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arXiv:2307.01842 · Journal of Statistical Mechanics Theory and Experiment

Universality in the tripartite information after global quenches: spin flip and semilocal charges

Vanja Marić

Published Nov 1, 2023• 3 citations
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Abstract

Abstract We study stationary states emerging after global quenches in which the time evolution is under local Hamiltonians that possess semilocal conserved operators. In particular, we study a model that is dual to quantum XY chain. We show that a localized perturbation in the initial state can turn an exponential decay of spatial correlations in the stationary state into an algebraic decay. We investigate the consequences on the behavior of the (Rényi-α) entanglement entropies, focusing on the tripartite information of three adjacent subsystems. In the limit of large subsystems, we show that in the stationary state with the algebraic decay of correlations the tripartite information exhibits a non-zero value with a universal dependency on the cross ratio, while it vanishes in the stationary state with the exponential decay of correlations.

DOI
10.1088/1742-5468/ad0636

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Abstract We study stationary states emerging after global quenches in which the time evolution is under local Hamiltonians that possess semilocal conserved operators. In particular, we study a model that is dual to quantum XY chain. We show that a localized perturbation in the initial state can turn an exponential decay of spatial correlations in the stationary state into an algebraic decay. We investigate the consequences on the behavior of the (Rényi-α) entanglement entropies, focusing on the tripartite information of three adjacent subsystems. In the limit of large subsystems, we show that in the stationary state with the algebraic decay of correlations the tripartite information exhibits a non-zero value with a universal dependency on the cross ratio, while it vanishes in the stationary state with the exponential decay of correlations.
We consider macroscopically large 3-partitions (A,B,C)(A,B,C)(A,B,C) of connected subsystems $A\ensuremath{\cup}B\ensuremath{\cup}C$ in infinite quantum spin chains and study the R\'enyi-$\ensuremath{\alpha}$ tripartite information ${I}_{3}^{(\ensuremath{\alpha})}(A,B,C)$. At equilibrium in clean 1D systems with local Hamiltonians it generally vanishes. A notable exception is the ground state of conformal critical systems, in which ${I}_{3}^{(\ensuremath{\alpha})}(A,B,C)$ is known to be a universal function of the cross ratio x=∣A∣∣C∣/[(∣A∣+∣B∣)(∣C∣+∣B∣)]x=|A||C|/[(|A|+|B|)(|C|+|B|)]x=∣A∣∣C∣/[(∣A∣+∣B∣)(∣C∣+∣B∣)], where ∣A∣|A|∣A∣ denotes AAA's length. We identify different classes of states that, under time evolution with translationally invariant Hamiltonians, locally relax to states with a nonzero (R\'enyi) tripartite information, which furthermore exhibits a universal dependency on xxx. We report a numerical study of ${I}_{3}^{(\ensuremath{\alpha})}$ in systems that are dual to free fermions, propose a field-theory description, and work out their asymptotic behavior for $\ensuremath{\alpha}=2$ in general and for generic $\ensuremath{\alpha}$ in a subclass of systems. This allows us to infer the value of ${I}_{3}^{(\ensuremath{\alpha})}$ in the scaling limit $x\ensuremath{\rightarrow}{1}^{\ensuremath{-}}$, which we call ``residual tripartite information''. If nonzero, our analysis points to a universal residual value $\ensuremath{-}log2$ independently of the R\'enyi index $\ensuremath{\alpha}$, and hence applies also to the genuine (von Neumann) tripartite information.
Abstract We consider the Rényi-α tripartite information I3(α) {I}_3^{\left(\alpha \right)} I3(α)​ I 3 α of three adjacent subsystems in the stationary state emerging after global quenches in noninteracting spin chains from both homogeneous and bipartite states. We identify settings in which I3(α) {I}_3^{\left(\alpha \right)} I3(α)​ I 3 α remains nonzero also in the limit of infinite lengths and develop an effective quantum field theory description of free fermionic fields on a ladder. We map the calculation into a Riemann-Hilbert problem with a piecewise constant matrix for a doubly connected domain. We find an explicit solution for α = 2 and an implicit one for α > 2. In the latter case, we develop a rapidly convergent perturbation theory that we use to derive analytic formulae approximating I3(α) {I}_3^{\left(\alpha \right)} I3(α)​ I 3 α with outstanding accuracy.
We investigate the effect of a single spin flip preceding a global quench between translationally invariant local Hamiltonians in spin-1/2 chains. The effect of the localized perturbation does not fade away however large the distance from the perturbation is. In particular, translational invariance is not restored and the infinite-time limit depends on whether the spin was flipped or not. We argue that this phenomenon is more general than the particular example considered and we conjecture that it is triggered by topological properties, specifically, the existence of "semilocal charges."
We investigate the effect of a single spin flip preceding a global quench between translationally invariant local Hamiltonians in spin-12\tfrac{1}{2}21​ chains. The effect of the localised perturbation does not fade away however large the distance from the perturbation is. In particular, translational invariance is not restored and the infinite time limit depends on whether the spin was flipped or not. We argue that this phenomenon is more general than the particular example considered and we conjecture that it is triggered by topological properties, specifically, the existence of "semilocal charges".
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
We consider the quantum XY model and study the effects of interacting perturbations on the time evolution of the von Neumann and Rényi entropies of spin blocks after global quenches. We show that the entropies are sensitive to perturbations that break hidden symmetries behind the integrability of the model. At times much larger than the characteristic time of the well-known linear increase of the entropies, we identify a time window characterized by a novel linear growth followed by saturation. The typical time of the phenomenon is inversely proportional to the perturbation strength and the behavior is trigger off by the extinction of an infinite number of local conservation laws following a non-abelian algebra. The universality of the crossover is revealed by a semi-classical picture that captures the leading behavior of the entropies. We check our theoretical predictions against iTEBD simulations. The good agreement between theory and numerics substantiates the method developed in [Bertini and Fagotti, J. Stat. Mech. (2015) P07012] for investigating a pre-relaxation limit in weakly interacting models.