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arXiv:2402.13558

Mutual information scrambling in Ising spin chain

Surbhi Khetrapal, Emil Tore Mærsk Pedersen

Published Feb 21, 2024• 0 citations
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Abstract

We consider a chain of spin-half particles of a finite length, evolved with the mixed-field Ising Hamiltonian and impose open boundary condition. We simulate the time evolution of entanglement entropy and mutual information following quench from the Néel state in this system using tensor networks. We find that the entanglement entropy for non-integrable systems saturates to a constant value at late times, however it continues to oscillate for integrable systems. We also find that mutual information peaks as a function of distance between intervals decay faster for non-integrable systems compared to integrable systems, in agreement with the conclusion of \cite{Alba:2019ybw} for XXZ chains. We compare the oscillations in entanglement entropy evolution obtained from simulations in the integrable case with analytic results from quasi-particle picture and find agreement.

DOI
10.48550/arxiv.2402.13558

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We consider a chain of spin-half particles of a finite length, evolved with the mixed-field Ising Hamiltonian and impose open boundary condition. We simulate the time evolution of entanglement entropy and mutual information following quench from the Néel state in this system using tensor networks. We find that the entanglement entropy for non-integrable systems saturates to a constant value at late times, however it continues to oscillate for integrable systems. We also find that mutual information peaks as a function of distance between intervals decay faster for non-integrable systems compared to integrable systems, in agreement with the conclusion of \cite{Alba:2019ybw} for XXZ chains. We compare the oscillations in entanglement entropy evolution obtained from simulations in the integrable case with analytic results from quasi-particle picture and find agreement.
Investigating the scrambling of quantum information (also known as quantum chaos) in many-body quantum systems provides valuable insights into the postquench evolution of a system. While various methods are available to understand the chaotic dynamics of these quantum systems, scrambling of mutual information following a quench has recently also emerged as a diagnostic of chaotic dynamics. In this work, we consider a chain of spin-1/21/21/2 particles of a finite length LLL evolving with the mixed-field Ising Hamiltonian and impose open boundary conditions. We simulate the time evolution of entanglement entropy and mutual information following a quench from the N\'eel state in this system using tensor networks and find that the entanglement entropy for nonintegrable systems saturates to a constant value at late times, while it continues to oscillate for integrable systems. We also find that mutual information peaks as a function of distance between intervals decay faster for nonintegrable systems compared to integrable systems, in agreement with the results in literature for XXZ chains. We compare the oscillations in entanglement entropy evolution obtained from simulations in the integrable case with analytic results from the quasiparticle picture and find agreement.
How quantum information is scrambled in the global degrees of freedom of nonequilibrium many-body systems is a key question to understand local thermalization. A consequence of scrambling is that in the scaling limit the mutual information between two intervals vanishes at all times, i.e., it does not exhibit a peak at intermediate times. Here we investigate the mutual information scrambling after a quantum quench in both integrable and nonintegrable one-dimensional systems. We study the mutual information between two intervals of finite length as a function of their distance. In integrable systems, the mutual information exhibits an algebraic decay with the distance between the intervals, signaling weak scrambling. This behavior may be qualitatively understood within the quasiparticle picture for the entanglement spreading. In the scaling limit of large intervals, times, and distances between the intervals, with their ratios fixed, this predicts a decay exponent equal to 1/21/21/2. Away from the scaling limit, the power-law behavior persists, but with a larger (and model-dependent) exponent. For nonintegrable models, a much faster decay is observed, which can be attributed to the finite lifetime of the quasiparticles: unsurprisingly, nonintegrable models are better scramblers.
