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arXiv:1704.08442 · Physical review. B./Physical review. B

Out-of-time-order correlators in finite open systems

Sergey Syzranov, Alexey Vyacheslavovich Gorshkov, Victor M. Galitski

Published Apr 26, 2018• 52 citations
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Abstract

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.

DOI
10.1103/physrevb.97.161114

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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 ${f}^{(r)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$ 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.

arXiv:1807.09799•Physical review. B./Physical review. B•Nov 14, 2018•52 citations
We study out-of-time-order correlators (OTOCs) of the form \ensuremath⟨A\ifmmode^\elseˆ\fi(t)B\ifmmode^\elseˆ\fi(0)C\ifmmode^\elseˆ\fi(t)D\ifmmode^\elseˆ\fi(0)\ensuremath⟩\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}\ensuremath⟨A\ifmmode^\elseˆ\fi​(t)B\ifmmode^\elseˆ\fi​(0)C\ifmmode^\elseˆ\fi​(t)D\ifmmode^\elseˆ\fi​(0)\ensuremath⟩ 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∝\ensuremath∑aie\ensuremath−t/\ensuremathτi+const\ensuremath{\propto}\ensuremath{\sum}{a}_{i}{e}^{\ensuremath{-}t/{\ensuremath{\tau}}_{i}}+\mathrm{const}\ensuremath∝\ensuremath∑ai​e\ensuremath−t/\ensuremathτi​+const to a constant value at t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞, 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 \ensuremathτi{\ensuremath{\tau}}_{i}\ensuremathτi​ 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.
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.
The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.
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.
Out-of-time-ordered correlators (OTOCs) are useful tools for investigating foundational questions such as thermalization in closed quantum systems because they can potentially distinguish between integrable and nonintegrable dynamics. Here, we discuss the properties of wavefronts of OTOCs by focusing on the region around the main wavefront at x=vBtx={v}_{B}tx=vB​t, where vB{v}_{B}vB​ is the butterfly velocity. Using a Heisenberg spin model as an example, we find that the leading edge of a propagating Gaussian with the argument $\ensuremath{-}m(x){(x\ensuremath{-}{v}_{B}t)}^{2}+b(x)t$ gives an excellent fit to the region around x=vBtx={v}_{B}tx=vB​t for both the free and chaotic cases. However, the scaling in these two regimes is very different: in the free case the coefficients m(x)m(x)m(x) and b(x)b(x)b(x) have an inverse power law dependence on xxx, whereas in the chaotic case they decay exponentially. We conjecture that this result is universal by using catastrophe theory to show that, on the one hand, the wavefront in the free case has to take the form of an Airy function and its local expansion shows that the power law scaling seen in the numerics holds rigorously and, on the other hand, an exponential scaling of the OTOC wavefront must be a signature of nonintegrable dynamics. We find that the crossover between the two regimes is smooth and characterized by an S\mathsf{S}S-shaped curve giving the lifting of Airy nodes as a function of a chaos parameter. This shows that the Airy form is qualitatively stable against weak chaos and consistent with the concept of a quantum Kolmogorov-Arnold-Moser theory.
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.
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.