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arXiv:1706.09160 · Physical review. E

Bound on the exponential growth rate of out-of-time-ordered correlators

Naoto Tsuji, Tomohiro Shitara, Masahito Ueda

Published Jul 30, 2018• 45 citations
arXivHTMLPDF

Abstract

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) $F(t)$ 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 $F(t)$ 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 $F(t)$ 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 $\ensuremath{\lambda}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$.

DOI
10.1103/physreve.98.012216

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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\ensuremathπkBT/\ensuremathℏ2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}2\ensuremathπkB​T/\ensuremathℏ. Here we introduce a one-parameter family of out-of-time-ordered correlators F\ensuremathγ(t){F}_{\ensuremath{\gamma}}(t)F\ensuremathγ​(t) (0\ensuremath≤\ensuremathγ\ensuremath≤10\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}10\ensuremath≤\ensuremathγ\ensuremath≤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⟨[A\ifmmode^\elseˆ\fi(t),B\ifmmode^\elseˆ\fi(0)]2\ensuremath⟩\ensuremath{\langle}{[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t),\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}(0)]}^{2}\ensuremath{\rangle}\ensuremath⟨[A\ifmmode^\elseˆ\fi​(t),B\ifmmode^\elseˆ\fi​(0)]2\ensuremath⟩ that diagnoses quantum many-body chaos, and coincides with F(t)F(t)F(t) at \ensuremathγ=1/2\ensuremath{\gamma}=1/2\ensuremathγ=1/2. We rigorously prove that if F\ensuremathγ(t){F}_{\ensuremath{\gamma}}(t)F\ensuremathγ​(t) shows a transient exponential growth for all \ensuremathγ\ensuremath{\gamma}\ensuremathγ in 0\ensuremath≤\ensuremathγ\ensuremath≤10\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}10\ensuremath≤\ensuremathγ\ensuremath≤1, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate \ensuremathλ\ensuremath{\lambda}\ensuremathλ does not depend on the regularization parameter \ensuremathγ\ensuremath{\gamma}\ensuremathγ and satisfies the inequality \ensuremathλ\ensuremath≤2\ensuremathπkBT/\ensuremathℏ\ensuremath{\lambda}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}\ensuremathλ\ensuremath≤2\ensuremathπkB​T/\ensuremathℏ.
A bstract We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillator. We numerically observe the exponential growth of the OTOC when the temperature is higher than a certain threshold. The Lyapunov exponent is found to be of the order of the classical Lyapunov exponent generated at the hilltop, and it remains non-vanishing even at high temperature. We adopt various shape of the potential and find these features universal. The study confirms that the exponential growth of the thermal OTOC does not necessarily mean chaos when the potential includes a local maximum. We also provide a bound for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in one dimension, which is of the same form as the chaos bound obtained by Maldacena, Shenker and Stanford.
We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.
Abstract Chaos, in quantum systems, can be diagnosed by certain out-of-time-order correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and Stanford (MSS). We begin by deriving a dispersion relation for this class of OTOCs, implying that they must satisfy many more constraints beyond the MSS bound. Motivated by this observation, we perform a systematic analysis obtaining an infinite set of constraints on the OTOC. This infinite set includes the MSS bound as the leading constraint. In addition, it also contains subleading bounds that are highly constraining, especially when the MSS bound is saturated by the leading term. These new bounds, among other things, imply that the MSS bound cannot be exactly saturated over any duration of time, however short. Furthermore, we derive a sharp bound on the Lyapunov exponent λ2 ≤ 6πβ \frac{6\pi }{\beta } β6π​ 6 π β of the subleading correction to maximal chaos.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate ΛΛΛ as a function of the number of degrees of freedom, fff, and inverse temperature, βββ. We show that the scaled growth rate, Λ/fΛ/fΛ/f, can be described by a universal function of fβf βfβ and displays a cross-over from classical to quantum behavior as we increase fff and/or lower the temperature. In the deep quantum regime of infinite fff, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.
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.
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.