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Journal of Physics A Mathematical and Theoretical

Trace formula for interacting spins

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

Published Dec 21, 2016• 8 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/aa5533

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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 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.
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.
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.
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.
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.
Abstract One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for <?CDATA N=2,3N = 2,3N=2,3?> N = 2 , 3 and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.
We study a model system composed of interacting quantum spins, with every spin coupled to every other, in the limit where the number of spins K and the angular momentum j of each spin are both large, aiming to explore the effect of large system size on the breakdown of the classical limit of quantum mechanics. We obtain the exact spectrum of the Hamiltonian, and hence the trace of the quantum-mechanical time-evolution operator. We examine the time dependence of , finding a simple approximation which is valid when j and K are large. At a time proportional to , this approximation breaks down, and the long-time behaviour is extremely complex. We use a renormalization scheme to investigate this complexity. The scheme is based upon a generalization of the Gauss continued-fraction map to the complex plane.