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arXiv:1807.09799 · Physical review. B./Physical review. B

Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators

Yunxiang Liao, Victor M. Galitski

Published Nov 14, 2018• 52 citations
arXivHTMLPDF

Abstract

The out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh nonlinear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs, defined as ${f}^{(r)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$ and ${f}^{(u)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$, respectively (where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}(\mathrm{r},t),{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}(0,0)}$ is the anticommutator of fermion field operators and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ is the thermal density matrix). The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the Altshuler-Aronov-Khmelnitskii dephasing rate in interacting metals, but here it involves two replicas of the system (two ``worlds''). The intraworld contributions reproduce the Altshuler-Aronov-Khmelnitskii dephasing (that would correspond to a decay of the correlator), while the interworld terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound, ${\ensuremath{\lambda}}^{(r)}\ensuremath{\le}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}$. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound, ${\ensuremath{\lambda}}^{(u)}\ensuremath{\gg}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}$. We argue that ${\ensuremath{\lambda}}^{(u)}$ is not a reliable indicator of many-body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an important general question of the physical meaning of the OTOCs often used in calculations and proofs. We briefly discuss possible connections of the OTOCs to observables in quantum interference effects and level statistics via a generalization of the Bohigas-Giannoni-Schmit conjecture to many-body chaotic systems.

