Gutzwiller Trace Formula Research Articles

Exact quantization conditions and full transseries structures for PT symmetric anharmonic oscillators

We study exact Wentzel–Kramers–Brillouin analysis (EWKB) for a PT symmetric quantum mechanics (QM) defined by the potential VPT(x)=ω2x2+gx2K(ix)ϵ with ω∈R≥0, g∈R>0 and K,ϵ∈N to clarify its perturbative/nonperturbative structure. In our analysis, we mainly consider the massless cases, i.e., ω=0, and derive the exact quantization conditions (QCs) for arbitrary (K,ϵ) including all perturbative/nonperturbative corrections. From the exact QCs, we clarify full transseries structure of the energy spectra with respect to the inverse energy level expansion, and then formulate the Gutzwiller trace formula, the spectral summation form, and the Euclidean path integral. For the massive cases, i.e., ω>0, we show the fact that, by requiring the existence of the solution of the exact QCs, the path of analytic continuation in EWKB is uniquely determined for a given N=2K+ϵ, and in consequence the exact QCs, the energy spectra, and the three formulas are all perturbative. Similarities to Hermitian QMs and resurgence are also discussed as additional remarks. Published by the American Physical Society 2024

AdS3/RMT2 duality

We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS3 quantum gravity. We present a 2d CFT trace formula, precisely analogous to the Gutzwiller trace formula for chaotic quantum systems, which originates from the SL(2, ℤ) spectral decomposition of the Virasoro primary density of states. An analogy to Berry’s diagonal approximation allows us to extract spectral statistics of individual 2d CFTs by coarse-graining, and to identify signatures of chaos and random matrix universality. This leads to a necessary and sufficient condition for a 2d CFT to display a linear ramp in its coarse-grained spectral form factor.Turning to gravity, AdS3 torus wormholes are cleanly interpreted as diagonal projections of squared partition functions of microscopic 2d CFTs. The projection makes use of Hecke operators. The Cotler-Jensen wormhole of AdS3 pure gravity is shown to be extremal among wormhole amplitudes: it is the minimal completion of the random matrix theory correlator compatible with Virasoro symmetry and SL(2, ℤ)-invariance. We call this MaxRMT: the maximal realization of random matrix universality consistent with the necessary symmetries. Completeness of the SL(2, ℤ) spectral decomposition as a trace formula allows us to factorize the Cotler-Jensen wormhole, extracting the microscopic object ZRMT(τ) from the coarse-grained product. This captures details of the spectrum of BTZ black hole microstates. ZRMT(τ) may be interpreted as an AdS3 half-wormhole. We discuss its implications for the dual CFT and modular bootstrap at large central charge.

A Gutzwiller Trace Formula for Dirac Operators on a Stationary Spacetime

A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2021) to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.

Trace formula for the magnetic Laplacian at zero energy level

The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of Gutzwiller's semiclassical trace formula and reduces to it in the case when the magnetic field form is exact. It differs slightly from the Guillemin-Uribe trace formula considered in a previous paper of the author and Taimanov. Moreover, in contrast to that paper, the focus is on the trace formula at the zero energy level, which is a critical energy level. An overview of the main notions and results related to the trace formula at the zero energy level is presented, various approaches to its proof are described, and concrete examples of its computation are given. In addition, a brief review of Gutzwiller's trace formula for regular and critical energy levels is presented. Bibliography: 88 titles.

A Gutzwiller trace formula for stationary space-times

We give a relativistic generalization of the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space ker⁡□ of the wave operator and define U(t) to be translation by the flow etZ of the timelike Killing vector field Z on □. The spectrum of Z on ker⁡□ is discrete and the singularities of TretZ|ker⁡□ occur at periods of periodic orbits of exp⁡tZ on the symplectic manifold of null geodesics. The trace formula gives a Weyl law for the eigenvalues of Z on ker⁡□.

Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant and Souriau. We express the associated evolution operators via parallel transport in the quantum spaces over the induced path of almost complex structures, and we establish various semi-classical estimates. In particular, we establish a Gutzwiller trace formula for the Kostant-Souriau operator and compute explicitly the leading term. We then describe a potential application to contact topology.

A Gutzwiller trace formula for large hermitian matrices

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schrödinger equation. In the framework of Weyl quantization, we construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantization condition for eigenvalues as well as some simple examples.

Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula with applications to graphene-type systems

We present a tractable and physically transparent semiclassical theory of matrix-valued Hamiltonians, i.e., those that describe quantum systems with internal degrees of freedoms, based on a generalization of the Gutzwiller trace formula for a $n\ifmmode\times\else\texttimes\fi{}n$ dimensional Hamiltonian $H(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{q}})$. The classical dynamics is governed by $n$ Hamilton-Jacobi (HJ) equations that act in a phase space endowed with a classical Berry curvature encoding anholonomy in the parallel transport of the eigenvectors of $H(\mathbf{p},\mathbf{q})$; these vectors describe the internal structure of the semiclassical particles. At the $\mathcal{O}({\ensuremath{\hbar}}^{1})$ level and for nondegenerate HJ systems, this curvature results in an additional semiclassical phase composed of (i) a Berry phase and (ii) a dynamical phase resulting from the classical particles ``moving through the Berry curvature''. We show that the dynamical part of this semiclassical phase will, generally, be zero only for the case in which the Berry phase is topological (i.e., depends only on the winding number). We illustrate the method by calculating the Landau spectrum for monolayer graphene, the four-band model of AB bilayer graphene, and for a more complicated matrix Hamiltonian describing the silicene band structure. Finally, we apply our method to an inhomogeneous system consisting of a strain engineered one-dimensional moir\'e in bilayer graphene, finding localized states near the Dirac point that arise from electron trapping in a semiclassical moir\'e potential. The semiclassical density of states of these localized states we show to be in perfect agreement with an exact quantum mechanical calculation of the density of states.

Geometric determination of classical actions of heteroclinic and unstable periodic orbits.

Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase-space areas bounded by segments of stable and unstable manifolds and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following from the exact relations are given with error estimates. They arise from asymptotic scaling of certain bounded phase-space areas. The actions of infinite subsets of periodic orbits are determined by their periods and the locations of the limiting homoclinic points on which they accumulate.

Local Scaling Asymptotics for the Gutzwiller Trace Formula in Berezin–Toeplitz Quantization

Let f be a smooth function on a quantizable compact Kahler manifold, with the special property that the associated Hamiltonian flow acts by isometries. We consider a naturally associated Berezin–Toeplitz operator, which generates a unitary action on the Hardy space of the quantization. In this setting, we produce local scaling asymptotics in the semiclassical regime for a Berezin–Toeplitz version of the Gutzwiller trace formula, in the spirit of the near-diagonal scaling asymptotics of Szego and Toeplitz kernels. More precisely, we consider an analogue of the ‘Gutzwiller–Toeplitz kernel’ previously introduced in this setting by Borthwick, Paul, and Uribe, and study how it asymptotically concentrates along the appropriate classical loci defined by the dynamics, with an explicit description of the exponential decay along normal directions. These local scaling asymptotics probe into the concentration behavior of the eigenfunctions of the quantized Hamiltonian flow. When globally integrated, they yield the analogue of the Gutzwiller trace formula.

Exceptional points in the elliptical three-disk scatterer using semiclassical periodic orbit quantization

The three-disk scatterer has served as a paradigm for semiclassical periodic orbit quantization of classical chaotic systems using Gutzwiller's trace formula. It represents an open quantum system, thus leading to spectra of complex eigenenergies. An interesting general feature of open quantum systems described by non-Hermitian operators is the possible existence of exceptional points where not only the complex eigenvalues but also their respective eigenvectors coincide. Using Gutzwiller's periodic orbit theory we show that exceptional points exist in a three-disk scatterer if the system's geometry is modified by extending the system from circular to elliptical disks. The extension is implemented in such a way that the system's characteristic symmetry is preserved. The two-dimensional parameter plane of the system is then spanned by the distance between and the excentricity of the elliptical disks. As typical signatures of exceptional points we observe the permutation of two resonances when an exceptional point is encircled in parameter space, and a non-Lorentzian resonance line shape in the weighted density of states.

Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential

We have derived an analytical trace formula for the level density of the Hénon–Heiles potential using the improved stationary phase method, based on extensions of Gutzwiller's semiclassical path integral approach. This trace formula has the correct limit to the standard Gutzwiller trace formula for the isolated periodic orbits far from all (critical) symmetry-breaking points. It continuously joins all critical points at which an enhancement of the semiclassical amplitudes occurs. We found a good agreement between the semiclassical and the quantum oscillating level densities for the gross shell structures and for the energy shell corrections, solving the symmetry breaking problem at small energies.

Equivariant spectral asymptotics for h-pseudodifferential operators

We prove equivariant spectral asymptotics for h-pseudodifferential operators for compact orthogonal group actions generalizing results of El Houakmi and Helffer [“Comportement semi-classique en présence de symétries: Action d'un groupe de Lie compact,” Asymp. Anal. 5(2), 91–113 (1991)] and Cassanas [“Reduced Gutzwiller formula with symmetry: Case of a Lie group,” J. Math. Pures Appl. 85(6), 719–742 (2006)]. Using recent results for certain oscillatory integrals with singular critical sets [P. Ramacher, “Singular equivariant asymptotics and Weyl's law: On the distribution of eigenvalues of an invariant elliptic operator,” J. Reine Angew. Math. (Crelles J.) (to be published)], we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow.

Phonon-induced superconductivity at high temperatures in electrical graphene superlattices

Due to the chiral nature of the Dirac equation, overlying of an electrical superlattice (SL) can open new Dirac points on the Fermi-surface of the energy spectrum. These lead to novel low-excitation physical phenomena. A typical example for such a system is neutral graphene with a symmetrical unidirectional SL. We show here that in smooth SLs, a semiclassical approximation provides a good mathematical description for particles. Due to the one-dimensional nature of the unidirectional potential, a wavefunction description leads to a generalized Bohr-Sommerfeld quantization condition for the energy eigenvalues. In order to pave the way for the application of semiclassical methods to two dimensional SLs in general, we compare these energy eigenvalues with those obtained from numerical calculations, and with the results from a semiclassical Gutzwiller trace formula via the beam-splitting technique. Finally, we calculate ballistic conductivities in general point-symmetric unidirectional SLs with one electron and one hole region in the fundamental cell showing only Klein scattering of the semiclassical wavefunctions.

Hadrons on the worldline, holography, and Wilson flow

Holographic principles have impacted the way we look at strong coupling phenomena in quantum chromodynamics, strongly interacting extensions of the standard model, and {condensed-matter} physics. In real world settings, however, we still lack understanding of why and when such an approach is justified. Therefore, here, without invoking any such principle a priori, we demonstrate how such a picture arises in the worldline formulation of quantum field theory. Among other connections to holographic models, a warped AdS5 geometry, a quantum mechanical picture, and hidden local symmetry emerge, as well as a Wilson flow (gradient flow), which extends the four-dimensional sources to five-dimensional fields and a link to the Gutzwiller trace formula. The worldline formulation also reproduces the non-relativistic case, which is important for condensed-matter physics.

