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arXiv:2504.15203

Fine-grained dynamics of entanglement in non-integrable quenches far across the Ising quantum critical point

Banerjee, Aditya

Published Apr 21, 2025• 0 citations
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Abstract

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 $\textit{fine-grained}$ (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 $1-$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-$\textit{un}$scrambling 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.

DOI
10.48550/arxiv.2504.15203

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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.
It is believed that an isolated and far-from-equilibrium quantum many-body system should try to attain equilibrium via a mechanism whereby any given subsystem acts as an open quantum system that is coupled to an environment, which is the complementary part of the full system, and undergoes a complicated equilibration process such that all the subsystems in the long-time limit attain equilibrium states compatible with the global equilibrium state. This picture begs the question whether the dynamics of any given subsystem is Markovian (monotonic loss of information and memory) or non-Markovian. In this work, by numerically probing the dynamical behaviour of the quantum distances between temporally-separated\textit{temporally-separated}temporally-separated states of small subsystems, we reveal the telltale signatures of (non-)Markovianity of the dynamics of subsystems of an isolated quantum spin system brought in the far-from-equilibrium regime, exemplified with the mixed-field Ising spin chain quenched between parameter regimes deep inside its magnetically ordered and disordered regimes. Additionally, remarkably systematic behaviour is seen in a measure of classical distances between the quantum states of the considered subsystems. These features strongly depend on the direction of quenching in the parameter space, with paramagnetic-to-ferromagnetic quenches offering considerably stronger signatures of subsystem non-Markovianity, for which we offer heuristic arguments.
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.
We study the out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter ggg driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of ggg around the critical value gc{g}_{c}gc​, and mainly address the question whether, and how, the quantum transition leaves traces in the evolution of the transverse and longitudinal magnetizations during such a deep out-of-equilibrium dynamics. We shed light on the emergence of singularities in the thermodynamic infinite-size limit, likely related to the integrability of the model. Finite systems in periodic and open boundary conditions develop peculiar power-law finite-size scaling laws related to revival phenomena, but apparently unrelated to the quantum transition, because their main features are generally observed in quenches to generic values of ggg. We also investigate the effects of dissipative interactions with an environment, modeled by a Lindblad equation with local decay and pumping dissipation operators within the quadratic fermionic model obtainable by a Jordan-Wigner mapping. Dissipation tends to suppress the main features of the unitary dynamics of closed systems. We finally address the effects of integrability breaking, due to further lattice interactions, such as in anisotropic next-to-nearest-neighbor Ising (ANNNI) models. We show that some qualitative features of the post-quench dynamics persist, in particular, the different behaviors when quenching to quantum ferromagnetic and paramagnetic phases, and the revival phenomena due to the finite size of the system.
We study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.
We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse (ggg) and longitudinal (hhh) magnetic fields. This model exhibits a continuous quantum transition (CQT) at g=gcg=g_cg=gc​ and h=0h=0h=0, and first-order quantum transitions (FOQTs) driven by hhh along the line h=0h=0h=0 (g<gcg<g_cg<gc​). In the integrable limit h=0h=0h=0, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value hi<0h_i < 0hi​<0 to a positive value hf>0h_f>0hf​>0. We focus on values of hfh_fhf​ such that the spectrum of the post-QQ Hamiltonian H(g,hf)H(g,h_f)H(g,hf​) lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of ggg, finding qualitatively distinct behaviors for g>gcg > g_cg>gc​ (where the chain is in the disordered phase), for g=gcg = g_cg=gc​ (QQ across the CQT), and for g<gcg<g_cg<gc​ (QQ across the FOQT line).
The transverse field Ising model of N fully connected spins provides a mathematical accessible, yet non-trivial, system to investigate the non-thermal dynamics after a sudden quantum quench. The focus in this thesis is three-fold. First, in a 1/N expansion the dynamics of the mean magnetization within its variance is obtained analytically in the large N limit. The variance constitutes a leading order correction to the mean field limit at infinite N. While mean field theories are one of the most accessible approximations to deal with the quantum complexity, its validity in time out of equilibrium has to be checked on a case by case basis. By studying the dynamics of the variance, we see that the mean field approximation can break down at surprisingly early times scaling with the square root of N. The underlying mechanism in the fully connected Ising model is identified as a dephasing effect between nearby effective orbits. The dynamics of the magnetization variance discriminates four qualitatively different regions in the dynamical phase diagram that are indistinguishable by just looking at the mean magnetization. Second, bipartite entanglement between disjoint groups of the spins in the fully connected Ising model is studied. In the large N limit the entanglement Hamiltonian is shown to be a harmonic oscillator. A quantitative relation between the angular frequency of the oscillator and a squeezing measure of the spin state is derived. Explicit expressions for the Renyi entanglement entropies are given. In contrast to the variance, which vanishes in the thermodynamic limit, the entanglement entropy saturate to an N-independent positive value. The dynamics of the entanglement Hamiltonian and the entanglement entropies is discussed. Linear growth as a function of time is linked to the unstable fixed point and the homoclinic orbit of a pitchfork bifurcation, while logarithmic increase is explained due to a dephasing mechanism of nearby effective orbits. A generalization to other models that permit a semiclassical effective description (on a possibly higher dimensional phase space) is given. As a byproduct the connection between entangled Gaussian states and the symplectic eigenvalues of its reduced covariance is derived. An upper bound on the von Neumann entanglement entropy of any bipartite pure state is given in terms of the symplectic properties of its reduced covariance. Third, the existence of dynamical phase transitions (DPTs) in the fully connected Ising model is argued by two different approaches. DPTs are defined as non-analyticities in the time-dependence of the return probability rate function in the thermodynamic limit. Like entanglement entropy, the behavior of the return rate function is an observable-independent quantity used to characterize non-equilibrium states. A numerically based connection between the DPT and the analytic property of the infinite time averaged order parameter as a function of quench strength is discussed. The concept of Feynman history states is used to devise the notion of history entanglement. An information theoretic interpretation of the history entanglement entropy as the precision of a quantum clock follows from Holevo’s bound. The dependence of this entanglement entropy on the observation time constitutes an operator-independent characterization of the quantum dynamics in closed systems. For observation times comparable to the Heisenberg time, the history entanglement becomes sensitive to the statistics of the energy gaps, and thus distinguishes integrable and non-integrable dynamics. This is demonstrated for two examples, a random matrix ensemble, and a one-dimensional hardcore boson model with integrability breaking next nearest neighbor interaction.