begin{align}\n\tH_n(\\xi) &= 2\\xi H_{n-1}(\\xi) - 2(n-1)H_{n-2}(\\xi) \\quad.\\label{eqn:Hermite:recursive}\n\\end{align}\n\n\\noindent The Schr\\\"odinger equation of a system, which is perturbed with the perturbation potential $v(x) = \\frac{1}{2}\\hbar\\omega u(\\xi)$, respective the corresponding dimensionless perturbation potential $u(\\xi)$, is\n\\begin{align}\n\t\\left[ \\mathcal{H}_0 + \\lambda u(\\xi) \\right] \\phi_k(\\xi) &= \\epsilon_k\\phi_k(\\xi) \\quad.\n\\end{align}\nTransforming the eigenstates into the basis of unperturbed eigenstates and expanding the eigenenergies and coefficients in Taylor series in $\\lambda$,\n\\begin{align}\n\t\\phi_k &= \\sum_nc_{kn}\\psi_n \\quad,\\quad \\epsilon_k = \\sum_p\\lambda^p\\epsilon_k^{(p)} \\quad,\\quad c_{km} = \\sum_p\\lambda^pc_{km}^{(p)} \\quad,\\quad u_{mn} = \\int\\limits_{-\\infty}^\\infty\\psi_m^\\ast(\\xi)u(\\xi)\\psi_n(\\xi)\\,\\text{d}\\xi \\quad,\n\\end{align}\nyields\n\\begin{align}\n\t&\\left(\\epsilon_m^{(0)}-\\epsilon_k^{(0)}\\right)c_{km}^{(0)} + \\sum_{p=1}^\\infty \\left[ \\vphantom{\\sum_{q=1}^{p-1}} \\left(\\epsilon_m^{(0)}-\\epsilon_k^{(0)}\\right)c_{km}^{(p)} - \\epsilon_k^{(p)}\\delta_{km} - \\sum_{q=1}^{p-1}\\epsilon_k^{(q)}c_{km}^{(p-q)} + \\sum_nc_{kn}^{(p-1)}u_{mn} \\right]\\lambda^p = 0 \\quad.\n\\end{align}\nThis gives recursion equations for the $p$th order corrections:\n\\begin{align}\n\t\\epsilon_k^{(p)} &= \\sum_{n\\neq k}u_{kn}c_{kn}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{kk}^{(p-q)} \\\\\n\tc_{km}^{(p)} &= \\frac{1}{\\epsilon_k^{(0)}-\\epsilon_m^{(0)}}\\left( \\sum_nu_{mn}c_{kn}^{(p-1)} - u_{kk}c_{km}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{km}^{(p-q)} \\right) \\quad.\n\\end{align}\nEspecially for the 1st, 2nd, and 3rd order corrections, one gets\n\\begin{align}\n\t\\epsilon_k^{(1)} &= u_{kk} \\quad,\\\\\n\t\\epsilon_k^{(2)} &= -\\sum_{n\\neq k}\\frac{u_{nk}u_{kn}}{\\epsilon_n^{(0)}-\\epsilon_k^{(0)}} \\quad,\\\\\n\t\\epsilon_k^{(3)} &= -\\sum_{n\\neq k}\\frac{u_{nk}u_{kk}u_{kn}}{\\left(\\epsilon_n^{(0)}-\\epsilon_k^{(0)}\\right)^2} + \\sum_{n\\neq k}\\sum_{l\\neq k}\\frac{u_{kn}u_{nl}u_{lk}}{\\left(\\epsilon_n^{(0)}-\\epsilon_k^{(0)}\\right)\\left(\\epsilon_l^{(0)}-\\epsilon_k^{(0)}\\right)} \\quad.\n\\end{align}\n\n\\noindent Expanding $u(\\xi)$ in a Taylor series in $\\xi$,\n\\begin{align}\n\tu(\\xi) &= \\sum_j u_j\\xi^j \\quad,\n\\end{align}\ngives\n\\begin{align}\n\tu_{mn} &= \\sum_j u_j u_{mn}^{(j)} \\quad,\\\\\n\tu_{mn}^{(j)} &= \\frac{1}{\\sqrt{2^{m+n}m!n!}\\sqrt{\\pi}} \\int\\limits_{-\\infty}^\\infty \\xi^j\\text{e}^{-\\xi^2}H_m(\\xi)H_n(\\xi) \\,\\text{d}\\xi \\quad.\n\\end{align}\nInserting~\\eqn{Hermite:recursive} gives the recursion\n\\begin{align}\n\tu_{mn}^{(j)} &= \\sqrt{\\frac{n+1}{2}}u_{m,n+1}^{(j-1)} + \\sqrt{\\frac{n}{2}}u_{m,n-1}^{(j-1)} \\quad.\n\\end{align}\n\n\\noindent The diagonal elements $u_{kk}^{(j)}$ are polynomials in $k$ of order $j\/2$.\nFor the smallest $j$ one gets\n\\begin{align}\n\tu_{kk}^{(0)} &= 1 \\quad,\\\\\n\tu_{kk}^{(2)} &= \\frac{1}{2} + k \\quad,\\\\\n\tu_{kk}^{(4)} &= \\frac{3}{4} + \\frac{3}{2}k + \\frac{3}{2}k^2 \\quad,\\\\\n\tu_{kk}^{(6)} &= \\frac{15}{8} + 5k + \\frac{15}{4}k^2 + \\frac{5}{2}k^3 \\quad,\\\\\n\tu_{kk}^{(8)} &= \\frac{105}{16} + \\frac{35}{2}k + \\frac{175}{8}k^2 + \\frac{35}{4}k^3 + \\frac{35}{8}k^4 \\quad.\n\\end{align}\nThe nondiagonal elements can be written as\n\\begin{align}\n\tu_{k,k-l}^{(j)} = \\sqrt{\\frac{k!}{2^l(k-l)!}}p_{kl}^{(j)} \\quad ,\n\\end{align}\nwhere $p_{kl}^{(j)}$ is a polynomial in $k$ of order $(j-l)\/2$.\nThus, $(u_{k,k-l}^{(j)})^2$ is a polynomial in $k$ of order $j$ for all $l$.\nFor the smallest $j$  and $l$ one gets\n\\begin{align}\n\tp_{k,1}^{(1)} &= 1 \\quad,\\\\\n\tp_{k,1}^{(3)} &= \\frac{3}{2}k \\quad,\\\\\n\tp_{k,1}^{(5)} &= \\frac{5}{4}(1+2k^2) \\quad,\\\\\n\tp_{k,1}^{(7)} &= \\frac{35}{8}k(2+k^2) \\quad,\\\\\n\tp_{k,2}^{(2)} &= 1 \\quad,\\\\\n\tp_{k,2}^{(4)} &= -1+2k \\quad,\\\\\n\tp_{k,2}^{(6)} &= \\frac{15}{4}(1-k+k^2) \\quad,\\\\\n\tp_{k,2}^{(8)} &= \\frac{7}{2}(-1+2k)(3-k+k^2) \\quad,\\\\\n\tp_{k,3}^{(3)} &= 1 \\quad,\\\\\n\tp_{k,3}^{(5)} &= \\frac{5}{2}(-1+k) \\quad,\\\\\n\tp_{k,3}^{(7)} &= \\frac{21}{4}(2-2k+k^2) \\quad,\\\\\n\tp_{k,3}^{(9)} &= \\frac{21}{4}(-1+k)(9-4k+2k^2) \\quad,\\\\\n\tp_{k,4}^{(4)} &= 1 \\quad,\\\\\n\tp_{k,4}^{(6)} &= \\frac{3}{2}(-3+2k) \\quad,\\\\\n\tp_{k,4}^{(8)} &= \\frac{7}{2}(7-6k+2k^2) \\quad,\\\\\n\tp_{k,4}^{(10)} &= \\frac{15}{4}(-3+2k)(13-6k+2k^2) \\quad.\n\\end{align}\nThe results are shown in \\fig{u_mn_j}.\n\n\\begin{figure}[tbp]\n\t\\includegraphics{supplement_graphics\/u_kk-d.pdf}\n\t\\caption{$u_{k,k-l}^{(j)}$ for different $l$ and for different perturbation potentials $u(\\xi)=\\xi^j$.}\n\t\\label{fig:u_mn_j}\n\\end{figure}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt  is long known that  the forward Compton scattering (CS) amplitudes can, by unitarity, causality and crossing, be expressed\nthrough integrals of the photoabsorption cross sections~\\cite{GellMann:1954db}. \nThe low-energy expansions of these expressions lead to a number of useful sum rules, most notably \nthose of Baldin \\cite{Baldin}, and Gerasimov, Drell and Hearn (GDH)~\\cite{Gerasimov:1965et,Drell:1966jv}. Given the photoabsorption cross sections,\none can thus provide a reliable assessment of some of the static electromagnetic \nproperties of the nucleon and nuclei, as well as of the forward CS amplitudes in general.\nFor the proton, the first such assessments was performed in the early \n1970s \\cite{Damashek,Armstrong}. Since then, the knowledge of the photoabsorption cross sections\nappreciably improved, and yet for the unpolarized case only the Baldin\nsum rule has been updated~\\cite{Schroder,Babusci,Olmos}. \nIn this work, we provide a re-assessment of the forward spin-independent amplitude of proton CS, and evaluate the associated sum rules involving the\n{\\em dipole and quadrupole polarizabilities of the proton}.\n\n\nSum rules are essentially the only way to gain empirical knowledge of \nthe forward CS amplitudes.  It is impossible to access the forward kinematics \nin real CS experiments. The measurement of the forward spin-independent CS amplitude can be done indirectly through the process of dilepton photoproduction ($\\gamma \\,p \\to  p \\,e^+ e^-$)~\\cite{Alvensleben}. The timelike CS, involved\nin the process of dilepton photoproduction, yields access to real CS given \nthe small virtuality of the outgoing photon, or equivalently,  the nearly vanishing invariant\nmass of the produced pair. The experimental result~\\cite{Alvensleben} \ncompared well with the aforementioned evaluations \\cite{Damashek,Armstrong}. \nDespite the substantial\nadditions to the database of total photoabsorption cross sections, the works of Damashek and Gilman (DG) \\cite{Damashek} as well as Armstrong {\\it et al.}~\\cite{Armstrong} remained to be, until now, the only evaluations\nof the full amplitude.\n\nThe newer data were used, however, in the most recent evaluations of the Baldin sum rule \\cite{Babusci,Olmos}, which \nyields the sum of the electric and magnetic dipole polarizabilities,\n\\Eqref{BaldinSR}. These recent analyses obtained somewhat lower value for the sum than DG, cf.~Table \\ref{sumruletest}. In this work we find that the difference between the early and the recent evaluations arises from systematic inconsistencies in the experimental database. \nWe also obtain the sum rule value for a combination of higher-order quadrupole polarizabilities and \ncompare it with several theoretical predictions.\n\nThis paper is organized as follows.\nIn Sect.\\ \\ref{sec:overview} we give a brief overview of \nthe Kramers-Kronig relation and sum rules for polarizabilities.\nIn Sect.\\ \\ref{sec:fitting} we discuss the fitting procedure\nfor the unpolarized total proton photoabsorption cross section data.\nThe sum rule evaluations of scalar polarizabilities\nand of the spin-independent forward CS amplitude are presented \nin Sect.\\ \\ref{sec:sumrules}.\nConclusions are given in Sect.\\ \\ref{sec:summary}. The Appendix\ndemonstrates the elastic-channel contribution to the sum rules and polarizabilities on the example of one-loop scalar QED.