Semiclassical approach to the physics of smooth superlattice potentials in graphene

Due to the chiral nature of the Dirac equation, governing the dynamics of electrons in graphene, overlying of an electrical superlattice (SL) can open new Dirac points on the Fermi surface of the energy spectrum. These lead to novel low-excitation physical phenomena. A typical example for such a system is neutral graphene with a symmetrical unidirectional SL. We show here that in smooth SLs, a semiclassical approximation provides a good mathematical description for particles. Due to the one-dimensional nature of the unidirectional potential, a wave-function description leads to a generalized Bohr-Sommerfeld quantization condition for the energy eigenvalues. In order to pave the way for the application of semiclassical methods to two-dimensional SLs in general, we compare these energy eigenvalues with those obtained from numerical calculations and with the results from a semiclassical Gutzwiller trace formula via the beam-splitting technique. Finally, we calculate ballistic conductivities in general point-symmetric unidirectional SLs with one electron and one hole region in the fundamental cell showing only Klein scattering of the semiclassical wave functions.

Semiclassical Quantization and Periodic Orbits of Dispersing Billiards

Gutzwiller's trace formula, the Riemann-Siegel lookalike formula, and Novel quantization are applied to the dispersing billiards. All three of these give remarkable agreement with quantum results. The statistical properties of periodic orbits are also found to be universal in strongly chaotic systems.

Comment on the Breakdown of Van Vleck's Propagator and Its Implications for the Energy Domain Green's Function

Van Vleck's propagator is perhaps one of the most fundamental objects of semiclassical theory. Its Laplace transform gives the energy Green's function G(x, x′, E), which is the starting point for the derivation of Gutzwiller trace formula. The validity of Van Vleck's formula for chaotic and mixed systems is therefore of crucial importance, especially since during the derivation of the trace formula it is assumed to be valid for infinite time. In the last two years there has been numerical evidence for the time-scale validity of the semiclassical propagator when the underlying dynamics is chaotic. The time-scales were longer that previously predicted by Tabor et al. An explanation for this long time validity based on the multidimensional stationary phase approximation was recently proposed by Sepúlveda-Tomsovic-Heller. In this contribution we would like to review these recent results and comment on their implications in the energy domain and particularly in the validity of the trace formula.

Edge effects in graphene nanostructures: Semiclassical theory of spectral fluctuations and quantum transport

We investigate the effect of different edge types on the statistical properties of both the energy spectrum of closed graphene billiards and the conductance of open graphene cavities in the semiclassical limit. To this end, we use the semiclassical Green's function for ballistic graphene flakes that we have derived in Reference 1. First we study the spectral two point correlation function, or more precisely its Fourier transform the spectral form factor, starting from the graphene version of Gutzwiller's trace formula for the oscillating part of the density of states. We calculate the two leading order contributions to the spectral form factor, paying particular attention to the influence of the edge characteristics of the system. Then we consider transport properties of open graphene cavities. We derive generic analytical expressions for the classical conductance, the weak localization correction, the size of the universal conductance fluctuations and the shot noise power of a ballistic graphene cavity. Again we focus on the effects of the edge structure. For both, the conductance and the spectral form factor, we find that edge induced pseudospin interference affects the results significantly. In particular intervalley coupling mediated through scattering from armchair edges is the key mechanism that governs the coherent quantum interference effects in ballistic graphene cavities.

Edge effects in graphene nanostructures: From multiple reflection expansion to density of states

We study the influence of different edge types on the electronic density of states of graphene nanostructures. To this end we develop an exact expansion for the single particle Green's function of ballistic graphene structures in terms of multiple reflections from the system boundary, that allows for a natural treatment of edge effects. We first apply this formalism to calculate the average density of states of graphene billiards. While the leading term in the corresponding Weyl expansion is proportional to the billiard area, we find that the contribution that usually scales with the total length of the system boundary differs significantly from what one finds in semiconductor-based, Schr\"odinger type billiards: The latter term vanishes for armchair and infinite mass edges and is proportional to the zigzag edge length, highlighting the prominent role of zigzag edges in graphene. We then compute analytical expressions for the density of states oscillations and energy levels within a trajectory based semiclassical approach. We derive a Dirac version of Gutzwiller's trace formula for classically chaotic graphene billiards and further obtain semiclassical trace formulae for the density of states oscillations in regular graphene cavities. We find that edge dependent interference of pseudospins in graphene crucially affects the quantum spectrum.