A simpler probe of the quantum Mpemba effect in closed systems

TL;DR

This work develops a practical diagnostic for local relaxation in closed quantum systems by analyzing the relative entropy between a subsystem's reduced state and its stationary limit, showing it is well approximated by the difference of entanglement entropies with the diagonal ensemble. By linking this metric to time-translation symmetry via the entanglement asymmetry, the authors provide a unified perspective on relaxation dynamics and symmetry restoration. They apply this framework to the quantum Mpemba effect across integrable models and random unitary circuits, deriving analytic quasiparticle-based expressions and membrane-picture results that reproduce known Mpemba cases and reveal new ones, while also identifying regimes where the effect is suppressed. The findings enable efficient, universal access to Mpemba-type relaxation phenomena and clarify the roles of symmetry, integrability, and entanglement structure in determining when a farther-from-equilibrium state can relax faster.

Abstract

We study the local relaxation of closed quantum systems through the relative entropy between the reduced density matrix and its long time limit. We show, using analytic arguments combined with numerical checks, that this relative entropy can be very well approximated by an entropy difference, affording a significant computational advantage. We go on to relate this to the entanglement asymmetry of the subsystem with respect to time translation invariance. In doing this, we obtain a simple probe of the relaxation dynamics of closed many-body systems and use it to re-examine the quantum Mpemba effect, wherein states can relax faster if they are initially further from equilibrium. We reproduce earlier instances of the effect related to symmetry restoration as well as uncover new cases in the absence of such symmetries. For integrable models, we obtain the criteria for this to occur using the quasiparticle picture. Lastly, we show that, in models obeying the entanglement membrane picture, the quantum Mpemba effect cannot occur for a large class of initial states.

Figures (3)

• Figure 1: Numerical check of the approximation in Eq. \ref{['eq:result']}. We compare the relative entropy (solid lines) and the difference between entanglement entropies (symbols) of $\rho_A(t)$ and the diagonal ensemble $\rho_{{\rm d}, A}$ in a subsystem $A$ of $\ell$ contiguous sites. Both quantities have been obtained with exact diagonalization. (a) Quench in the XXZ spin-$1/2$\ref{['eq:xxz']} with $\Delta=0.5$ and $h=0.1$ from two tilted ferromagnetic configurations \ref{['eq:ferro']} (b) Quench in the spin-$1/2$ chain described by the next-nearest neighbor Hamiltonian \ref{['eq:nnn']} with $\Delta=1.5$, $\Delta_2=0.5$, and $J_2=1$ from two tilted Néel states \ref{['eq:neel']}. (c) Quench in the Mixed-Field Ising chain \ref{['eq:mf_ising']} with the parameters indicated in the main text and prepared in different tilted ferromagnetic states \ref{['eq:ferro']}. (d) Quench in the long-range XX spin-$1/2$ chain \ref{['eq:lr_xx']} with $\alpha=1$ and $J_0=1/4$, starting from two tilted ferromagnetic states \ref{['eq:ferro']}.
• Figure 2: Time evolution of the relative entropy between $\rho_A(t)$ and $\rho_{{\rm d}, A}$ after different quenches in integrable models as a function of $\zeta=t/\ell$. The solid curves in all panels were obtained using the approximation of Eq. \ref{['eq:result']} and the quasiparticle picture \ref{['eq:qp']}. (a) Quench from different ground states of the XY spin chain \ref{['eq:xy']} with parameters $(\gamma_0, h_0)$ to the XX spin chain $(\gamma=0, h=0)$. (b) Quench in the quantum Ising chain $(\gamma_0=\gamma=1)$ from several values of the external magnetic field $h_0$ to $h=0.2$. (c-d) Quench starting from tilted ferromagnetic states \ref{['eq:ferro']} with tilting angle $\theta$ in the XXZ spin chain \ref{['eq:xxz']}, taking $h=0$ and $\Delta=0.5$ (gapless chain, panel (c)) and $\Delta=2$ (gapped chain, panel (d)).
• Figure 3: (a) Schematic representation of the membrane picture. (b) Difference between the Rényi-2 entropies of $\rho_A(t)$ and $\rho_A(\infty)$ as a function of $\zeta=t/ \ell$ in a brickwork random unitary circuit initialized in a state with Rényi-2 entanglement entropy $s_0\log(q)\ell$ and $q=2$. The curves are the prediction of the membrane picture \ref{['eq:membrane']}, taking as the membrane tension Eq. \ref{['eq:tension']}.