\n\n\n\n\\begin{figure*}[hbt]\n\\includegraphics[width=0.8\\textwidth]{figs\/cs_tot.pdf}\n\\caption{\n(Color online)\nFits of experimental data for the total photoabsorption cross section on the proton.  Fit I is obtained using MAID \\cite{MAID} results below the $2\\pi$-production, and data from LEGS \\cite{BNL} and Armstrong {\\it et al.}~\\cite{Armstrong} above it. Fit II uses SAID \\cite{SAID} and the data of MacCormick {\\it et al.}~\\cite{MacCormick}. Both fits use Bartalini {\\it et al.} \\cite{Bartalini} and the high-energy data \\cite{Caldwell,Aid,Chekanov} displayed in the insert.\n}\n\\label{fig:CS_tot}\n\\end{figure*}\n\\section{Forward Compton Amplitude and Sum Rules}\n\\label{sec:overview}\n\n\n\nFor a spin-1\/2 target, such as the proton, \nthe forward CS amplitude is given by\n\\begin{equation} \nT_{fi} = f(\\nu) \\, \\boldsymbol{\\varepsilon}^{\\prime\\,\\ast }\\cdot \\boldsymbol{\\varepsilon} \\, + \\, g(\\nu) \\,i \\, (\\boldsymbol{\\varepsilon}^{\\prime\\,\\ast }\\! \\times \\boldsymbol{\\varepsilon})\\cdot\\boldsymbol{\\sigma},\n\\eqlab{definition}\n\\end{equation} \nwhere $f$ and $g$ are scalar functions of the photon lab energy $\\nu$;\nvectors $\\boldsymbol{\\varepsilon}$ and $\\boldsymbol{\\sigma}$ represent the photon and proton polarizations, respectively. The crossing symmetry implies that the spin-independent amplitude $f$ is an even\nand the spin-dependent amplitude $g$ is an odd function of $\\nu$.\n\nThe optical theorem (unitarity) relates the imaginary\npart of the amplitudes to the total photoabsorption cross sections:\n\\begin{subequations}\n\\begin{eqnarray}\n\\im f(\\nu) & = & \\frac{\\nu}{8\\pi} \\left[\\sigma_{1\/2}(\\nu)+\\sigma_{3\/2}(\\nu)\\right] , \\\\\n\\im g(\\nu) & = & \\frac{\\nu}{8\\pi}\n\\left[\\sigma_{1\/2}(\\nu)-\\sigma_{3\/2}(\\nu)\\right] .\n\\eqlab{unitarity}\n\\end{eqnarray}\n\\end{subequations}\nHere $\\sigma_{\\lambda} \\def\\La{{\\it\\Lambda}}(\\nu) $ is the doubly-polarized cross section\nwith $\\lambda$ representing the\ncombined helicity of the initial $\\gamma} \\def\\Ga{{\\it\\Gamma} p$ state.  Averaging over the\npolarization of initial particles gives the  unpolarized\nphotoabsorption cross section, $\\sigma =\\mbox{\\small{$\\frac{1}{2}$}} ( \\sigma_{1\/2}+\\sigma_{3\/2})$.\n\n\n\n\nIn the present article we focus on relations involving  the spin-independent amplitude $f$, and the unpolarized cross section\n$\\sigma} \\def\\Si{{\\it\\Sigma}$.\nThe Kramers-Kronig relation between these quantities exploits the optical theorem, causality and crossing symmetry, to yield for the proton~\\cite{GellMann:1954db}:\n\\begin{equation}\n\\eqlab{KK}\n\\re f(\\nu) =-\\,\\frac{\\alpha}{M_p} + \\frac{\\nu^2}{2\\pi^2}\\fint_0^\\infty\\! \\frac{{\\rm d} \\nu'}{\\nu^{\\prime \\, 2}-\\nu^2}\\, \\sigma} \\def\\Si{{\\it\\Sigma}(\\nu'),\\end{equation}\nwhere $\\alpha=e^2\/4\\pi$ is the fine-structure constant and $M_p$ is the proton mass; the slashed integral denotes the principal-value integration. \n\nWe next would like to consider the low-energy expansion of $f$.\nAt this point it is important to note that the elastic scattering, i.e.\\ the CS process itself, \nis one of the photoabsorption processes. The total CS cross section does\nnot vanish for $\\nu\\to 0$ but goes to a constant --- the Thomson cross section: \n\\begin{equation} \n\\sigma} \\def\\Si{{\\it\\Sigma}(0 ) = \\frac{8\\pi \\alpha^2}{3 M_p^2}.\n\\end{equation} \nThis means \\Eqref{KK} does not admit a Taylor-series expansion\naround $\\nu=0$ (each coefficient in that expansion is infrared divergent, cf. Appendix).\nSuch expansion is nonetheless important for establishing the polarizability sum rules.\nWe hence prefer to take the CS out of the total cross section, i.e.:\n\\begin{equation} \n\\sigma} \\def\\Si{{\\it\\Sigma}(\\nu) = \\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{CS}}(\\nu)+\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu), \n\\end{equation} \nwhere $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}$ can be assumed to be dominated\nby hadron-production processes, for which there is a threshold \nat some $\\nu_0 > m_\\pi$. \n\n\n\nThe amplitude $f$ can be decomposed accordingly into the elastic\nand inelastic terms,\n\\begin{eqnarray} \nf(\\nu) & = & f_{\\mathrm{el}}(\\nu) + f_{\\mathrm{inel}}(\\nu)  , \\\\\nf_{\\mathrm{el}}(\\nu) &=& -\\,\\frac{\\alpha}{M_p} + \\frac{\\nu^2}{2\\pi^2}\\fint_0^\\infty\\! \\frac{{\\rm d} \\nu'}{\\nu^{\\prime \\, 2}-\\nu^2}\\, \\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{CS}}(\\nu'),\n\\eqlab{fel}\\\\\nf_{\\mathrm{inel}}(\\nu) &=& \\frac{\\nu^2}{2\\pi^2}\\fint_{\\nu_0}^\\infty\\! \\frac{{\\rm d} \\nu'}{\\nu^{\\prime \\, 2}-\\nu^2}\\, \\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu').\n\\eqlab{fbar}\n\\end{eqnarray} \nThe details on dealing with $f_{\\mathrm{el}}$ can be found in \nAppendix \\ref{sec:scalarQEDExample}. In what follows, however, we neglect\nthe contribution from $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{CS}}$, as it is suppressed by an extra order of $\\alpha$. Hence we set $f_{\\mathrm{el}}(\\nu) = -\\alpha\/M_p$, as is usually done. \n\nConsidering $f_{\\mathrm{inel}}$,\nthe low-energy expansion of both sides of \\Eqref{fbar} leads to the sum rules for polarizabilities. At the leading order [$O(\\nu^2)$], one\nobtains the Baldin sum rule \\cite{Baldin} for the\nsum of electric ($\\alpha_{E1}$) and magnetic ($\\beta_{M1}$) dipole polarizabilities:\n\\begin{equation}\n\\eqlab{BaldinSR}\n\\alpha_{E1} + \\beta_{M1} =\\frac{1}{2\\pi^2} \\int_{\\nu_0}^\\infty \\!{\\rm d}\\nu \\,\\frac{\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu)}{\\nu^2}.\n\\end{equation}\nAt $O(\\nu^4)$ we obtain `the 4\\textsuperscript{th}-order sum rule': \n\\begin{eqnarray}\n\\eqlab{4thSR}\n\\alpha_{E\\nu} + \\beta_{M\\nu} + \\frac{1}{12}\\,(\\alpha_{E2} + \\beta_{M2}) =\\frac{1}{2\\pi^2} \\int_{\\nu_0}^\\infty \\!{\\rm d}\\nu \\,\\frac{\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu)}{\\nu^4},\\nonumber\\\\\n\\end{eqnarray}\nwhich involves the quadrupole  polarizabilities\n$\\alpha_{E2}$, $\\beta_{M2}$, as well as the leading dispersive\ncontribution to the dipole polarizabilities denoted as $\\alpha_{E\\nu}$, $\\beta_{M\\nu}$, see \\cite{Babusci:1998ww} for more details.\n\n\nOur aim here is to provide an empirical fit of the available\ndata for $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}$ and evaluate the various sum rules. \n\n\\section{Fits of the Photoabsorption Cross Section}\n\\label{sec:fitting}\n\nThe presently available experimental data, together with\nthe results of the empirical analyses MAID and SAID, as well as our fits are displayed in \\Figref{CS_tot}. In our fitting we\ndistinguish the following \nthree regions:\n\\begin{itemize}\n \\item {\\it low-energy},   $\\nu \\in [\\nu_0   \\,,\\;  \\nu_1  ) $;\n \\item {\\it medium-energy},   $\\nu \\in [\\nu_1   \\,,\\;  2\\;\\mathrm{GeV}) $;\n \\item {\\it high-energy}, $\\nu \\in [2\\;\\mathrm{GeV} \\,,\\;  \\infty ) $;\n\\end{itemize}\nwhere $\\nu_0$ ($\\simeq 0.145$ GeV) and $\\nu_1$ ($\\simeq 0.309$ GeV) are respectively the thresholds for the single- and double-pion photoproduction on the proton.\n\nIn the {\\it low-energy} region we use the pion production \n($\\pi^+  n$ and $\\pi^0 p$) cross sections from the \nMAID \\cite{MAID} and SAID \\cite{SAID} partial-wave analyses. \nIn our error estimate we assign a 2\\% uncertainty on these values.\n\nIn the {\\it medium-energy} region we fit the actual experimental data, using a sum of Breit-Wigner resonances and a background. \nFollowing \\cite{Babusci}, we take \nsix Breit-Wigner resonances, each parameterized \nas\n\\begin{equation}\n\\label{res_fit}\n\\sigma} \\def\\Si{{\\it\\Sigma}_R(W)=A \\cdot \\frac{\\Gamma^2\/4}{(W-M)^2 + \\Gamma^2\/4},\n\\end{equation}\nwhere $W=\\sqrt{s}$ is the total energy of the $\\gamma p$ system.\nThe background function is from \\cite{Armstrong}:\n\\begin{equation}\n\\label{bgd_fit}\n\\sigma} \\def\\Si{{\\it\\Sigma}_B(W)=\\sum_{k=-2}^2 C_k (W-W_0)^k,\n\\end{equation}\nwhere $W_0=M_p+m_{\\pi}$ corresponds with\nthe pion photoproduction threshold.