A unified approach to periodic orbits formulas : interference in semiclassical regime of quantum system with chaotic classical limit

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.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.

quant-pharXiv:2605.19019May 20, 2026

Semiclassical aspects of (chaotic) quantum systems

Klaus Richter

Mesoscopic Physics in Complex MediaDec 1, 20100 citations

The role of classical periodic orbits in quantum many-body systems

Daniel Waltner, Boris Gutkin

Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in $\textit{single}$ -particle or at least $\textit{few}$ -particle systems. Challenges on the way to an extension to $\textit{many}$ -body systems result from the exponential proliferation of the number of classical orbits in chaotic systems and the exponential growth of the quantum Hilbert-space dimension with the particle number. To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number.Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in $\textit{single}$-particle or at least $\textit{few}$-particle systems. Challenges on the way to an extension to $\textit{many}$-body systems result from the exponential proliferation of the number of classical orbits in chaotic systems and the exponential growth of the quantum Hilbert-space dimension with the particle number. To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number.

quant-pharXiv:2604.14015Apr 16, 20260 citations

Semiclassical roots of universality in many-body quantum chaos

Klaus Richter, Juan Diego Urbina, Steven Tomsovic

Abstract Quantum chaos of many-body (MB) systems has been swiftly developing into a vibrant research area at the interface between various disciplines, ranging from statistical physics to condensed matter to quantum information and to cosmology. In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between classically chaotic dynamics and corresponding universal features at the quantum level. Recently, single-particle (SP) techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ → 0 have begun to be transformed into the field theoretical domain of N -particle systems in the analogous semiclassical limit ℏ e f f = 1 / N → 0 , thereby accounting for genuine MB quantum interference. This semiclassical MB theory provides a unified framework for understanding random-matrix correlations of both SP and MB quantum chaotic systems. Certain braided bundles of classical orbits, and of mean field modes, govern interference, respectively, and provide the key to the foundation of universality. Case studies presented include an MB version of Gutzwiller’s trace formula for the spectral density and out-of-time-order correlators along with brief remarks on where further progress may be forthcoming.Abstract Quantum chaos of many-body (MB) systems has been swiftly developing into a vibrant research area at the interface between various disciplines, ranging from statistical physics to condensed matter to quantum information and to cosmology. In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between classically chaotic dynamics and corresponding universal features at the quantum level. Recently, single-particle (SP) techniques dealing with ergodic wave interference in the usual semiclassical limit ℏ → 0 have begun to be transformed into the field theoretical domain of N -particle systems in the analogous semiclassical limit ℏ e f f = 1 / N → 0 , thereby accounting for genuine MB quantum interference. This semiclassical MB theory provides a unified framework for understanding random-matrix correlations of both SP and MB quantum chaotic systems. Certain braided bundles of classical orbits, and of mean field modes, govern interference, respectively, and provide the key to the foundation of universality. Case studies presented include an MB version of Gutzwiller’s trace formula for the spectral density and out-of-time-order correlators along with brief remarks on where further progress may be forthcoming.

arXiv:2205.02867Journal of Physics A Mathematical and TheoreticalNov 1, 202237 citations

Interference of Quantum Chaotic Systems in Phase Space

徐酉阳

理论物理通讯：英文版Jan 1, 20130 citations

Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in

\textit{single}

-particle or at least

\textit{few}

-particle systems. Challenges on the way to an extension to

\textit{many}

-body systems result from the exponential proliferation of the number of classical orbits in chaotic systems and the exponential growth of the quantum Hilbert-space dimension with the particle number. To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number.