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arXiv:2210.02427 · Journal of High Energy Physics

Absence of operator growth for average equal-time observables in charge-conserved sectors of the Sachdev-Ye-Kitaev model

Alessio Paviglianiti, Soumik Bandyopadhyay, Philipp Uhrich, Philipp Hauke

Published Mar 17, 2023• 4 citations
arXivHTMLPDF

Abstract

Abstract Quantum scrambling plays an important role in understanding thermalization in closed quantum systems. By this effect, quantum information spreads throughout the system and becomes hidden in the form of non-local correlations. Alternatively, it can be described in terms of the increase in complexity and spatial support of operators in the Heisenberg picture, a phenomenon known as operator growth. In this work, we study the disordered fully-connected Sachdev-Ye-Kitaev (SYK) model, and we demonstrate that scrambling is absent for disorder-averaged expectation values of observables. In detail, we adopt a formalism typical of open quantum systems to show that, on average and within charge-conserved sectors, operators evolve in a relatively simple way which is governed by their operator size. This feature only affects single-time correlation functions, and in particular it does not hold for out-of-time-order correlators, which are well-known to show scrambling behavior. Making use of these findings, we develop a cumulant expansion approach to approximate the evolution of equal-time observables. We employ this scheme to obtain analytic results that apply to arbitrary system size, and we benchmark its effectiveness by exact numerics. Our findings shed light on the structure of the dynamics of observables in the SYK model, and provide an approximate numerical description that overcomes the limitation to small systems of standard methods.

DOI
10.1007/jhep03(2023)126

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Abstract Quantum scrambling plays an important role in understanding thermalization in closed quantum systems. By this effect, quantum information spreads throughout the system and becomes hidden in the form of non-local correlations. Alternatively, it can be described in terms of the increase in complexity and spatial support of operators in the Heisenberg picture, a phenomenon known as operator growth. In this work, we study the disordered fully-connected Sachdev-Ye-Kitaev (SYK) model, and we demonstrate that scrambling is absent for disorder-averaged expectation values of observables. In detail, we adopt a formalism typical of open quantum systems to show that, on average and within charge-conserved sectors, operators evolve in a relatively simple way which is governed by their operator size. This feature only affects single-time correlation functions, and in particular it does not hold for out-of-time-order correlators, which are well-known to show scrambling behavior. Making use of these findings, we develop a cumulant expansion approach to approximate the evolution of equal-time observables. We employ this scheme to obtain analytic results that apply to arbitrary system size, and we benchmark its effectiveness by exact numerics. Our findings shed light on the structure of the dynamics of observables in the SYK model, and provide an approximate numerical description that overcomes the limitation to small systems of standard methods.
The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding of thermalization in isolated quantum many-body systems. Recently, significant progress has been made analytically by modeling non-integrable systems as periodically driven systems, lacking a Hamiltonian picture, while honest Hamiltonian dynamics are frequently limited to small system sizes due to computational constraints. In this paper, we address this by investigating the role of conservation laws (including energy conservation) in the thermalization process from an information-theoretic perspective. For general non-integrable models, we use the equilibrium approximation to show that the maximal amount of information is scrambled (as measured by the tripartite mutual information of the time-evolution operator) at late times even when a system conserves energy. In contrast, we explicate how when a system has additional symmetries that lead to degeneracies in the spectrum, the amount of information scrambled must decrease. This general theory is exemplified in case studies of holographic conformal field theories (CFTs) and the Sachdev-Ye-Kitaev (SYK) model. Due to the large Virasoro symmetry in 1+1D CFTs, we argue that, in a sense, these holographic theories are not maximally chaotic, which is explicitly seen by the non-saturation of the second Rényi tripartite mutual information. The roles of particle-hole and U(1) symmetries in the SYK model are milder due to the degeneracies being only two-fold, which we confirm explicitly at both large- and small-N. We reinterpret the operator entanglement in terms of the growth of local operators, connecting our results with the information scrambling described by out-of-time-ordered correlators, identifying the mechanism for suppressed scrambling from the Heisenberg perspective.
We investigate operator growth in a Brownian spin Sachdev--Ye--Kitaev (SYK) model with random all-to-all interactions, focusing on the full operator-size distribution. For Hamiltonians containing interactions of order two up to LLL, we derive a closed master equation for the Pauli-string expansion coefficients and recast their dynamics into a generating-function formulation suitable for the large-NNN limit. This approach allows us to diagonalize the leading-order evolution operator explicitly and obtain exact solutions for arbitrary initial operator distributions, including the effects of decoherence. Going beyond leading order, we develop a systematic 1/N1/N1/N expansion that captures higher-order corrections to the operator-size dynamics and the late-time behavior. Our results demonstrate that higher-order effects play a crucial role in operator scrambling and that the full operator-size distribution provides a more refined probe of quantum chaos in Brownian and open quantum systems.
The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, N Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For t≲sqrt[N], after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At t∼sqrt[N], the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading 1/N effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at t∼2N, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at t≲sqrt[N] indicate lack of thermalization-a desired feature for better performance of quantum information devices.
Abstract In generic closed quantum systems, the complexity of operators increases under time evolution governed by the Heisenberg equation, reflecting the scrambling of local quantum information. However, when systems interact with an external environment, the system-environment coupling allows operators to escape from the system, inducing a dynamical transition between the scrambling phase and the dissipative phase. This transition is known as the environment-induced information scrambling transition, originally proposed in Majorana fermion systems. In this work, we advance this discovery by investigating the transition in charge-conserved systems with space-time randomness. We construct solvable Brownian Sachdev-Ye-Kitaev models of complex fermions coupled to an environment, enabling the analytical computation of operator growth. We determine the critical dissipation strength, which is proportional to n(1−n)n(1-n)n(1−n) n ( 1 - n ) with n being the density of the complex fermions, arising from the suppression in the quantum Lyapunova exponent due to the Pauli blockade in the scattering process. We further analyze the density dependence of maximally scrambled operators at late time. Our results shed light on the intriguing interplay between information scrambling, dissipation, and conservation laws.
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-qqq model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.
In closed generic many-body systems, unitary evolution disperses local quantum information into highly nonlocal objects, resulting in thermalization. Such a process is called information scrambling, whose swiftness is quantified by the operator size growth. However, the impact of couplings to the environment on the process of information scrambling remains unexplored for quantum systems embedded within an environment. Here we predict a dynamical transition in quantum systems with all-to-all interactions accompanied by an environment, which separates two phases. In the dissipative phase, information scrambling halts as the operator size decays with time, while in the scrambling phase, dispersion of information persists, and the operator size grows and saturates to an O(N) value in the long-time limit with N the number of degrees of freedom of the systems. The transition is driven by the competition between the system's intrinsic and environment propelled scramblings and the environment-induced dissipation. Our prediction is derived from a general argument based on epidemiological models and demonstrated analytically via solvable Brownian Sachdev-Ye-Kitaev models. We provide further evidence which suggests that the transition is generic to quantum chaotic systems when coupled to an environment. Our study sheds light on the fundamental behavior of quantum systems in the presence of an environment.