\n\n\n\n\\begin{table}[bth]\n{\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n&$M$ [MeV] & $\\Gamma$ [MeV] & $A$ [$\\upmu$b]\\\\\n\\hline\n\\hline\n\\multirow{6}{*}{\\begin{sideways} \\hspace{-0.4cm} fit I\\end{sideways}}&$1213.6 \\pm  0.1$ & $117.6 \\pm  1.9$ &  $522.7 \\pm  17.0$\\\\\n&$1412.8 \\pm  5.9$ & $ 82.8 \\pm 26.8$ &  $ 40.1 \\pm 33.8$\\\\\n&$1496.0 \\pm  2.8$ & $136.5 \\pm 11.1$ &  $161.8 \\pm 32.4$\\\\\n&$1649.4 \\pm  4.1$ & $135.3 \\pm 15.3$ &  $ 83.2 \\pm 22.7$\\\\\n&$1697.5 \\pm  2.6$ & $ 18.8 \\pm 12.6$ &  $ 18.2 \\pm 26.0$\\\\\n&$1894.3 \\pm 15.6$ & $302.0 \\pm 41.3$ &  $ 31.5 \\pm  8.7$\\\\\n\\hline\n\\hline\n\\multirow{6}{*}{\\begin{sideways}\\hspace{-0.4cm}  fit II\\end{sideways}}&$1214.8 \\pm  0.1$ & $99.0 \\pm  1.1$ & $502.3 \\pm 12.3$\\\\\n&$1403.9 \\pm  6.2$ & $118.2 \\pm 19.6$ & $ 51.8 \\pm 23.8$\\\\\n&$1496.9 \\pm  2.1$ & $133.4 \\pm  9.4$ & $162.0 \\pm 29.2$\\\\\n&$1648.0 \\pm  4.4$ & $135.2 \\pm 15.9$ & $ 83.6 \\pm 23.8$\\\\\n&$1697.2 \\pm  2.7$ & $ 21.2 \\pm 13.2$ & $ 18.7 \\pm 25.9$\\\\\n&$1893.7 \\pm 17.4$ & $323.5 \\pm 45.3$ & $ 31.7 \\pm  9.1$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Fitting parameters for the resonances (\\ref{res_fit})\nobtained for fit I and II.}\n\\label{table:res_par_2}\n\\end{table}\n\n\n\\begin{table}[bth]\n{\n\\begin{tabular}{|c|c|c|}\n\\hline\n& fit I & fit II \\\\\n\\hline\n$C_{-2}$, [$\\upmu$b $\\cdot$ GeV$^2$] & $ 0.44  \\pm  0.22$ & $ 0.26 \\pm  0.17$\\\\\n$C_{-1}$, [$\\upmu$b $\\cdot$ GeV]     & $-11.06 \\pm  3.69$ & $-7.97 \\pm  2.89$\\\\\n$C_0$, [$\\upmu$b]                    & $ 74.38 \\pm 20.16$ & $57.27 \\pm 16.09$\\\\\n$C_1$, [$\\upmu$b $\\cdot$ GeV$^{-1}$] & $ 22.18 \\pm 37.71$ & $54.26 \\pm 31.07$\\\\\n$C_2$, [$\\upmu$b $\\cdot$ GeV$^{-2}$] & $ 37.69 \\pm 21.48$ & $19.51 \\pm 18.17$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Fitting parameters for the background (\\ref{bgd_fit})\nobtained for fit I and II in the resonance region.}\n\\label{table:bgd_par}\n\\end{table}\n\n\n\nObserving a significant discrepancy between SAID and MAID \naround the $\\De$(1232)-resonance peak and a similar discrepancy\nbetween two sets of experimental data, we have made two\ndifferent fits:\n\\begin{itemize}\n\\item[I.] MAID\\cite{MAID} + LEGS \\cite{BNL} + Armstrong {\\it et al.}~\\cite{Armstrong}, \n\\item[II.] SAID \\cite{SAID} + MacCormick {\\it et al.}~\\cite{MacCormick}.\n\\end{itemize}\nThey are shown in \\Figref{CS_tot} by the red solid and blue dashed lines, respectively. The corresponding values of parameters are\ngiven in Tables~\\ref{table:res_par_2} and \\ref{table:bgd_par}.\nIn both fits we have also made\nuse of the GRAAL 2007 data \\cite{Bartalini}, shown in the figure by\nlight-blue squares. These data had not been available at the time of the previous sum rule evaluations. \n\n\nFinally, for the {\\it high-energy region} we use the\nstandard Regge form \\cite[p.~191]{Barnett}:\n\\begin{equation}\n\\label{regge_fit}\n\\sigma} \\def\\Si{{\\it\\Sigma}_\\mathrm{Regge}(W)=c_1 \\, W^{p_1} + c_2 \\, W^{p_2}.\\\\\n\\end{equation}\nFor $W$ in GeV and the cross section in $\\upmu$b, we obtain\nthe following parameters (for both of our fits):\n\\begin{gather*}\nc_1 = 62.0 \\pm 8.1, \\quad c_2 = 126.3 \\pm 4.3, \\\\\np_1 = 0.184 \\pm 0.032, \\quad p_2 = -0.81 \\pm 0.12.\n\\end{gather*}\nWe also tried the high-energy parameterization used\nin \\cite{Babusci}, but obtained a worse fit and abandoned it. \n\n\n\n\n\n\\begin{figure*}[bht]\n\t\\includegraphics[width=0.8\\textwidth]{figs\/re_f.pdf}\n\t\\caption{(Color online) Our evaluation of $\\re f$ based on the two fits\nof the photoabsorption cross section, compared with previous evaluations \\cite{Damashek,Olmos,Armstrong}. The experimental data point is from Ref.~\\cite{Alvensleben}.}\n\t\\label{fig:Re_f}\n\\end{figure*}\n\n\\begin{table*}[tbh]\n\\begin{ruledtabular}\n\\centering \n\\caption{\n\tEmpirical evaluations of sum rules \n    and verification of the\n    Kramers-Kronig relation for the proton.}\n    \\begin{tabular}{|r|l|l|l|l|}\n\t&  Baldin& 4\\textsuperscript{th} order &  6\\textsuperscript{th} order\\footnote{$ \\int_{\\nu_0}^\\infty \\!{\\rm d}\\nu\\, \\nu^{-6} \\,\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu)\/(2\\pi^2)$} & $\\re f$($2.2$ GeV) \\\\\n    \t&[$10^{-4}$ fm$^3$]&[$10^{-4}$ fm$^5$]&[$10^{-4}$ fm$^7$]&[$\\upmu$b $\\cdot$ GeV]\\\\\n\t\\hline\n\tDamashek--Gilman \\cite{Damashek}      & $14.2\\pm 0.3$  &  &  & $-11.5$\\footnote{Interpolated value.}\\\\\n    Armstrong {\\it et al.} \\cite{Armstrong} &              &  &  & $-10.8$\\\\\n\tSchr{\\\"o}der \\cite{Schroder}            & $14.7\\pm0.7$  &$6.4$&  & \\\\\n\tBabusci {\\it et al.} \\cite{Babusci}     & $13.69\\pm0.14$ &  &  & \\\\\n\tA2 Collaboration \\cite{Olmos} & $13.8\\pm0.4$   &  &  & $-10.5$\\footnote{Based on the cross-section parametrization from \\cite{Ahrens}.}\\\\\n\tMAID ($\\pi$ chan.)~\\cite{MAID}  & $11.63$\\footnote{Integrated from threshold to $\\nu_\\mathrm{max}=1.663$ GeV.}     &  &  & \\\\\n\tSAID ($\\pi$ chan.)~\\cite{SAID}             & $11.5$\\footnote{Integrated from threshold to $\\nu_\\mathrm{max}=2$ GeV.}           &  &  & \\\\\n\t\\hline\n    This work &&&&\\\\\n    \t fit I & $14.29\\pm 0.27$ & $6.08 \\pm 0.12 $ & $ 4.36 \\pm 0.09 $ & $-10.35$ \\\\\n\t fit II  & $13.85\\pm 0.22$ & $6.01 \\pm 0.11 $ & $ 4.42 \\pm 0.08 $ & $-9.97 $ \\\\\n     \t\\hline\n    Experiment &&&&\\\\\n      Alvensleben {\\it et al.} \\cite{Alvensleben} &          &  &  & $-12.3\\pm2.4$\\\\\n\t\\end{tabular}\n\t\\label{sumruletest}\n   \\end{ruledtabular}\n\\end{table*}\n\n\\twocolumngrid \n\nThe fitting was done with the help of the {\\it SciPy} package for Python.\nThe resulting chi-square, evaluated as\n\\begin{equation}\n\\eqlab{chisq}\n\\chi^2 = \\sum_i \\frac{\\left(\\sigma_i^{\\mathrm{fit}}  - \\sigma_i^{\\mathrm{exp}}\\right)^2}{(\\De\\sigma_i^{\\mathrm{exp}})^2},\n\\end{equation}\nis of about the same quality for the two fits. In the intermediate region,\nwe obtain $\\chi^2\/\\mbox{point}=0.7$ for fit I, and $\\chi^2\/\\mbox{point}=0.6$\nfor fit II. In the high-energy region $\\chi^2\/\\mbox{point}=1.2$ in both cases.\nAgain, the low-energy region is not fitted but is borrowed from, respectively, the MAID and SAID\nanalyses.\n\n\n\\section{Sum Rule Evaluations}\n\\label{sec:sumrules}\n\n\n\n\nHaving obtained the fits of the total photoabsorption cross section $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}$, we evaluate the integrals \nin Eqs.~\\eref{fbar}, \\eref{BaldinSR} and \\eref{4thSR}; the\nresults are presented in \\Figref{Re_f} and Table~\\ref{sumruletest}.\n\nTables \\ref{BaldinSR_regions} and \\ref{4thSR_regions} show contributions of each region to the Baldin and the 4\\textsuperscript{th}-order sum rule, respectively. \nThe uncertainty in calculating an integral\n$I_n = \\int{\\rm d}\\nu\\,  \\nu^{-n} \\sigma(\\nu) $ has been evaluated as follows:\n\\begin{equation}\n\\Delta I_n = \\sum_i \\frac{\\Delta \\nu_i}{\\nu_i^n} \\chi_i^2 \\,\\Delta\\sigma_i^{\\mathrm{exp}},\n\\end{equation}\nwhere $\\chi_i^2$ is the chi-square at the point $i$, cf.~\\Eqref{chisq}.\n\n\n\n\\begin{table}[tbh]{\n\\caption{Contributions of different regions to the Baldin sum rule\nfor the two fits in \\Figref{CS_tot}.}\n   \\label{BaldinSR_regions}\n$\\alpha_{E1} + \\beta_{M1}$\\; [$10^{-4}$ fm$^3$]\\\\\n\t\\centering \n\t\\begin{tabular}{|c|c|c|c|}\n\t\\hline\n\\diagbox{Fit}{Region}& {\\it low-energy} & {\\it medium-energy}  & {\\it high-energy}  \\\\\n\t\\hline\n\tI & $6.12 \\pm 0.12$ & $7.53 \\pm 0.13$ & $0.64 \\pm 0.02$ \\\\\n\tII& $6.06 \\pm 0.12$ & $7.15 \\pm 0.08$ & $0.64 \\pm 0.02$ \\\\\n\t\\hline\n\t\\end{tabular}\n\t\\par}\n \n\\end{table}\n\n\\begin{table}[tbh]{\n\\caption{ Contributions of different regions to the 4\\textsuperscript{th}-order sum rule\nfor the two fits in \\Figref{CS_tot}.