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Universe

Correlation Functions of Quantum Artin System

H. Babujian, Rubik Poghossian, George K. Savvidy

Published Jun 30, 2020• 1 citation
Publisher

Abstract

It was conjectured by Maldacena, Shenker and Stanford that the classical chaos can be diagnosed in thermal quantum systems by using an out-of-time-order correlation function. The Artin dynamical system defined on the fundamental region of the modular group SL(2,Z) represents a well defined example of a highly chaotic dynamical system in its classical regime. We investigated the influence of the classical chaotic behaviour on the quantum–mechanical properties of the Artin system calculating the corresponding out-of-time-order thermal quantum–mechanical correlation functions. We demonstrated that the two- and four-point correlation functions of the Liouiville-like operators decay exponentially with temperature dependent exponents and that the square of the commutator of the Liouiville-like operators separated in time grows exponentially.

DOI
10.3390/universe6070091

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It was conjectured by Maldacena, Shenker and Stanford that the classical chaos can be diagnosed in thermal quantum systems by using an out-of-time-order correlation function. The Artin dynamical system defined on the fundamental region of the modular group SL(2,Z) represents a well defined example of a highly chaotic dynamical system in its classical regime. We investigated the influence of the classical chaotic behaviour on the quantum–mechanical properties of the Artin system calculating the corresponding out-of-time-order thermal quantum–mechanical correlation functions. We demonstrated that the two- and four-point correlation functions of the Liouiville-like operators decay exponentially with temperature dependent exponents and that the square of the commutator of the Liouiville-like operators separated in time grows exponentially.
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.
We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form ⟨W(t)VW(t)V⟩⟨W(t)VW(t)V⟩⟨W(t)VW(t)V⟩. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large ccc Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of $\ensuremath{\sim}{t}_{*}\ensuremath{-}(\ensuremath{\beta}/2\ensuremath{\pi})\mathrm{log}{\ensuremath{\beta}}^{2}{E}_{w}{E}_{v}$, where t∗{t}_{*}t∗​ is the fast scrambling time $(\ensuremath{\beta}/2\ensuremath{\pi})\mathrm{log}c$ and Ew,Ev{E}_{w},{E}_{v}Ew​,Ev​ are the energy scales of the W,VW,VW,V operators.
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.
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 examine three billiard systems -- the cardioid, diamond (Superman), and Sinai billiards -- all of which are known to be classically chaotic. We compute their classical Lyapunov exponents, and using out-of-time-order correlators (OTOCs) in the semi-classical regime, we also derive their quantum Lyapunov exponents. We observe that the classical and quantum Lyapunov exponents are in agreement, strengthening the role of OTOCs as a diagnostic for quantum chaos in billiard systems. At very low temperatures, the OTOC of the Sinai billiard shows sharp growth, a phenomenon absent in the other two billiards. We identify the source of this anomalous behaviour in the geometry of the ground state wave function of the Sinai billiard, which is more sensitive to the curvature of the billiard compared to the other billiards. We also remark on the late-time behaviour of the OTOCs and how the scrambling/Ehrenfest time is related to the temperature of quantum billiards.
The out-of-time-order correlator (OTOC), recently analyzed in several physical contexts, is studied for low-dimensional chaotic systems through semiclassical expansions and numerical simulations. The semiclassical expansion for the OTOC yields a leading-order contribution in ${\ensuremath{\hbar}}^{2}$ that is exponentially increasing with time within an intermediate, temperature-dependent, time window. The growth-rate in such a regime is governed by the Lyapunov exponent of the underlying classical system and scales with the square-root of the temperature.