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arXiv:2205.05663 · Physical review. D/Physical review. D.

Classical and quantum butterfly effect in nonlinear vector mechanics

Nikita Kolganov, Dmitrii A. Trunin

Published Jul 8, 2022• 14 citations
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Abstract

We establish the correspondence between the classical and quantum butterfly effects in nonlinear vector mechanics with the broken $O(N)$ symmetry. On one hand, we analytically calculate the out-of-time-ordered correlation functions and the quantum Lyapunov exponent using the augmented Schwinger-Keldysh technique in the large-$N$ limit. On the other hand, we numerically estimate the classical Lyapunov exponent in the high-temperature limit, where the classical chaotic behavior emerges. In both cases, Lyapunov exponents approximately coincide and scale as $\ensuremath{\kappa}\ensuremath{\approx}1.3\sqrt[4]{\ensuremath{\lambda}T}/N$ with temperature $T$, number of degrees of freedom $N$, and coupling constant $\ensuremath{\lambda}$.

DOI
10.1103/physrevd.106.025003

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We establish the correspondence between the classical and quantum butterfly effects in nonlinear vector mechanics with the broken O(N)O(N)O(N) symmetry. On one hand, we analytically calculate the out-of-time-ordered correlation functions and the quantum Lyapunov exponent using the augmented Schwinger-Keldysh technique in the large-NNN limit. On the other hand, we numerically estimate the classical Lyapunov exponent in the high-temperature limit, where the classical chaotic behavior emerges. In both cases, Lyapunov exponents approximately coincide and scale as \ensuremathκ\ensuremath≈1.3\ensuremathλT4/N\ensuremath{\kappa}\ensuremath{\approx}1.3\sqrt[4]{\ensuremath{\lambda}T}/N\ensuremathκ\ensuremath≈1.34\ensuremathλT​/N with temperature TTT, number of degrees of freedom NNN, and coupling constant \ensuremathλ\ensuremath{\lambda}\ensuremathλ.
We suggest a new indicator of quantum chaos based on the logarithmic out-of-time-order correlator. On the one hand, this indicator correctly reproduces the average classical Lyapunov exponent in the semiclassical limit and directly links the definitions of quantum chaos and classical K system. On the other hand, it can be analytically calculated using the replica trick and the Schwinger-Keldysh diagram technique on a 2n2n2n-fold Keldysh contour. To illustrate this approach, we consider several one-dimensional systems, including the quantum cat map, and three paradigmatic large-NNN models, including the Sachdev-Ye-Kitaev model. Furthermore, we find that correlations between replicas can reduce the magnitude of the Lyapunov exponent compared to estimates based on conventional out-of-time-order correlators.
The majority of classical dynamical systems are chaotic and exhibit the butterfly effect: a minute change in initial conditions has exponentially large effects later on. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. In isolated systems in the absence of decoherence, there is such a correspondence in dynamics, but it usually persists only over a short time window, after which quantum interference washes out classical chaos. We demonstrate that quantum mechanics can also play the opposite role and generate exponential instabilities in classically nonchaotic systems within this early-time window. Our calculations employ the out-of-time-ordered correlator (OTOC)-a diagnostic that reduces to the Lyapunov exponent in the classical limit but is well defined for general quantum systems. We show that certain classically nonchaotic models, such as polygonal billiards, demonstrate a Lyapunov-like exponential growth of the OTOC at early times with Planck's-constant-dependent rates. This behavior is sharply contrasted with the slow early-time growth of the analog of the OTOC in the systems' classical counterparts. These results suggest that classical-to-quantum correspondence in dynamics is violated in the OTOC even before quantum interference develops.
The commutator [x(t),p][x(t),p][x(t),p] in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. It reduces to a c-number and does not show any quantum fluctuations for arbitrary states. Related to this nature, the quantum Lyapunov exponent computed through the out-of-time-order correlator (OTOC) ⟨[x(t),p]2⟩⟨[x(t),p{]}^{2}⟩⟨[x(t),p]2⟩ precisely agrees with the classical one. Hence, the OTOC may be regarded as an ideal indicator of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the commutator [x(t),p][x(t),p][x(t),p] and the OTOCs might be seen in various situations, too. In order to clarify this point, as a first step, we investigate OTOCs in one-dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growth reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using a suitably localized wave packet near the peak, and the second one is taking a limit akin to the large-NNN limit in the noncritical string theories.
We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, $\ensuremath{\mu}$, and the butterfly speed, vb{v}_{b}vb​, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, DDD, and spin-autocorrelation time, $\ensuremath{\tau}$. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form $D\ensuremath{\sim}{v}_{b}^{2}/\ensuremath{\mu}$ and ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\sim}T$. The vanishing of $\ensuremath{\mu}\ensuremath{\sim}{T}^{0.48}$ is parametrically slower than that of the corresponding quantum bound, $\ensuremath{\mu}\ensuremath{\sim}T$, raising interesting questions regarding the semiclassical limit of such spin systems.
Time-dependent density-functional theory is mathematically formulated through nonlinear coupled time-dependent three-dimensional partial differential equations, and it is natural to expect a strong sensitivity of its solutions to variations of the initial conditions, akin to the butterfly effect ubiquitous in classical dynamics. Since the Schr\"odinger equation for an interacting many-body system is, however, linear and mathematically the exact equations of the density-functional theory reproduce the corresponding one-body properties, it would follow that the Lyapunov exponents are also vanishing within a density-functional theory framework. Whether for realistic implementations of the time-dependent density-functional theory the question of the absence of the butterfly effect and whether the dynamics provided is indeed a predictable theory was never discussed. At the same time, since the time-dependent density-functional theory is a unique tool allowing us to study the nonequilibrium dynamics of strongly interacting many-fermion systems, the question of predictability of this theoretical framework is of paramount importance. Our analysis, for a number of quantum superfluid many-body systems (unitary Fermi gas, nuclear fission, and heavy-ion collisions) with a classical equivalent number of degrees of freedom O(1010)O({10}^{10})O(1010) and larger, suggests that its maximum Lyapunov exponents are negligible for all practical purposes.
Lyapunov exponents, a purely classical quantity, play an important role in the evolution of quantum chaotic systems in the semiclassical limit. We conjecture the existence of an upper bound on the Lyapunov exponents that contribute to the quantum motion, namely, even in the semiclassical limit only a limited range of Lyapunov exponents, bounded from above, are important for the quantum evolution. This is a universal feature in any quantum system or quantum field theory, including those with a gravity dual. It has its origin in the finite size of the Hilbert space that is available to an initial quasi-classical configuration. An upper bound also exists in the limit of an infinite Hilbert space provided that the system is in contact with an environment, for instance a thermal bath. An important consequence of this result is a universal quantum bound on the maximum growth rate of the entanglement entropy at zero and finite temperature.