} \\label{4thSR_regions}\n$\\alpha_{E\\nu} + \\beta_{M\\nu} + \\frac{1}{12}(\\alpha_{E2} + \\beta_{M2})$ \\; [$10^{-4}$ fm$^5$]\\\\\n\t\\centering \n\t\\begin{tabular}{|c|c|c|c|}\n\t\\hline\n\t\\diagbox{Fit}{Region} & {\\it low-energy} & {\\it medium-energy}   & {\\it high-energy}  \\\\\n\t\\hline\n\tI  & $4.50 \\pm 0.09$ & $1.58 \\pm 0.03$ & $(219 \\pm 8)\\times 10^{-5}$ \\\\\n\tII  & $4.53 \\pm 0.09$ & $1.48 \\pm 0.01$ & $(219 \\pm 8)\\times 10^{-5}$ \\\\\n\t\\hline\n\t\\end{tabular}\n\t\\par}\n\\end{table}\n\nThe corresponding full results (sum of the three regions) \nare given in \nTable~\\ref{sumruletest}, and compared with the results of previous works. In this table we also give the result for the 6th-order\nintegral, and for the full amplitude $f$ at $\\nu=2.2$ GeV.\nThe real part of $f$ is plotted in \\Figref{Re_f} over a broad\nenergy range and compared with previous evaluations and \nthe experimental number from the 1973 DESY experiment at \n2.2 GeV. Although none of the evaluations really contradicts the experiment, \nthere is a clear tendency to a higher central value. \n\n\nThe new\ndilepton photoproduction experiments planned  at the Mainz Microtron (MAMI) could perhaps provide experimental values in the lower energy range. Obviously, the regions of the extrema (e.g., the $\\De(1232)$ region or the interval between 0.6 and 0.7 GeV) are most interesting as the different evaluations seem to differ there the most. In\nthe region around 0.6 GeV, for example, one of our evaluations (fit I) is nearly \nidentical with Armstrong's~\\cite{Armstrong}, while the other one (fit II) is aligning with DG~\\cite{Damashek} and A2 Coll.~\\cite{Olmos}. An appropriately precise\nexperiment could tell which of the groups is correct, if any.\n\n\\begin{figure}[htb]\n\t\\includegraphics[width=0.95\\columnwidth]{figs\/f_chipt_hw.pdf}\n\t\\caption{(Color online) Our evaluations of $f(\\nu)$ compared to the $\\chi$PT calculation\nof Ref.~\\cite{Lensky:2009uv}.}\n\t\\label{fig:ReIm_f}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\t\\includegraphics[width=0.92\\columnwidth]{figs\/nosr_hw.pdf}\n\t\\caption{(Color online) The 4\\textsuperscript{th}-order sum rule constraint for $\\alpha_{E\\nu} +\\, \\frac{1}{12} \\alpha_{E2}$ and $\\beta_{M\\nu} +\\, \\frac{1}{12} \\beta_{M2}$ combinations of polarizabilities, compared to results from dispersion relation approaches (DR) \\cite{Babusci:1998ww,Drechsel:2002ar}, baryon chiral perturbation theory (B$\\chi$PT) \\cite{Lensky:2015}, and heavy baryon chiral perturbation theory (HB$\\chi$PT) \\cite{Holstein:1999uu}. The errors of the $\\chi$PT derive from our  crude estimate of the next-order corrections.}\n\t\\label{fig:4thSR}\n\\end{figure}\n\n\nFigure~\\ref{fig:ReIm_f} shows both the real and imaginary\npart of $f$ at lower energies, where it can be compared\nwith a calculation done within chiral perturbation theory ($\\chi$PT) \\cite{Lensky:2009uv}. A rather nice agreement between theory and empirical evaluations is observed for energies up to\nabout the pion-production threshold.  \n\n\nFor very low energy this comparison can\nbe made more quantitative by looking at the polarizabilities. While for the Baldin\nsum rule the situation was extensively discussed in the literature (cf.~\\cite{Griesshammer:2012we} for a recent review), the \n4\\textsuperscript{th}-order sum rule was not studied at all. \nIt can,\nhowever, be very useful in unraveling the higher-order polarizabilities,\nas illustrated by \\Figref{4thSR}. This is the plot of a combination of proton magnetic\npolarizabilities versus electric, where the various theory predictions are compared with our 4\\textsuperscript{th}-order sum rule evaluation. The band representing the sum rule covers the \ninterval between the two values given in \nTable~\\ref{sumruletest} (rows `fit I' and `fit II'). The sum rule clearly provides a\nmodel-independent constraint on these polarizabilities and a rather stringent test for the theoretical approaches.  \n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:summary}\nThe fundamental relation between the photon absorption and scattering, encompassed in the Kramers-Kronig type of relations, allows us\nto evaluate the forward Compton scattering off protons \nusing the empirical knowledge of the total photoabsorption cross sections.\nThe present database of the unpolarized photoabsorption cross section is not entirely\nconsistent and so as to reflect that we obtain two distinct fits to it.\nThe two fits yield slightly different results for the spin-independent\namplitude $f(\\nu)$, and hence for its low-energy expansion characterized\nby the scalar  polarizabilities of the proton. Our two results for the\nsum of dipole polarizabilities (or, Baldin sum rule) correspond nicely\nwith the  results of previous evaluations, which too can be separated into\ntwo groups: \nthe {\\it old} \\cite{Damashek,Schroder}, with the value slightly above 14 (in units of $10^{-4}$ fm\\textsuperscript{3}), and \nthe {\\it new}~\\cite{Babusci,Olmos}, with the value slightly below 14.\nThe 1996 DAPHNE@MAMI experiment \\cite{MacCormick}, superseding\nthe 1972 experiment of \nArmstrong {\\it et al.}~\\cite{Armstrong}, is clearly responsible\nfor this difference. Neglecting the older data in favor of the newer ones, yields the lower value of the Baldin sum rule, and vice versa. \nWhile one can take a preference in one of the two fits and corresponding\nresults, we prefer to think of their difference\nas a systematic uncertainty in the present \nevaluation of the polarizabilities  and of the forward spin-independent amplitude of the proton. \n\nAs far as polarizabilities are concerned,\nonly the Baldin sum rule is appreciably affected by the inconsistency\nin the photoabsorption database. Nevertheless, the two results\n(fit I and II in Table~\\ref{sumruletest}) are not in conflict with\neach other,\ngiven the overlapping error bars. It is customary to take a statistical average in such cases.\nTaking a weighted average\\footnote{For the weighted average, $\\bar x\\pm \\bar \\sigma} \\def\\Si{{\\it\\Sigma}$, over \na set $\\{ x_i \\pm \\sigma} \\def\\Si{{\\it\\Sigma}_i \\}$, we use \\cite[p.~120]{Parratt:1961}: \n$$\\bar x = \\frac{\\sum_i x_i\/\\sigma} \\def\\Si{{\\it\\Sigma}_i^2}{\\sum_j 1\/\\sigma} \\def\\Si{{\\it\\Sigma}_j^2},\\; \n\\bar \\sigma} \\def\\Si{{\\it\\Sigma} = \\Big(\\frac{\\sum_i (x_i-\\bar x)^2\/\\sigma} \\def\\Si{{\\it\\Sigma}_i^2}{\\sum_j 1\/\\sigma} \\def\\Si{{\\it\\Sigma}_j^2}\\Big)^{1\/2}.$$ } over our two values for the Baldin sum rule  we obtain:\n$\\alpha_{E1}^{(\\mathrm{p})}+\\beta_{M1}^{(\\mathrm{p})}=(14.0\\pm 0.2)\\times 10^{-4}\\,\\mathrm{fm}^3.$\nThe error bar here does directly not include\nthe aforementioned systematic uncertainty of the cross section database.\nHowever, since the two results are fairly well surmised by the weighted\naverage, the latter should be less prone to \nthe systematic uncertainty of the database.\n\nWe have presented a first study of the sum rule\ninvolving the quadrupole polarizabilities, \\Eqref{4thSR},\nhere referred to as 'the 4\\textsuperscript{th}-order sum rule'. \nOur weighted average value for this sum rule, in the proton case, \nis $6.04(4) \\times 10^{-4}\\,\\mathrm{fm}^5$. It agrees\nvery nicely with the state-of-the-art calculations \nof these polarizabilities based\non fixed-$t$ dispersion relations and chiral perturbation theory,\nsee \\Figref{4thSR}. We note that, while the calculations\ndemonstrate significant differences in the values of individual\nhigher-order polarizabilities, these differences apparently \ncancel out from the\nforward combination of these polarizabilities which enters the sum rule.\n\n\nIn the subsequent paper we will discuss the evaluation  of the forward {\\it spin-dependent} amplitude\n$g(\\nu)$ and  related sum rules for the forward spin polarizabilities\nof the proton. The knowledge of the two amplitudes will allow us\nto reconstruct the observables for the proton Compton scattering\nat zero angle.\n\n\n\\begin{acknowledgments}\nWe thank J\\\"urgen Ahrens for kindly supplying us with a database of total photoabsorption cross sections.\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 1044 [The Low-Energy Frontier of the Standard Model], and the Graduate School DFG\/GRK 1581\n[Symmetry Breaking in Fundamental Interactions].\n\\end{acknowledgments}\n\n\n\\onecolumngrid\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{5sec:Intro}Introduction}\n\nA quantum point contact (QPC) is the simplest mesoscopic device that\ndirectly shows quantum mechanical properties. It is a short\nballistic transport channel between two electron reservoirs, which\nshows quantized conductance as a function of the width of the\nchannel \\cite{5Wees,5Wharam}.  