Ergodicity is a fundamental principle of statistical mechanics underlying the behavior of generic quantum many-body systems.However, how this universal many-body quantum chaotic regime emerges due to interactions remains largely a puzzle.This paper demonstrates using both heuristic arguments and a microscopic calculation that a dephasing mechanism, similar to Altshuler-Aronov-Khmelnitskii dephasing in the theory of localization, underlies this transition to chaos.We focus on the behavior of the spectral form factor (SFF) as a function of "time" t, which characterizes level correlations in the many-body spectrum.The SFF can be expressed as a sum over periodic classical orbits and its behavior hinges on the interference of trajectories related to each other by a time translation.In the absence of interactions, time-translation symmetry is present for each individual particle, which leads to a fast exponential growth of the SFF and correspondingly loss of correlations between many-body levels.Interactions lead to dephasing, which disrupts interference, and breaks the massive time-translation symmetry down to a global time-translation/energy conservation.This in turn gives rise to the hallmark linear-in-t ramp in the SFF reflecting Wigner-Dyson level repulsion.This general picture is supported by a microscopic analysis of an interacting many-body model.Specifically, we study the complex SYK 2 + SYK 2 2 model, which allows to tune between an integrable and chaotic regime.It is shown that the dephasing mass vanishes in the former case, which maps to the noninteracting complex SYK 2 model via a time reparameterization.In contrast, the chaotic regime gives rise to dephasing, which suppresses the exponential ramp of the noninteracting theory and induces correlations between many-body levels.
Ergodicity is a fundamental principle of statistical mechanics underlying the behavior of generic quantum many-body systems.However, how this universal many-body quantum chaotic regime emerges due to interactions remains largely a puzzle.This paper demonstrates using both heuristic arguments and a microscopic calculation that a dephasing mechanism, similar to Altshuler-Aronov-Khmelnitskii dephasing in the theory of localization, underlies this transition to chaos.We focus on the behavior of the spectral form factor (SFF) as a function of "time" t, which characterizes level correlations in the many-body spectrum.The SFF can be expressed as a sum over periodic classical orbits and its behavior hinges on the interference of trajectories related to each other by a time translation.In the absence of interactions, time-translation symmetry is present for each individual particle, which leads to a fast exponential growth of the SFF and correspondingly loss of correlations between many-body levels.Interactions lead to dephasing, which disrupts interference, and breaks the massive time-translation symmetry down to a global time-translation/energy conservation.This in turn gives rise to the hallmark linear-in-t ramp in the SFF reflecting Wigner-Dyson level repulsion.This general picture is supported by a microscopic analysis of an interacting many-body model.Specifically, we study the complex SYK 2 + SYK 2 2 model, which allows to tune between an integrable and chaotic regime.It is shown that the dephasing mass vanishes in the former case, which maps to the noninteracting complex SYK 2 model via a time reparameterization.In contrast, the chaotic regime gives rise to dephasing, which suppresses the exponential ramp of the noninteracting theory and induces correlations between many-body levels.
It is suggested that many-body quantum chaos appears as the spontaneous symmetry breaking of unitarity in interacting quantum many-body systems. It has been shown that many-body level statistics, probed by the spectral form factor (SFF) defined as $K(\ensuremath{\eta},t)=\ensuremath{\langle}|\mathrm{Tr}exp(\ensuremath{-}\ensuremath{\eta}H+itH){|}^{2}\ensuremath{\rangle}$, is dominated by a diffuson-type mode in a field theory analysis. The key finding of this Letter is that the ``unitary'' $\ensuremath{\eta}=0$ case is different from the $\ensuremath{\eta}\ensuremath{\rightarrow}{0}^{\ifmmode\pm\else\textpm\fi{}}$ limit, with the latter leading to a finite mass of these modes due to interactions. This mass suppresses a rapid exponential ramp in the SFF, which is responsible for the fast emergence of Poisson statistics in the noninteracting case, and gives rise to a nontrivial random matrix structure of many-body levels. The interaction-induced mass in the SFF shares similarities with the dephasing rate in the theory of weak localization and the Lyapunov exponent of the out-of-time-ordered correlators.
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).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (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 the '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 a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t 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.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
We consider a noninteracting many-fermion system populating levels of a unitary random matrix ensemble (equivalent to the q=2 complex Sachdev-Ye-Kitaev model)-a generic model of single-particle quantum chaos. We study the corresponding many-particle level statistics by calculating the spectral form factor analytically using algebraic methods of random matrix theory, and match it with an exact numerical simulation. Despite the integrability of the theory, the many-body spectral rigidity is found to have a surprisingly rich landscape. In particular, we find a residual repulsion of distant many-body levels stemming from single-particle chaos, together with islands of level attraction. These results are encoded in an exponential ramp in the spectral form factor, which we show to be a universal feature of nonergodic many-fermion systems embedded in a chaotic medium.
We consider systems of fermions evolved by noninteracting unitary circuits with correlated on-site potentials. When these potentials are drawn from the eigenvalue distribution of a circular random matrix ensemble, the single-particle sector exhibits chaotic dynamics. The corresponding many-body spectral form factors (SFFs) can be computed exactly, revealing signatures of single-particle chaos in the many-body spectral statistics. Due to the absence of interactions the SFF grows exponentially in time, a result which we demonstrate through simple arguments, scaling collapses, and closed-form evaluation of the SFF. Upon introducing interactions, the SFF crosses over to a linear growth regime consistent with many-body random matrix universality. Our exact results for the SFF provide a baseline for future studies of the crossover between single-particle and many-body random matrix behavior.
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). Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in the spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges. For single-particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc. R. Soc. A 400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of the spectral pair correlation function) from semiclassics has been completed by Müller et al. [Phys. Rev. Lett. 93, 014103 (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 to 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 the "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 a clear theoretical explanation for these observations. We compute K(t) in the leading two orders in t and show its agreement with RMT for nonintegrable, 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. In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t; hence, the leading-order RMT behavior K(t)≃2t is a consequence of translation and reflection symmetry of the Ising partition function.Received 5 February 2018Revised 12 April 2018DOI:https://doi.org/10.1103/PhysRevX.8.021062Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosQuantum statistical mechanicsStatistical PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics
We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$ of a spatial subsystem A. We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.