DOI
10.1103/physrevb.98.205124

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The out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh nonlinear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs, defined as f(r)(t)=Tr[\ensuremathρ\ifmmode^\elseˆ\fiA\ifmmode^\elseˆ\fi(t)\ensuremathρ\ifmmode^\elseˆ\fiA\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi(t)]{f}^{(r)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]f(r)(t)=Tr​\ensuremathρ\ifmmode^\elseˆ\fi​​A\ifmmode^\elseˆ\fi​(t)\ensuremathρ\ifmmode^\elseˆ\fi​​A\ifmmode^\elseˆ\fi​\ifmmode†\else\textdagger\fi(t)​ and f(u)(t)=Tr[\ensuremathρ\ifmmode^\elseˆ\fiA\ifmmode^\elseˆ\fi(t)A\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi(t)]{f}^{(u)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]f(u)(t)=Tr[\ensuremathρ\ifmmode^\elseˆ\fi​A\ifmmode^\elseˆ\fi​(t)A\ifmmode^\elseˆ\fi​\ifmmode†\else\textdagger\fi(t)], respectively (where A\ifmmode^\elseˆ\fi(t)=\ensuremathψ\ifmmode^\elseˆ\fi(r,t),\ensuremathψ\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi(0,0)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}(\mathrm{r},t),{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}(0,0)}A\ifmmode^\elseˆ\fi​(t)=\ensuremathψ\ifmmode^\elseˆ\fi​(r,t),\ensuremathψ\ifmmode^\elseˆ\fi​\ifmmode†\else\textdagger\fi(0,0) is the anticommutator of fermion field operators and \ensuremathρ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}\ensuremathρ\ifmmode^\elseˆ\fi​ is the thermal density matrix). The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the Altshuler-Aronov-Khmelnitskii dephasing rate in interacting metals, but here it involves two replicas of the system (two ``worlds''). The intraworld contributions reproduce the Altshuler-Aronov-Khmelnitskii dephasing (that would correspond to a decay of the correlator), while the interworld terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound, \ensuremathλ(r)\ensuremath≤2\ensuremathπkBT/\ensuremathℏ{\ensuremath{\lambda}}^{(r)}\ensuremath{\le}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}\ensuremathλ(r)\ensuremath≤2\ensuremathπkB​T/\ensuremathℏ. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound, \ensuremathλ(u)\ensuremath≫2\ensuremathπkBT/\ensuremathℏ{\ensuremath{\lambda}}^{(u)}\ensuremath{\gg}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}\ensuremathλ(u)\ensuremath≫2\ensuremathπkB​T/\ensuremathℏ. We argue that \ensuremathλ(u){\ensuremath{\lambda}}^{(u)}\ensuremathλ(u) is not a reliable indicator of many-body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an important general question of the physical meaning of the OTOCs often used in calculations and proofs. We briefly discuss possible connections of the OTOCs to observables in quantum interference effects and level statistics via a generalization of the Bohigas-Giannoni-Schmit conjecture to many-body chaotic systems.
Recently, the out-of-time-ordered correlator (OTOC) has gained much attention as an indicator of quantum chaos. In the semi-classical limit, its exponential growth rate resembles the classical Lyapunov exponent. The quantum–classical correspondence has been supported for the one-body chaotic systems as well as realistic systems with interactions, as in the Dicke model, a model of multi-two-level atoms and cavity field interactions. To this end, we calculate the OTOC for different variations of the Dicke model in an open quantum system setting. The connection between the superradiant phase transition of the Dicke model and the OTOC is studied. Further, we establish a relation between the OTOC and the second-order coherence function. This becomes important for the experimental studies of the OTOC and quantum chaos in the models of quantum optics.
It has been conjectured by Maldacena, Shenker, and Stanford [J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 08 (2016) 106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) F(t)F(t)F(t) has a universal upper bound $2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$. Here we introduce a one-parameter family of out-of-time-ordered correlators ${F}_{\ensuremath{\gamma}}(t)$ ($0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$), which has as good properties as F(t)F(t)F(t) as a regularization of the out-of-time-ordered part of the squared commutator $\ensuremath{\langle}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t),\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}(0)]}^{2}\ensuremath{\rangle}$ that diagnoses quantum many-body chaos, and coincides with F(t)F(t)F(t) at $\ensuremath{\gamma}=1/2$. We rigorously prove that if ${F}_{\ensuremath{\gamma}}(t)$ shows a transient exponential growth for all $\ensuremath{\gamma}$ in $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate $\ensuremath{\lambda}$ does not depend on the regularization parameter $\ensuremath{\gamma}$ and satisfies the inequality $\ensuremath{\lambda}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$.
We study out-of-time-order correlators (OTOCs) of the form $\ensuremath{\langle}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}(0)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}(t)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{D}(0)\ensuremath{\rangle}$ for a quantum system weakly coupled to a dissipative environment. Such an open system may serve as a model of, e.g., a small region in a disordered interacting medium coupled to the rest of this medium considered as an environment. We demonstrate that for a system with discrete energy levels the OTOC saturates exponentially $\ensuremath{\propto}\ensuremath{\sum}{a}_{i}{e}^{\ensuremath{-}t/{\ensuremath{\tau}}_{i}}+\mathrm{const}$ to a constant value at $t\ensuremath{\rightarrow}\ensuremath{\infty}$, in contrast with quantum-chaotic systems which exhibit exponential growth of OTOCs. Focusing on the case of a two-level system, we calculate microscopically the decay times ${\ensuremath{\tau}}_{i}$ and the value of the saturation constant. Because some OTOCs are immune to dephasing processes and some are not, such correlators may decay on two sets of parametrically different time scales related to inelastic transitions between the system levels and to pure dephasing processes, respectively. In the case of a classical environment, the evolution of the OTOC can be mapped onto the evolution of the density matrix of two systems coupled to the same dissipative environment.
Regularized out-of-time-ordered correlators (OTOCs) are frequently used as a measure of chaos, i.e., nonintegrability, in quantum many-body systems. Here, the authors compute the regularized OTOCs of both fermion and boson operators at the fixed point of the spin-fermion model found by Schlief, Lunts, and Lee. They find that, despite being a strongly coupled theory, the degrees of nonintegrability of the fermion and boson sectors are parametrically different, and both are parametrically smaller than the upper bound for the growth rate of regularized OTOCs.
Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τ_{E} in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τ_{E}=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τ_{E}, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems.
Abstract We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.