A widely applied approach for\nimplementing QPCs is using a split-gate structure on the surface of\na heterostructure with a two-dimensional electron gas (2DEG) at\nabout 100~nm beneath its surface. The conventional design of such a\nsplit-gate QPC has two metallic gate fingers\n(Fig.~\\ref{fig1SEMpic}a). Operating this device with a negative gate\nvoltage $V_g$ results in the formation of a barrier with a small\ntunable opening between two 2DEG reservoirs, because the 2DEG below\nthe gate fingers gets depleted over a range that depends on $V_g$.\nFor electrons in the 2DEG, this appears as an electrostatic\npotential $U$ that is a large barrier with a small opening in the\nform of a saddle-point potential (Fig.~\\ref{fig3SaddlePotential}).\nThe saddle-point potential gives transverse confinement in the\nchannel that is roughly parabolic, which results for this transverse\ndirection in a discrete set of electronic energy levels. For\nelectron transport along the channel this gives a discrete set of\nsubbands with one-dimensional character. Quantized conductance\nappears because each subband contributes $G_{0} = 2e^2\/h$ to the\nchannel's conductance \\cite{5Wees,5Wharam}, where $e$ is the\nelectron charge and $h$ is Planck's constant.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{SEMimage.pdf}\n\\caption{(a) SEM image of a conventional split-gate quantum point\ncontact (QPC). It has two gate fingers (QPC$_{\\rm 2F}$ device). The\nlength of the QPC channel is fixed and can be parameterized by the\nlithographic length $L_{litho}$ of the gate structure. The diagram\nalso illustrates the measurement scheme (see main text for details).\n(b) SEM image of a length tunable QPC with 6 gate fingers (QPC$_{\\rm\n6F}$ device). Here the effective length $L_{eff}$ of the QPC can be\ntuned by changing the ratio of the gate voltages on the central\ngates ($V_{g1}$) and side gates ($V_{g2}$).} \\label{fig1SEMpic}\n\\end{figure}\n\n\n\n\n\n\nWe present here the design and experimental characterization of QPCs\nwhich offer additional control over the shape of the saddle-point\npotential. We focused on developing devices for which the effective\nlength of the saddle-point potential (along the transport direction)\ncan be tuned \\textit{in situ}. The additional control is implemented\nwith a symmetric split-gate design based on 6 gate fingers\n(Fig.~\\ref{fig1SEMpic}b). Such devices will be denoted as QPC$_{\\rm\n6F}$, and conventional devices with 2 gate fingers\n(Fig.~\\ref{fig1SEMpic}a) as QPC$_{\\rm 2F}$. These QPC$_{\\rm 6F}$ are\noperated with the gate voltage on the outer fingers ($V_{g2}$) less\nnegative than the gate voltage on the central fingers ($V_{g1}$) to\navoid quantum dot formation. Sweeping $V_{g1}$ from more to less\nnegative values opens the QPC$_{\\rm 6F}$. By co-sweeping $V_{g2}$ at\nfixed ratio $V_{g2}\/V_{g1}$ it behaves as a QPC with a certain\nlength for the saddle-point potential, and this length can be chosen\nby setting $V_{g2}\/V_{g1}$: It is shortest for $V_{g2}\/V_{g1}\n\\approx 0$ and longest for $V_{g2}\/V_{g1} \\lessapprox 1$. For our\ndesign the effective length could be tuned from about 200~nm to\n600~nm.\n\n\nThe motivation for developing these length-tunable QPCs comes from\nstudies of electron many-body effects in QPCs. A well-known\nmanifestation of these many-body effects is the so-called\n0.7~anomaly \\cite{5Thomas1996}, which is an additional shoulder at\n$0.7~G_{0}$ in quantized conductance traces. These many-body effects\nare, despite many experimental and theoretical studies since 1996\n\\cite{5Review0.7}, not yet fully understood. Recent theoretical work\n\\cite{5Meir} suggested that many-body effects cause the formation of\none or more self-consistent localized states in the QPC channel, and\nthat these effects result in the 0.7~anomaly and the other\nsignatures of many-body physics. This theoretical work predicted a\nclear dependence on the length of the QPC channel, and testing this\ndirectly requires experiments where this length is varied.\n\nThe work by Koop \\textit{et al.} \\cite{5Koop} already explored the\nrelation between the device geometry and parameters that describe\nthe many-body effects in a large set of QPC$_{\\rm 2F}$ devices. This\nwork compared nominally identical devices, and devices for which the\nlithographic length $L_{litho}$ (see Fig.~\\ref{fig1SEMpic}a) and\nwidth of the channel in the split-gate structure was varied. These\nresults were, however, not conclusive. The parameters that describe\nthe many-body effects showed large, seemingly random variation, not\ncorrelated with the device geometry. At the same time, the devices\nshowed (besides the 0.7 anomaly) clean quantized conductance traces,\nand the parameters that reflect the non-interacting electron physics\ndid show the variation that one expects when changing the geometry\n(for example, the channel pinch-off gate voltage $V_{po}$ and\nsubband spacing $\\hbar \\omega_{12}$). This confirms that these QPCs\nhad saddle-point potentials that were smooth enough for showing\nquantized conductance, while it also shows that the many-body\neffects are very sensitive to small static fluctuations on these\nsaddle-point potentials or to nanoscale device-to-device variations\nin the dimensions of the potentials. This picture was confirmed by\nshifting the channel position inside a particular QPC$_{\\rm 2F}$\ndevice. This can be implemented by operating a QPC$_{\\rm 2F}$ with a\ndifference $\\Delta V_g$ between the values of $V_g$ on the two gate\nfingers in Fig.~\\ref{fig1SEMpic}a. Such a channel shift did not\nchange the quantized conductance significantly, but did cause strong\nvariation in the signatures of many-body physics. Earlier work had\nestablished that such device-to-device fluctuations can be due to\nremote defects or impurities, a slight variation in electron\ndensity, or due to the nanoscale variation in devices that is\ninherent to the nanofabrication process\n\\cite{5Koop,5DefectsInteractions,5JohnAltDavies}. Consequently,\nstudying how the many-body effects depend on the length of the QPC\nchannel requires QPCs for which the channel length can be tuned\ncontinuously \\textit{in situ}, and where this can be operated\nwithout a transverse displacement of the QPC channel in the\nsemiconductor material. The work that we report here aimed at\nrealizing such devices.\n\n\n\n\nThis article is organized as follows:\nSection~\\ref{5sec:designConsids} starts with a short overview of the\noptions and the choices we made for realizing the QPC$_{\\rm 6F}$\ndevices. Next, in Section~\\ref{5sec:Simulations}, we present the\nresults of electrostatic simulations. In\nSection~\\ref{5sec:sampleFabAndTech}, we describe the sample\nfabrication and measurement techniques. This is followed by\ncomparing results from simulations and experiments for QPC$_{\\rm\n6F}$ devices in Section \\ref{5sec:ExperimentalRelization}, and\nSection~\\ref{5sec:Conclusion} summarizes our conclusions.\n\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{DesignQPC6F.pdf}\n\\caption{(a) Design of the geometry of the 6 gate fingers for a\nQPC$_{\\rm 6F}$ device with 6 rectangular gate fingers. (b) Design of\na QPC$_{\\rm 6F}$ device with the 4 outer gates in a shape that\nexplicitly induces a funnel shape for the entry and exit of the QPC\ntransport channel.} \\label{fig2DesignQPC6F}\n\\end{figure}\n\n\n\n\n\\section{\\label{5sec:designConsids}Design considerations}\nWe designed our QPC$_{\\rm 6F}$ devices with 6 rectangular gate\nfingers, in a symmetric layout with two sets of 3 parallel gate\nfingers (Fig.~\\ref{fig1SEMpic}b and Fig.~\\ref{fig2DesignQPC6F}a).\nSEM inspection of fabricated devices yields that the central gate\nfinger is 200~nm wide (as measured along the direction of channel\nlength $L_{eff}$). The outer gate fingers are 160~nm wide, and the\nnarrow gaps between gate fingers are 44~nm wide. This yields $(200 +\n2 \\cdot 160 + 2 \\cdot 44)~{\\rm nm} = 608~{\\rm nm}$ for the total\ndistance between the outer sides of the 3 parallel gate fingers. The\nlithographic width of the QPC channel (distance between the two sets\nof 3 gate fingers) is 350~nm.