The task of exploring and understanding various aspects of far-from-equilibrium dynamics of closed and generic quantum many-body systems has received a thrust of attention in recent years, driven partly by remarkable advances in ultracold experimental technologies. In this work, for the paradigmatic Ising spin chain with transverse and longitudinal fields and partly motivated by the practice of site-resolved control in contemporary ultracold experiments, we present numerical observations of several fine-grained\textit{fine-grained}fine-grained (small-subsystem level) features of far-from-equilibrium dynamics from a quantum informational point of view, induced by quantum quenches far across the Ising critical point between states deep inside the para- and ferro-magnetic regimes. Rather featureless dynamics is seen for ferromagnetic to paramagnetic quenches, but paramagnetic to ferromagnetic quenches exhibit rich behaviour, including recurrences of an approximately Page-like dynamics of entanglement entropies of one- and two-spin subsystems, periodic but short-lived occurrences of approximately 1−1-1−uniform states, a series of sudden deaths and revivals of entanglement between two spins in the system's bulk, non-analytic cusps in single-copy entanglement entropy for three-spin and bigger subsystems, insufficient mixedness and a series of scrambling-un\textit{un}unscrambling of local mutual information between neighboring spins. Moreover, essentially indistinguishable dynamics is seen at very early times between the integrable limit (zero longitudinal field) and non-integrable cases, with the former eventually showing signatures of better mixing and faster approach to equilibration than the latter. These features are expected to hold for quench dynamics across Ising quantum critical points in more complicated systems.
Abstract We consider a quench in a free-fermion chain by joining two homogeneous half-chains via a defect. The time evolution of the entanglement negativity is studied between adjacent segments surrounding the defect. In case of equal initial fillings, the negativity grows logarithmically in time and essentially equals one-half of the Rényi mutual information with index α = 1/2 in the limit of large segments. In sharp contrast, in the biased case one finds a linear increase followed by the saturation at an extensive value for both quantities, which is due to the backscattering from the defect and can be reproduced in a quasiparticle picture. Furthermore, a closer inspection of the subleading corrections reveals that the negativity and the mutual information have a small but finite difference in the steady state. Finally, we also study a similar quench in the XXZ spin chain via density-matrix renormalization group methods and compare the results for the negativity to the fermionic case.
Understanding how and whether local perturbations can affect the entire quantum system is a fundamental step in understanding non-equilibrium phenomena such as thermalization. This knowledge of non-equilibrium phenomena is applicable for quantum computation, as many quantum computers employ non-equilibrium processes for computations. In this paper, we investigate the evolution of bi- and tripartite operator mutual information of the time-evolution operator and the Pauli spin operators in the one-dimensional Ising model with magnetic field and the disordered Heisenberg model to study the properties of quantum circuits. In the Ising model, the early-time evolution qualitatively follows an effective light cone picture, and the late-time value is well described by Page’s value for a random pure state. In the Heisenberg model with strong disorder, we find that many-body localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian that describes the time evolution of bi- and tripartite operator mutual information for the Heisenberg model in the large disorder regime.
We study dynamics of entanglement in the thermalization process of an isolated quantum many-body system. We propose a simple setup for measuring the propagation speed of entanglement entropy (EE) in numerical simulations and apply it to the integrable/non-integrable spin models in 1D - the transverse Ising (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising (ECI) model. We find that two distinct time-scales t∗t^\astt∗ and tdifft_{\rm {diff}}tdiff​ arise in the dynamics of EE in the thermalization process: the former represents the time-scale for the saturation of EE and the latter characterizes spreading of EE over the entire system. Evaluating the propagation speed of entanglement from tdifft_{\rm diff}tdiff​, we find that entanglement propagates ballistically with a constant velocity irrespective of the integrability of the model. The propagation speed of entanglement is found to coincide with the maximum group velocity of quasi-particle excitations in the TI model. We also evaluate the propagation speed of entanglement by mutual information and find the characteristic time-scale tMIt_{\mathrm{MI}}tMI​. We show that the propagation speeds of entanglement evaluated by tMIt_{\mathrm{MI}}tMI​ and tdifft_{\rm {diff}}tdiff​ agree well. We discuss the condition for thermalization based on the numerical results and propose that scrambling of the entire system has to take place before saturation of EE for thermalization.