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arXiv:1806.04156 · Physical review. B./Physical review. B

Operator growth and eigenstate entanglement in an interacting integrable Floquet system

Sarang Gopalakrishnan

Published Aug 14, 2018• 108 citations
arXivHTMLPDF

Abstract

We analyze a simple model of quantum dynamics, which is a discrete-time deterministic version of the Fredrickson-Andersen model. This model is integrable, with a quasiparticle description related to the classical hard-rod gas. Despite the integrability of the model, commutators of physical operators grow with a diffusively broadening front, in this respect resembling generic chaotic models. In addition, local operators behave consistently with the eigenstate thermalization hypothesis (ETH). However, large subsystems violate ETH; as a function of subsystem size, eigenstate entanglement first increases linearly and then saturates at a scale that is parametrically smaller than half the system size.

DOI
10.1103/physrevb.98.060302

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We analyze a simple model of quantum dynamics, which is a discrete-time deterministic version of the Fredrickson-Andersen model. This model is integrable, with a quasiparticle description related to the classical hard-rod gas. Despite the integrability of the model, commutators of physical operators grow with a diffusively broadening front, in this respect resembling generic chaotic models. In addition, local operators behave consistently with the eigenstate thermalization hypothesis (ETH). However, large subsystems violate ETH; as a function of subsystem size, eigenstate entanglement first increases linearly and then saturates at a scale that is parametrically smaller than half the system size.
We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.
Local observables in generic periodically driven closed quantum systems are known to relax to values described by periodic infinite temperature ensembles. At the same time, ergodic static systems exhibit anomalous thermalization of local observables and satisfy a modified version of the eigenstate thermalization hypothesis (ETH), when disorder is present. This raises the question, how does the introduction of disorder affect relaxation in periodically driven systems? In this Rapid Communication, we analyze this problem by numerically studying transport and thermalization in an archetypal example. We find that thermalization is anomalous and is accompanied by subdiffusive transport with a disorder-dependent dynamical exponent. Distributions of matrix elements of local operators in the eigenbases of a family of effective time-independent Hamiltonians, which describe the stroboscopic dynamics of such systems, show anomalous departures from predictions of ETH, signaling that only a modified version of ETH is satisfied. The dynamical exponent is shown to be related to the scaling of the variance of these distributions with system size.
The eigenstate thermalization hypothesis (ETH) is a conjecture on the nature of isolated quantum systems that guarantees the thermal behavior of subsystems when it is satisfied. ETH has been tested in various forms on a number of local many-body interacting systems. Here we examine the validity of ETH in a class of nonlocal disordered many-body interacting systems --- the Sachdev-Ye-Kitaev Majorana (SYK) models --- that may be tuned from chaotic behavior to integrability. Our analysis shows that SYK4_44​ (with quartic couplings) which is maximally chaotic in the large system size limit satisfies the standard ETH scaling while SYK2_22​ (with quadratic couplings), which is integrable, does not. We show that the low-energy and high-energy properties are affected drastically differently when the two Hamiltonians are mixed.
In a many-body quantum system, local operators in the Heisenberg picture O(t)=e^{iHt}Oe^{-iHt} spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement-the entanglement entropy in operator space-is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time t in chaotic systems, we present evidence that it grows only logarithmically in generic interacting integrable systems. Although this logarithmic growth has been previously established for noninteracting fermions, there has been no progress on interacting integrable systems to date. In this Letter we provide an analytical upper bound on operator entanglement for all local operators in the "Rule 54" qubit chain, a cellular automaton model introduced in the 1990s [Bobenko et al., CMP 158, 127 (1993)CMPHAY0010-361610.1007/BF02097234], and recently advertised as the simplest representative of interacting integrable systems. Physically, the logarithmic bound originates from the fact that the dynamics of the models is mapped onto the one of stable quasiparticles that scatter elastically. The possibility of generalizing this scenario to other interacting integrable systems is briefly discussed.
Entanglement and magic are resources that reveal complementary aspects of quantum many-body systems. Their interplay is captured by nonlocal magic, the magic that survives arbitrary local changes of basis. Yet their markedly different dynamical behavior leaves open how this irreducible component of magic spreads. Here we connect nonlocal magic to the capacity of entanglement, a tractable quantity measuring fluctuations of the entanglement Hamiltonian. This connection enables analytically controlled predictions across a wide range of many-body dynamics, from chaotic to integrable systems, which we investigate also using large-scale numerical simulations. In chaotic systems, nonlocal magic exhibits a transient buildup, with logarithmic growth in time followed by decay to a size-independent value. For integrable systems, we develop a quasiparticle picture in quantitative agreement with numerics, showing that the same initial growth instead leads to saturation at a value logarithmic in subsystem size. These contrasting behaviors have an operational consequence for entanglement embezzlement: the strongest scramblers embezzle only transiently, whereas free-fermionic dynamics can sustain universal embezzlement in the steady state.