\n\nAn example of alternative designs for the gate geometries that we\nconsidered  is in Fig.~\\ref{fig2DesignQPC6F}b. This design has a\ntwo-sided funnel shape for the channel and this could result in\nlength-tunable QPC operation that better maintains a regular shape\nfor the saddle-point potential. However, the electrostatic\nsimulations in Section~\\ref{5sec:Simulations} show that the\nrectangular gate fingers as in Fig.~\\ref{fig2DesignQPC6F}a also give\na length-tunable saddle-point potential that maintains a regular\nshape while tuning the length. This observation holds for a range of\ndevice dimensions similar to our design. For our particular design,\nthe lithographic length and width (350~nm) of the channel are\ncomparable, and the 2DEG is as far as 110~nm distance below the\nsurface (and the part in the center of the channel that actually\ncontains electrons is very narrow, about 20~nm). In this regime, the\nsaddle-point potential is strongly rounded with respect to the\nlithographic shapes of the gates (see for example\nFig.~\\ref{fig3SaddlePotential}c,d). An important advantage of the\nrectangular design is that it provides two clear points for\ncalibrating the effective channel length $L_{eff}$: Operating at\n$V_{g2}\/V_{g1} = 0$ gives $L_{eff}=L_{litho}$ for the central gate\nfinger alone (200~nm, see Fig.~\\ref{fig1SEMpic}b), while operating\nat $V_{g2}\/V_{g1} = 1$ gives $L_{eff}$ equal to the lithographic\ndistance between the outer sides of the 3 parallel gate fingers\n(608~nm).\n\n\n\nA point of concern for this design that deserves attention is\nwhether the narrow gaps between the 3 parallel gate fingers induce\nsignificant structure on the saddle-point potential. The\nelectrostatic simulations show that this is not the case (see again\nthe examples in Fig.~\\ref{fig3SaddlePotential}c,d). The part of the\nchannel that contains electrons is relatively far away from the gate\nelectrodes, and the potential $U$ at this location is strongly\nrounded. Notably, the full height of the potentials in\nFig.~\\ref{fig3SaddlePotential} is about 1~eV, while the occupied\nsubbands are at a height of only about 10~meV above the stationary\npoint of the saddle-point potential (in the center of the channel).\nSuch gaps between parallel gate electrodes can be much narrower when\ndepositing a wider gate on top of the central gate, with an\ninsulating layer between them. We chose against applying this idea\nsince we also aimed to have devices with a very low level of noise\nand instabilities from charge fluctuations at defect and impurity\nsites in the device materials. In this respect, we expect better\nbehavior when all gate fingers are deposited in a single fabrication\ncycle, and when deposition of an insulating oxide or polymer layer\ncan be omitted.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{\\label{5sec:Simulations}Electrostatic simulations}\n\nThis section presents results of electrostatic simulations of the\nsaddle-point potentials that define the QPC channel. The focus is on\nthe design with 6 rectangular gate fingers\n(Fig.~\\ref{fig2DesignQPC6F}a), with gate dimensions as mentioned in\nthe beginning of Section~\\ref{5sec:designConsids}. The simulations\nare based on the modeling approach that was introduced by Davies\n\\textit{et al.} \\cite{5Davies}.\n\n\n\\subsection{Davies' method for simulating 2DEG electrostatics}\n\nDavies \\textit{et al.} \\cite{5Davies} introduced a method for\nmodeling the electrostatics of gated 2DEG. It calculates the\nelectrostatic potential $U$ for electrons in the 2DEG regions around\nthe gates (the approach only applies to the situation where the 2DEG\nunderneath the gates is depleted due to a negative voltage on gate\nelectrodes). There are other models and approaches\n\\cite{5JohnAltDavies,5SniderAltDavies,5MinhanAltDavies,5LauxAltDavies,5John1AlteDavies}\nfor calculating such potential landscapes, but these are all more\ncomplicated and computationally more demanding. The approach by\nDavies \\textit{et al.} is relatively simple. It does not account for\nelectrostatic screening effects, and, notably, it does not account\nfor the electron many-body interactions that were mentioned earlier.\nStill, it was shown that it is well suited for calculating a valid\npicture of a QPC saddle-point potential near the channel pinch-off\nsituation \\cite{5Koop}.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Saddlepotential.pdf}\n\\caption{(a),(b) Saddle-point potentials that represent the\nelectrostatic potential $U$ felt by electrons in the 2DEG plane. The\nplots represent an area of 1000$\\times$1000~${\\rm nm^2}$, centered\nat the middle of a QPC channel with a length $L_{litho}$ of 200~nm\n(a) and 600~nm (b) of a QPC$_{\\rm 2F}$ device with a lithographic\nchannel width of 350~nm. It is calculated for the material\nparameters that are valid for the measured devices. See\nFig.~\\ref{fig1SEMpic}b for relating the $x$- and $y$-direction to\nthe gate geometry. (c),(d) Similar saddle-point potentials $U$\ncalculated for QPC$_{\\rm 6F}$ devices (with material parameters and\ngeometry as the measured devices). The effective channel length is\nshorter for the case that is calculated for $V_{g2}\/V_{g1}=0.2$ (c)\nthan for the case $V_{g2}\/V_{g1}=0.8$ (d) (also note that QPC$_{\\rm\n6F}$ results for $V_{g2}\/V_{g1}=0$ are the same as plot (a)). Panel\n(c) and (d) also show that the narrow gaps between 3 parallel gate\nfingers do not induce significant structure at low energies in the\nsaddle-point passage (it only induces a weak fingerprint off to the\nside in the channel, at energies that are much higher than the\noccupied electron levels, see panel (d)).}\n\\label{fig3SaddlePotential}\n\\end{figure}\n\n\n\n\n\nThe negative voltage on a gate that is needed to exactly deplete\n2DEG underneath a large gate is called the threshold voltage\n$V_{t}$, and it is to a good approximation given by\n\\begin{equation}\nV_{t} = \\frac{-e n_{2D}d}{\\epsilon_r\\epsilon_0}. \\label{5equ:Vt}\n\\end{equation}\nHere $n_{2D}$ is the electron density in the 2DEG (at zero gate\nvoltage), $d$ is the depth of the 2DEG, $\\epsilon_r$ is the relative\ndielectric constant of the material below the gate, and $\\epsilon_0$\nis the dielectric constant of vacuum (for details see\nRef.~\\onlinecite{5Koop,5Davies}). The value of $V_t$ for a certain\n2DEG material defines the value $U_0$ where the electrostatic\npotential $U$ for electrons in the 2DEG becomes higher than the\nchemical potential of the 2DEG. In turn, this can be used to define\nin an arbitrary potential landscape $U$ (for arbitrary gates shapes\nand for arbitrary gate voltages) the positions where $U=U_0$. That\nis, one can calculate the positions in a gated device structure\nwhere there is a boundary between depleted and non-depleted 2DEG,\nand also calculate the electrostatic potential $U$ around such\npoints. When the center of the QPC has $U=U_0$, the channel is at\npinch-off and no electrons can pass through the QPC. The gate\nvoltage at which this happens is called the pinch-off voltage\n$V_{po}$. Notably, the calculated value of $U$ at a certain position\nis simply the superposition of all the contributions to $U$ from\ndifferent gate electrodes, and it is linear in the gate voltage on\neach of these electrodes \\cite{5Davies}.\n\nFigure~\\ref{fig3SaddlePotential} presents examples of saddle-point\npotentials $U$ that are calculated with Davies' method, both for\nQPC$_{\\rm 2F}$ and QPC$_{\\rm 6F}$ devices. The calculations are for\nmaterial parameters and geometries of measured devices (as described\nin detail in the next sections).\nFigures~\\ref{fig3SaddlePotential}c,d show that the length of the\ntransport channel depends on the applied ratio $V_{g2}\/V_{g1}$, and\nthat the narrow gaps between 3 parallel electrodes in QPC$_{\\rm 6F}$\ndevices do not give significant structure on the saddle-point\npotential in the operation regime that we consider.