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arXiv:2010.06592 · Physical Review A

Quantum operator growth bounds for kicked tops and semiclassical spin chains

Chao Yin, Andrew Lucas

Published Apr 15, 2021• 35 citations
arXivHTMLPDF

Abstract

We present a framework for understanding the dynamics of operator size, and bounding the growth of out-of-time-ordered correlators, in models of large-$S$ spins. Focusing on the dynamics of a single spin, we show the finiteness of the Lyapunov exponent in the large-$S$ limit; our bounds are tighter than the best known Lieb-Robinson-type bounds on these systems. We numerically find our upper bound on Lyapunov exponents is within an order of magnitude of numerically computed values in classical and quantum kicked top models. Generalizing our results to coupled large-$S$ spins on lattices, we show that the butterfly velocity, which characterizes the spatial speed of quantum information scrambling, is finite as $S\ensuremath{\rightarrow}\ensuremath{\infty}$. We emphasize qualitative differences between operator growth in semiclassical large-spin models and quantum holographic systems including the Sachdev-Ye-Kitaev model.

DOI
10.1103/physreva.103.042414

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We present a framework for understanding the dynamics of operator size, and bounding the growth of out-of-time-ordered correlators, in models of large-SSS spins. Focusing on the dynamics of a single spin, we show the finiteness of the Lyapunov exponent in the large-SSS limit; our bounds are tighter than the best known Lieb-Robinson-type bounds on these systems. We numerically find our upper bound on Lyapunov exponents is within an order of magnitude of numerically computed values in classical and quantum kicked top models. Generalizing our results to coupled large-SSS spins on lattices, we show that the butterfly velocity, which characterizes the spatial speed of quantum information scrambling, is finite as S\ensuremath→\ensuremath∞S\ensuremath{\rightarrow}\ensuremath{\infty}S\ensuremath→\ensuremath∞. We emphasize qualitative differences between operator growth in semiclassical large-spin models and quantum holographic systems including the Sachdev-Ye-Kitaev model.
We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify "information scrambling" due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system's dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival's conjecture to partition the eigenstates of the Floquet map into "regular" and "chaotic." Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.
Scrambling of information in a quantum many-body system, quantified by the out-of-time-ordered correlator (OTOC), is a key manifestation of quantum chaos. A regime of exponential growth in the OTOC, characterized by a Lyapunov exponent, has so far mostly been observed in systems with a high-dimensional local Hilbert space and in weakly-coupled systems. Here, we propose a general criterion for the existence of a well-defined regime of exponential growth of the OTOC in spatially extended systems with local interactions. In such systems, we show that a parametrically long period of exponential growth requires the butterfly velocity to be much larger than the Lyapunov exponent times a microscopic length scale, such as the lattice spacing. As an explicit example, we study a random unitary circuit with tunable interactions. In this model, we show that in the weakly interacting limit, the above criterion is satisfied, and there is a prolonged window of exponential growth. Our results are based on numerical simulations of both Clifford and universal random circuits supported by an analytical treatment.
A hypothesis is presented for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate alpha\\alphaalpha in generic systems, with an extra logarithmic correction in 1d. The rate alpha\\alphaalpha --- an experimental observable --- governs the exponential growth of operator complexity in a sense that is made precise. This exponential growth prevails beyond semiclassical or large-NNN limits. Moreover, alpha\\alphaalpha upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, a sharp bound is obtained on Lyapunov exponents lambdaLleq2alpha\\lambda_L \\leq 2 \\alphalambdaL​leq2alpha, which complements and improves the known universal low-temperature bound lambdaLleq2piT\\lambda_L \\leq 2 \\pi TlambdaL​leq2piT. Our results are illustrated in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally hypothesis is used in in conjunction with the recursion method to develop a technique for computing diffusion constants.
Local out-of-time-order correlators (OTOCs) measure the growth of initially small commutators. For bounded local operators, this growth ends when the local commutator becomes of order one, but in an extended system the region over which it is large can continue to expand. We study this spatial proliferation using three related observables: the local squared commutator Cr(t)C_r(t)Cr​(t), its spatial sum Cˉ(t)=∑rCr(t)\bar C(t) = \sum_r C_r(t)Cˉ(t)=∑r​Cr​(t), and the symmetrically regularised connected dOTOC D(t)D(t)D(t) of two extensive observables. We show that Lieb--Robinson locality gives Cˉ(t)≲td\bar C(t) \lesssim t^dCˉ(t)≲td. We then prove that, in a Gibbs state with uniform exponential clustering, the full dOTOC obeys the same bound, D(t)≲tdD(t) \lesssim t^dD(t)≲td, improving our earlier suboptimal t3dt^{3d}t3d bound. We illustrate these results in two quantum many-body systems: a Brownian coupled-cluster model and a finite-temperature SYK chain. In both cases, an exact disorder symmetry relates D(t)D(t)D(t) to Cˉ(t)\bar C(t)Cˉ(t) and allows us to follow the same extensive observable from the pre-scrambling Lyapunov regime into the post-scrambling growth associated with an expanding saturated region. This late-time power-law growth is the concrete realisation of the weak quantum chaos introduced in our earlier work.
We study quantum many-body systems with a global U(1) conservation law, focusing on a theory of N interacting fermions with charge conservation, or N interacting spins with one conserved component of total spin. We define an effective operator size at finite chemical potential through suitably regularized out-of-time-ordered correlation functions. The growth rate of this density-dependent operator size vanishes algebraically with charge density; hence we obtain new bounds on Lyapunov exponents and butterfly velocities in charged systems at a given density, which are parametrically stronger than any Lieb-Robinson bound. We argue that the density dependence of our bound on the Lyapunov exponent is saturated in the charged Sachdev-Ye-Kitaev model. We also study random automaton quantum circuits and Brownian Sachdev-Ye-Kitaev models, each of which exhibit a different density dependence for the Lyapunov exponent, and explain the discrepancy. We propose that our results are a cartoon for understanding Planckian-limited energy-conserving dynamics at finite temperature.