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arXiv:2411.10691

Exact Quantum Trace Formula from Complex Periodic Orbits

Chaoming Song

Published Nov 16, 2024• 0 citations
arXivHTMLPDF

Abstract

The Gutzwiller trace formula establishes a profound connection between the quantum spectrum and classical periodic orbits. However, its application is limited by its reliance on the semiclassical saddle point approximation. In this work, we explore the full quantum version of the trace formula using the Lefschetz thimble method by incorporating complexified periodic orbits. Upon complexification, classical real periodic orbits are transformed into cycles on compact Riemann surfaces. Our key innovation lies in the simultaneous complexification of the periods of cycles, resulting in a fully quantum trace formula that accounts for all contributions classified by the homology classes of the associated Riemann surfaces. This formulation connects the quantum spectrum to contributions across all complex time directions, encompassing all relevant homology classes. Our approach naturally unifies and extends two established methodologies: periodic orbits in real time, as in Gutzwiller's original work, and quantum tunneling in imaginary time, as in the instanton method.

DOI
10.48550/arxiv.2411.10691

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The Gutzwiller trace formula establishes a profound connection between the quantum spectrum and classical periodic orbits. However, its application is limited by its reliance on the semiclassical saddle point approximation. In this work, we explore the full quantum version of the trace formula using the Lefschetz thimble method by incorporating complexified periodic orbits. Upon complexification, classical real periodic orbits are transformed into cycles on compact Riemann surfaces. Our key innovation lies in the simultaneous complexification of the periods of cycles, resulting in a fully quantum trace formula that accounts for all contributions classified by the homology classes of the associated Riemann surfaces. This formulation connects the quantum spectrum to contributions across all complex time directions, encompassing all relevant homology classes. Our approach naturally unifies and extends two established methodologies: periodic orbits in real time, as in Gutzwiller's original work, and quantum tunneling in imaginary time, as in the instanton method.
Abstract It is well known that quantum tunneling can be described by instantons in the imaginary-time path integral formalism. However, its description in the real-time path integral formalism has been elusive. Here we establish a statement that quantum tunneling can be characterized in general by the contribution of complex saddle points, which can be identified by using the Picard-Lefschetz theory. We demonstrate this explicitly by performing Monte Carlo simulations of simple quantum mechanical systems, overcoming the sign problem by the generalized Lefschetz thimble method. We confirm numerically that the contribution of complex saddle points manifests itself in a complex “weak value” of the Hermitian coordinate operator x^ \hat{x} x^ x ̂ evaluated at time t, which is a physical quantity that can be measured by experiments in principle. We also discuss the transition to classical dynamics based on our picture.
We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.
Abstract We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.
Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.
Abstract Using the path integral formalism, quantum-mechanical tunneling is described by instantons—solutions to the equations of system motion in imaginary time. However, in imaginary time, this formalism is suitable only for studying the characteristics of a system in thermodynamic equilibrium and is not applicable to significantly nonequilibrium phenomena. Therefore, it is necessary to develop an approach that takes into account instanton solutions within the framework of the real-time path integral. This paper takes a step in this direction by generalizing the Gutzwiller trace formula, derived within the real-time formalism, for the density of states of a quantum-mechanical system in a tunnel-type potential. It is shown that to obtain real values of the level splitting in a symmetric double-well potential, it is necessary to take into account approximate solutions to the equations of motion in imaginary time, corresponding to multi-instanton configurations.
Abstract Out-of-time-order correlators (OTOCs) quantify quantum information scrambling, but their connection to localized phase-space structures—such as chemical transition states—has remained largely unexplored. We derive a formal leading-order semiclassical expansion for the microcanonical OTOC in systems with an index-1 saddle point, expressing its oscillatory, geometry-resolving contribution as a coherent sum over resonant invariant tori of the normally hyperbolic invariant manifold (NHIM), each a family of periodic orbits, transversely unstable in the full phase space. Although formally subleading in ℏ to the smooth Weyl contribution, this orbit sum carries the localized exponent 2 Λ γ , compared with the smooth exponent Λ max , and it is the only contribution that resolves the periodic-orbit geometry of the transition state: within the local normal-form description, the NHIM is the one structure on the energy shell whose trajectories never escape the saddle neighborhood. Valid in the semiclassical limit and the intermediate-time regime before the Ehrenfest time, our derivation utilizes the normal form theory of the transition state, which transforms the Hamiltonian near the saddle into an integrable (though generally non-separable) form dependent on conserved actions. We outline the derivation of the microcanonical trace, the semiclassical propagator for integrable systems, the factorization of the stability matrix, and the explicit Schur complement reduction of the stationary phase approximation. Our result extends periodic-orbit trace methods to scrambling observables, recovering a local instability exponent Λ ( J ) governing the leading semiclassical growth window. Finally, we discuss how the dependence of the instability on transverse actions suggests a mechanism for mode-selective control of scrambling, and outline a numerical evaluation strategy to test these predictions.