\n\n\n\n\n\\subsection{Definition and tuning of the effective length $L_{eff}$}\n\nThe focus of this work is on realizing QPC channels with a tunable\nlength. The channels are in fact saddle-point potentials (see\nFig.~\\ref{fig3SaddlePotential}), and it is for such a smooth shape\nnot obvious what the value is of the channel length. We therefore\ncharacterize this channel length with the parameter $L_{eff}$, which\ncorresponds to the value of the lithographic length $L_{litho}$ of a\nQPC$_{\\rm 2F}$ type device (with rectangular gate electrodes, see\nFig.~\\ref{fig1SEMpic}a) that gives effectively the same saddle-point\npotential.\n\n\n\\begin{figure}[!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Lalpha.pdf}\n\\caption{(a) Schematic representation of a QPC$_{\\rm 6F}$ device,\nillustrating length variables that are introduced in the main text.\n(b) The calculated length $L_{\\alpha}$ for a range of values of the\nlithographic length of QPC$_{\\rm 2F}$ devices, for three values of\n$\\alpha$. (c) The calculated effective length $L_{\\alpha}$ for\n$\\alpha=1.1$ for a QPC$_{\\rm 6F}$ device, as a function of the ratio\n$V_{g2}\/V_{g1}$.} \\label{5fig:Lalpha}\n\\end{figure}\n\n\n\n\nWe implemented this as follows. We calculated the saddle-point\npotential $U(x,y)$ for the pinch-off situation (see\nFig.~\\ref{fig1SEMpic}b and Fig.~\\ref{fig3SaddlePotential}a for how\nthe $x$- and $y$-directions are defined). The transverse confinement\nin the middle of the QPC (defined as $x=0$, $y=0$) is parabolic to a\nvery good approximation. When moving out of the channel along the\n$x$-direction, the transverse confinement becomes weaker, but\nremains at first parabolic. Notably, the energy eigenstates for\nconfinement in such a parabolic potential, described as\n\\begin{equation}\nU(y)= \\frac12 m^* \\omega_0^2 y^2,\n\\label{5equ:parabollaUy}\n\\end{equation}\nhave a width that is (for all levels) proportional to\n$\\omega_0^{-1\/2}$. In this expression $m^*$ is the effective mass of\nthe electron and $\\omega_0$ is the angular frequency of natural\noscillations in this potential. The parameter $\\omega_0$ defines\nhere the steepness of $U(y)$, and we obtain $\\omega_0 (x)$ values\nfrom fitting Eq.~\\ref{5equ:parabollaUy} to potentials $U(x,y)$\nobtained with Davies's method. We use this and investigate the width\n$\\Delta y(x)$ in $y$-direction for the lowest energy eigenstate, at\nall positions $x$ along the channel (see Fig.~\\ref{5fig:Lalpha}a).\nFor parabolic confinement this wavefunction in $y$-direction has a\nGaussian shape and has a width\n\\begin{equation}\n\\Delta y(x) = \\sqrt{\\frac{h}{4 \\pi \\; m^* \\; \\omega_0(x)}}.\n\\label{5equ:widthDeltaY}\n\\end{equation}\n\n\n\n\n\nWith this approach we analyzed that the distance from $x=0$ to the\n$x$-position $x_{\\alpha}$ where the value $\\Delta y(x)$ increased by\na factor $\\alpha \\approx 1.1$ defines a suitable point for defining\nthe value of $L_{eff}$. That is, we define\n\\begin{equation}\nL_{\\alpha} = 2 \\; x_{\\alpha},\n\\label{5equ:DEFLalpha}\n\\end{equation}\nand find $x_{\\alpha}$ by solving\n\\begin{equation}\n\\Delta y(x=x_{\\alpha}) = \\alpha \\cdot \\Delta y(x=0)\n\\label{5equ:SolveXalpha}\n\\end{equation}\nfor a certain $\\alpha$. Subsequently, $L_{eff}$ is defined by using the suitable $\\alpha$ value,\n\\begin{equation}\nL_{eff} = L_{\\alpha} \\; \\;{\\rm for} \\; \\; \\alpha=1.1.\n\\label{5equ:DEFLeffalpha}\n\\end{equation}\nWe came to this parameterization as follows. We used this\n\\textit{ansatz} first in simulations of QPC$_{\\rm 2F}$ devices.\nHere, we explored for different values of $\\alpha$ the relation\nbetween $L_{litho}$ and $L_{\\alpha}$. Results of this for $\\alpha =\n1.05$, 1.1 and 1.2 are presented in Fig.~\\ref{5fig:Lalpha}b. For the\nrange of $L_{litho}$ values that is of interest to our study\n($\\sim$100~nm to $\\sim$500~nm), we find the most reasonable overall\nagreement between the actual value for $L_{litho}$ (input to the\nsimulation) and the value $L_{\\alpha}$ (derived from the simulation)\nfor $\\alpha=1.1$. The agreement is not perfect, but we analyzed that\nthe deviation is within an uncertainty that we need to assume\nbecause the exact shapes of saddle-point potentials in different\ndevice geometries do show some variation, and because the limited\nvalidity of Davies' method. Nevertheless, it provides a reasonable\nrecipe for assigning a value $L_{eff}$ to any saddle-point\npotential, with at most 20\\% error.\n\nFig.~\\ref{5fig:Lalpha}c presents results of calculating\n$L_{\\alpha}=L_{eff}$ for $\\alpha=1.1$ from simulations of a\nQPC$_{\\rm 6F}$ device, operated at different values for\n$V_{g2}\/V_{g1}$. The results show a clear monotonic trend, with\n$L_{eff}=210~{\\rm nm}$ for $V_{g2}\/V_{g1}=0$ to $L_{eff}=525~{\\rm\nnm}$ for $V_{g2}\/V_{g1}=1$. This is for a QPC$_{\\rm 6F}$ device for\nwhich we expect $L_{eff}=200~{\\rm nm}$ for $V_{g2}\/V_{g1}=0$ and\n$L_{eff}=608~{\\rm nm}$ for $V_{g2}\/V_{g1}=1$ (see\nSection~\\ref{5sec:designConsids}). In\nSection~\\ref{5sec:ExperimentalRelization} we discuss how this latter\npoint is used for applying a small correction to the simulated\nvalues for $L_{eff}$. These simulations show that the QPC$_{\\rm 6F}$\nthat we consider allows for tuning $L_{eff}$ by about a factor 3.\n\n\n\nIt is worthwhile to note that our current design showed optimal\nbehavior in the sense that it can tune $L_{eff}$ from about 200~nm\nto 600~nm, while the dependence of $L_{eff}$ on $V_{g2}\/V_{g1}$ is\nclose to linear. We also simulated QPC$_{\\rm 6F}$ devices with wider\ngate electrodes for the outer gates, and (as mentioned in\nSection~\\ref{5sec:designConsids}) devices with gate geometries as in\nFig.~\\ref{fig2DesignQPC6F}b. These devices showed a steeper slope\nfor part of the relation between $V_{g2}\/V_{g1}$ and $L_{eff}$,\nwhich is not desirable.\n\n\n\n\n\\section{\\label{5sec:sampleFabAndTech}Sample fabrication and measurement techniques}\n\nWe fabricated QPC devices with a GaAs\/${\\rm Al}_{0.35}{\\rm\nGa}_{0.65}{\\rm As}$ MBE-grown heterostructure, which has a 2DEG at\n110~nm depth below its surface from modulation doping. The layer\nsequence and thickness of the materials from top to bottom\n(\\textit{i.e.} going into the material) starts with a $5$ nm GaAs\ncapping layer, then a 60~nm ${\\rm Al}_{0.35}{\\rm Ga}_{0.65}{\\rm As}$\nlayer with Si doping at about $1 \\times 10^{18} \\text{cm}^{-3}$,\nwhich is followed by an undoped spacer layer of 45~nm. The 2DEG is\nlocated in a heterojunction quantum well at the interface with the\nnext layer, which is a 650~nm undoped GaAs layer. This\nheterostructure was grown on a commercial semi-insulating GaAs\nwafer, after first growing a sequence of 10 GaAs\/AlAs layers for\nsmoothing the surface and trapping impurities. The 2DEG had an\nelectron density $n_{2D}=1.6 \\cdot 10^{15}~{\\rm m^{-2}}$ and a\nmobility $\\mu = 118~{\\rm m^2 V^{-1} s^{-1}}$. We fabricated both\nconventional QPC$_{\\rm 2F}$ devices and QPC$_{\\rm 6F}$ devices by\nstandard electron-beam lithography and clean-room techniques. The\ngate fingers were deposited using 15~nm Au on top of a 5~nm Ti\nsticking layer. For measuring transport through the QPCs we realized\nohmic contacts to the 2DEG reservoirs by annealing of a AuGe\/Ni\/Au\nstack that was deposited on the wafer surface\n\\cite{5OhmicKoopIqbal}. The geometries of the fabricated devices\nwere already described in the beginning of\nSection~\\ref{5sec:designConsids}.\n\nThe measurements were performed in a He-bath cryostat and in a\ndilution refrigerator, thus getting access to effective electron\ntemperatures from 80~mK to 4.2~K. We used standard lock-in\ntechniques with an a.c.~excitation voltage $V_{bias}=10~{\\rm \\mu V}$\nRMS at $387$~Hz. Fig.~\\ref{fig1SEMpic}a shows the 4-probe\nvoltage-biased measurement scheme, where both the current and the\nactual voltage drop $V_{sd}$ across the QPC channel are measured\nsuch that any influence of series resistances could be removed\nunambiguously. The gate voltages are applied with respect to a\nsingle grounded point in the loop that carries the QPC current.