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arXiv:1910.09539 · Journal of Mathematical Physics

Non-perturbative dynamics of the operator size distribution in the Sachdev–Ye–Kitaev model

Andrew Lucas

Published Aug 1, 2020• 22 citations
arXivHTMLPDF

Abstract

We prove non-perturbative bounds on the time evolution of the probability distribution of operator size in the q-local Sachdev–Ye–Kitaev model with N fermions for any even integer q > 2 and any positive even integer N > 2q. If the couplings in the Hamiltonian are independent and identically distributed Rademacher random variables, the infinite temperature many-body Lyapunov exponent is almost surely finite as N → ∞. In the limit q → ∞, N → ∞, and q6+δ/N → 0, the shape of the size distribution of a growing fermion, obtained by leading order perturbation calculations in 1/N and 1/q, is similar to a distribution that locally saturates our constraints. Our proof is not based on Feynman diagram resummation; instead, we note that the operator size distribution obeys a continuous time quantum walk with bounded transition rates to which we apply concentration bounds from classical probability theory.

DOI
10.1063/1.5133964

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We prove non-perturbative bounds on the time evolution of the probability distribution of operator size in the q-local Sachdev–Ye–Kitaev model with N fermions for any even integer q > 2 and any positive even integer N > 2q. If the couplings in the Hamiltonian are independent and identically distributed Rademacher random variables, the infinite temperature many-body Lyapunov exponent is almost surely finite as N → ∞. In the limit q → ∞, N → ∞, and q6+δ/N → 0, the shape of the size distribution of a growing fermion, obtained by leading order perturbation calculations in 1/N and 1/q, is similar to a distribution that locally saturates our constraints. Our proof is not based on Feynman diagram resummation; instead, we note that the operator size distribution obeys a continuous time quantum walk with bounded transition rates to which we apply concentration bounds from classical probability theory.
A bstract We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with q -body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite q and analytically at large q . With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large q limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large q , leading to classical dynamics of the operator size growth with dissipation.
Many-body chaos has emerged as a powerful framework for understanding thermalization in strongly interacting quantum systems. While recent analytic advances have sharpened our intuition for many-body chaos in certain large N theories, it has proven challenging to develop precise numerical tools capable of exploring this phenomenon in generic Hamiltonians. To this end, we utilize massively parallel, matrix-free Krylov subspace methods to calculate dynamical correlators in the Sachdev-Ye-Kitaev model for up to N=60 Majorana fermions. We begin by showing that numerical results for two-point correlation functions agree at high temperatures with dynamical mean field solutions, while at low temperatures finite-size corrections are quantitatively reproduced by the exactly solvable dynamics of near extremal black holes. Motivated by these results, we develop a novel finite-size rescaling procedure for analyzing the growth of out-of-time-order correlators. Our procedure accurately determines the Lyapunov exponent, λ, across a wide range in temperatures, including in the regime where λ approaches the universal bound, λ=2π/β.
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.
Abstract We study operator dynamics in Brownian quantum many-body models with q -local interactions. The operator dynamics are characterized by the time-dependent size distribution, for which we derive an exact master equation in both the Brownian Majorana Sachdev–Ye–Kitaev model and the spin model for general q . This equation can be solved numerically for large systems. Additionally, we obtain the analytical size distribution in the large N limit for arbitrary initial conditions and q . The distributions for both models take the same form, related to the χ -squared distribution by a change of variable, and strongly depend on the initial condition. For small initial sizes, the operator dynamics are characterized by a broad distribution that narrows as the initial size increases. When the initial operator size is below q − 2 for the Majorana model or q − 1 for the spin model, the distribution diverges in the small size limit at all times. The mean size of all operators, which can be directly measured by the out-of-time ordered correlator, grows exponentially during the early time. In the late time regime, the mean size for a single Majorana or Pauli operator for all q decays exponentially as t e − t , much slower than all other operators, which decay as e − t . At finite N , the size distribution exhibits modulo-dependent branching within a symmetry sector for the q ⩾ 8 Majorana model and the q ⩾ 4 spin model. Our results reveal universal features of operator dynamics in q -local quantum many-body systems.
A bstract We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of [1]. Our results are based on the study of the dissipative q -body Sachdev-Ye-Kitaev (SYK q ) model, governed by the Markovian dynamics. We introduce a notion of “operator size concentration” which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ( a n and b n ) in the large q limit. Our results corroborate with the semi-analytics in finite q in the large N limit, and the numerical Arnoldi iteration in finite q and finite N limit. As a result, Krylov complexity exhibits exponential growth following a saturation at a time that grows logarithmically with the inverse dissipation strength. The growth of complexity is suppressed compared to the closed system results, yet it upper bounds the growth of the normalized out-of-time-ordered correlator (OTOC). We provide a plausible explanation of the results from the dual gravitational side.