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arXiv:1609.01191

Trace formula for spin chains

Daniel Waltner, Petr A. Braun, Maram Akila, Thomas Guhr

Published Sep 5, 2016• 1 citation
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Abstract

While detailed information about the semiclassics for single-particle systems is available, much less is known about the connection between quantum and classical dynamics for many-body systems. As an example, we focus on spin chains which are of considerable conceptual and practical importance. We derive a trace formula for coupled spin $j$ particles which relates the quantum energy levels to the classical dynamics. Our derivation is valid in the limit $j\rightarrow\infty$ with $j\hbar={\rm const.}$ and applies to time-continuous as well as to periodically driven dynamics. We provide a simple explanation why the Solari-Kochetov phase can be omitted if the correct classical Hamiltonian is chosen.

DOI
10.48550/arxiv.1609.01191

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While detailed information about the semiclassics for single-particle systems is available, much less is known about the connection between quantum and classical dynamics for many-body systems. As an example, we focus on spin chains which are of considerable conceptual and practical importance. We derive a trace formula for coupled spin jjj particles which relates the quantum energy levels to the classical dynamics. Our derivation is valid in the limit j→∞j\rightarrow\inftyj→∞ with jℏ=const.j\hbar={\rm const.}jℏ=const. and applies to time-continuous as well as to periodically driven dynamics. We provide a simple explanation why the Solari-Kochetov phase can be omitted if the correct classical Hamiltonian is chosen.
Abstract While detailed information about the semiclassics for single-particle systems is available, much less is known about the connection between quantum and classical dynamics for many-body systems. As an example, we focus on interacting spins, which are of considerable conceptual and practical importance. We derive a trace formula for coupled spin j particles which relates the quantum energy levels to the classical dynamics. Our derivation is valid in the limit j → ∞ with j ħ = c o n s t . and applies to time-continuous as well as to periodically driven dynamics. We provide a simple explanation why the Solari–Kochetov phase can be omitted if the correct classical Hamiltonian is chosen.
Many-body quantum chaos at ℏ∼1\hbar\sim 1ℏ∼1: Analytic connection to random matrix theory
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t)K(t)K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute K(t)K(t)K(t) explicitly in the leading two orders in ttt and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.
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.
We present a detailed derivation of the trace formula for a general Hamiltonian with two degrees of freedom where one of them is canonical and the other a spin. Our derivation starts from the semiclassical formula for the propagator in a basis formed by the product of a canonical and a spin coherent states and is valid in the limit ℏ→0, j→∞ with jℏ constant. The trace formula, obtained by taking the trace and the Fourier transform of the coherent state propagator, is compared to others found in the literature.
Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in single\textit{single}single-particle or at least few\textit{few}few-particle systems. Challenges on the way to an extension to many\textit{many}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.
We investigate the Dirac equation in the semiclassical limit $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$. A semiclassical propagator and a trace formula are derived and are shown to be determined by the classical orbits of a relativistic point particle. In addition, two phase factors enter, one of which can be calculated from the Thomas precession of a classical spin transported along the particle orbits. For the second factor we provide an interpretation in terms of dynamical and geometric phases.