\n\n\n\n\n\n\\section{\\label{5sec:ExperimentalRelization}Experimental realization of length-tunable QPCs}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{IsoGfinal.pdf}\n\\caption {QPC linear conductance as a function of $V_{g1}$ and\n$V_{g2}$ for a QPC$_{\\rm 6F}$ device, presented in the form of\niso-conductance lines at integer $G_0$ levels. The conductance was\nmeasured at 4.2~K where the quantized conductance is nearly fully\nwashed out by temperature. The two operational regimes above and\nbelow the line $V_{g1}=V_{g2}$ yield QPC and quantum dot behavior,\nrespectively.} \\label{5fig:Isoconductance2D}\n\\end{figure}\n\nThis section presents an experimental characterization of the\nQPC$_{\\rm 6F}$ devices that we designed (Fig~\\ref{fig1SEMpic}b) and\nwe compare the results to our simulations.\nFigure~\\ref{5fig:Isoconductance2D} presents measurements of the\nconductance $G$ as a function of $V_{g1}$ and $V_{g2}$. Several\nlabels in the plot illustrate relevant concepts, which were partly\ndiscussed before. For the area in this plot with $V_{g2}$ more\nnegative than $V_{g1}$ we expect some quantum-dot like localization\nin the middle of the channel and this regime should therefore be\navoided in studies of QPC behavior. Further, the plot illustrates\nthat operation for a particular value of $L_{eff}$ requires\nco-sweeping of $V_{g1}$ and $V_{g2}$ from a particular point below\npinch-off in a straight line to the pivot point. This corresponds to\nopening the QPC at a fixed ratio for $V_{g2}\/V_{g1}$. The pivot\npoint is the point where the gate voltages do not alter the original\nelectron density of the 2DEG. For this measurement this is for\n$V_{g1}=V_{g2}= 0~{\\rm V}$, but this is different for the case of\nbiased cool downs. We carried out biased cool downs for suppressing\nnoise from charge instabilities in the donor layer\n\\cite{5PRBBCool,5PRLBCool}. For such experiments the QPCs were\ncooled down with a positive voltage on the gates. We typically used\n+0.3~V, and observed indeed better stability with respect to charge\nnoise. The effect of such a cool down can be described as a\ncontribution to the gate voltage of -0.3~V that is frozen into the\nmaterial \\cite{5PRBBCool,5PRLBCool}. Consequently, co-sweeping of\n$V_{g1}$ and $V_{g2}$ for maintaining a fixed channel length must\nnow be carried out with respect to the pivot point $V_{g1}=V_{g2}= +\n0.3~{\\rm V}$ instead of $V_{g1}=V_{g2}= 0~{\\rm V}$.\n\n\n\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{LsimuVsMeasured.pdf}\n\\caption{(a) Experimentally determined relation between the\npinch-off gate voltage $V_{po}$ and the lithographic length of\nQPC$_{\\rm 2F}$ devices. Points are experimental results. The solid\nline is a phenomenological expression that was used for\nparameterizing the relation between $V_{po}$ and the lithographic\nlength. (b) Comparison between measured and simulated values of the\neffective channel length for a QPC$_{\\rm 6F}$ device.}\n\\label{5fig:LalphaVsmeasured}\n\\end{figure}\n\n\n\nThe theory behind the Davies method illustrates why operation at\nfixed effective lenght requires a fixed ratio $V_{g2}\/V_{g1}$. All\npoints in the potentials landscapes $U$ for QPC$_{\\rm 2F}$ devices\nas in Fig.~\\ref{fig3SaddlePotential}a,b have a height that scales\nlinear with the gate voltage $V_g$. Thus, when opening the QPC the\nfull saddle-point potential changes height at a fixed shape.\nMimicking this situation with QPC$_{\\rm 6F}$ devices requires a\nfixed ratio $V_{g2}\/V_{g1}$, again because $V_{g1}$ and $V_{g2}$\ninfluence $U$ in a linear manner. The plot also illustrates the two\nspecial operation lines where the effective length of the channel is\nunambiguous, and we used these points to better calibrate the\nrelation between $V_{g2}\/V_{g1}$ and $L_{eff}$. The first case is\nthe line at $V_{g2}=0$, which yields $L_{eff}=200~{\\rm nm}$, as\ndefined by the central gates alone. The second case is the line\n$V_{g1}=V_{g2}$. Here $L_{eff}$ is 608~nm, as defined by the full\nlithographic length of the 3 gate fingers.\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{LinearG.pdf}\n\\caption{The linear conductance $G$ as a function of $V_{g1}$ for a\nQPC$_{\\rm 6F}$ device, measured at 80~mK. Different traces are for\nfixed ratio $V_{g2}\/V_{g1}=0$ to $V_{g2}\/V_{g1}=1$ (left to right,\ntraces \\textit{not} offset), which corresponds to increasing the\neffective channel length $L_{eff}$ from 200~nm to 608~nm.}\n\\label{5fig:LinearG}\n\\end{figure}\n\n\n\nWe improved and further checked our calibration of the relation\nbetween $V_{g2}\/V_{g1}$ and $L_{eff}$ as follows. We used the trend\nthat came out of the simulations (Fig.~\\ref{5fig:Lalpha}c), but\npinned the curve at 200~nm for $V_{g2}\/V_{g1}=0$ and at 608~nm for\n$V_{g2}\/V_{g1}=1$ (black line in Fig.~\\ref{5fig:LalphaVsmeasured}b).\nThis trace shows good agreement with results from an independent\ncheck (dashed line) that used the pinch-off gate voltage $V_{po}$ as\nan identifier for the effective length. This independent check used\ndata from a set of QPC$_{\\rm 2F}$ devices for calibrating the\nrelation between $L_{litho}$ and $V_{po}$\n(Fig.~\\ref{5fig:LalphaVsmeasured}a). This shows the trend that\nshorter QPC$_{\\rm 2F}$ devices require a more negative gate voltage\nto reach pinch-off \\cite{5VpvsL}. We related this to the pinch-off\nvalues in QPC$_{\\rm 6F}$ devices. In particular, we analyzed the\npinch-off points on the $V_{g1}$ axis, and its dependence on\n$V_{g2}\/V_{g1}$ (see also Fig.~\\ref{5fig:LinearG}). The results of\nusing this for assigning a certain $L_{eff}$ to each $V_{g2}\/V_{g1}$\nis the dashed line in Fig.~\\ref{5fig:LalphaVsmeasured}b, and shows\ngood agreement with the values that were obtained from simulations.\nWe can thus assign a value to $L_{eff}$ for each $V_{g2}\/V_{g1}$\nwith an absolute error that is at most 50~nm. Notably, the relative\nerror when describing the increase in $L_{eff}$ upon increasing\n$V_{g2}\/V_{g1}$ is much smaller.\n\n\n\nThe results in Fig.~\\ref{5fig:LinearG} provide an example of linear\nconductance measurements on a QPC$_{\\rm 6F}$ device at 80~mK. The\ntraces show clear quantized conductance plateaus for all settings of\n$L_{eff}$. Several of these linear conductance traces also show the\n0.7~anomaly, and the strength of its (here rather weak) expression\nshows a modulation as a function of $L_{eff}$ over about 3 periods.\nA detailed study of this type of periodicity can be found in\nRef.~\\onlinecite{5IqbalPerio}. This example illustrates the validity\nand importance of our type of QPCs in studies of length-dependent\ntransport properties.\n\n\n\n\n\n\n\n\n\\section{\\label{5sec:Conclusion}Conclusions}\n\nWe have developed and characterized length-tunable QPCs that are\nbased on a symmetric split-gate geometry with 6 gate fingers. Gate\nstructures with different shapes and dimensions can be designed\ndepending upon the required range for length tuning and for\noptimizing the tuning curve. For our purpose (QPCs with an effective\nchannel length between about 200~nm and 600~nm, and 350~nm channel\nwidth) we found that simple rectangular gate fingers are an\nattractive choice. Our simulations and experimental results are in\nclose agreement. We were able to tune the effective length by about\na factor 3, from 200~nm to 608~nm. QPCs are the simplest devices\nthat show clear signatures of many-body physics, as for example the\n$0.7$ anomaly and the Zero-Bias Anomaly (ZBA) \\cite{Cronenwett2002}.\nOur length-tunable QPCs provide an interesting platform for\nsystematically investigating these many-body effects. In particular,\nthese QPCs provide a method for studying the influence of the QPC\ngeometry without suffering from device-to-device fluctuations that\nhamper such studies in conventional QPCs with 2 gate fingers.\nStudies in this direction are presented in\nRef.~\\onlinecite{5IqbalPerio}.\n\n\\section*{ACKNOWLEDGMENTS}\nWe thank Y.~Meir for discussions, B.~H.~J.~Wolfs for technical\nassistance, and the German programs DFG-SPP 1285, Research school\nRuhr-Universit\\\"{a}t Bochum and BMBF QuaHL-Rep 01BQ1035 for\nfinancial support. MJI acknowledges a scholarship from the Higher\nEducation Commission of Pakistan.\n\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}