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Journal of Physics A Mathematical and Theoretical

Integrability-breaking-induced Mpemba effect in spin chains

Adam J. McRoberts

Published Jun 5, 2026• 0 citations
Publisher

Abstract

Abstract We show that there are two distinct mechanisms that can cause the symmetry-restoration Mpemba effect in spin chains with weakly broken integrability, such that the asymptotic equilibration is diffusive, but the lifetime of anomalously fast spin hydrodynamics at low temperature is parametrically large. In particular, we consider isotropic classical spin chains quenched out of equilibrium by suppressing the z -components, without inducing any net magnetisation. Initially, the restoration of isotropy is faster in hotter systems—because they have more phase space available to scramble their initial conditions—which may cause the equilibration curves to cross at early times in both integrable and non-integrable systems. At later times, however, the equilibration is effectively hydrodynamic, and the colder systems start to equilibrate faster as the lifetime over which they evince superdiffusive spin hydrodynamics is parametrically larger—but only in non -integrable models. Depending on the details of the temperatures and the extent of the initial symmetry-breaking, two isotropy-restoration curves may have a crossing at early time, late time, neither, or both.

DOI
10.1088/1751-8121/ae7920

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We show that there are two distinct mechanisms that can cause the symmetry-restoration Mpemba effect in spin chains with \textit{weakly broken} integrability, such that the asymptotic equilibration is diffusive, but the lifetime of anomalously fast spin hydrodynamics at low temperature is parametrically large. In particular, we consider isotropic spin chains quenched out of equilibrium by suppressing the $z$-components, without inducing any net magnetisation. Initially, the restoration of isotropy is faster in hotter systems -- because they have more phase space available to scramble their initial conditions -- which may cause the equilibration curves to cross at early times in both integrable and non-integrable systems. At later times, however, the equilibration is effectively hydrodynamic, and the \textit{colder} systems start to equilibrate faster as the lifetime over which they evince superdiffusive spin hydrodynamics is parametrically larger -- but only in \textit{non}-integrable models. Depending on the details of the temperatures and the extent of the initial symmetry-breaking, two isotropy-restoration curves may have a crossing at early time, late time, neither, or both.

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arXiv:2405.04436•Journal of Statistical Mechanics Theory and Experiment•Oct 31, 2024•51 citations
Abstract We show that there are two distinct mechanisms that can cause the symmetry-restoration Mpemba effect in spin chains with weakly broken integrability, such that the asymptotic equilibration is diffusive, but the lifetime of anomalously fast spin hydrodynamics at low temperature is parametrically large. In particular, we consider isotropic classical spin chains quenched out of equilibrium by suppressing the z -components, without inducing any net magnetisation. Initially, the restoration of isotropy is faster in hotter systems—because they have more phase space available to scramble their initial conditions—which may cause the equilibration curves to cross at early times in both integrable and non-integrable systems. At later times, however, the equilibration is effectively hydrodynamic, and the colder systems start to equilibrate faster as the lifetime over which they evince superdiffusive spin hydrodynamics is parametrically larger—but only in non -integrable models. Depending on the details of the temperatures and the extent of the initial symmetry-breaking, two isotropy-restoration curves may have a crossing at early time, late time, neither, or both.
We show that there are two distinct mechanisms that can cause the symmetry-restoration Mpemba effect in spin chains with \textit{weakly broken} integrability, such that the asymptotic equilibration is diffusive, but the lifetime of anomalously fast spin hydrodynamics at low temperature is parametrically large. In particular, we consider isotropic spin chains quenched out of equilibrium by suppressing the zzz-components, without inducing any net magnetisation. Initially, the restoration of isotropy is faster in hotter systems -- because they have more phase space available to scramble their initial conditions -- which may cause the equilibration curves to cross at early times in both integrable and non-integrable systems. At later times, however, the equilibration is effectively hydrodynamic, and the \textit{colder} systems start to equilibrate faster as the lifetime over which they evince superdiffusive spin hydrodynamics is parametrically larger -- but only in \textit{non}-integrable models. Depending on the details of the temperatures and the extent of the initial symmetry-breaking, two isotropy-restoration curves may have a crossing at early time, late time, neither, or both.
In classical settings, the Mpemba effect occurs when a hotter system cools faster than an initially colder one. In quantum systems, this effect can be reinterpreted exploiting the concept of symmetries, with the asymmetry of a subsystem playing the role of temperature. A quantum Mpemba effect arises when a more asymmetric state restores the symmetry faster than a less asymmetric one. Previous work mainly focuses on closed systems characterized by thermal equilibration and Hamiltonian symmetries. In this paper, we analyze the dynamics of asymmetry in an open quantum many-body system featuring symmetry breaking and uncover dynamical behavior that appears to be unique to these settings. In the symmetric phase, we demonstrate the existence of a quantum Mpemba effect, which emerges as a direct consequence of a non-monotonic evolution of the asymmetry. In the broken-symmetry phase, we analyze the imbalance between the system's ability to increase or to decrease its asymmetry. Our results extend the notion of quantum Mpemba effects to open quantum many-body systems exhibiting symmetry-breaking phase transitions and establish them as a platform for observing and controlling anomalous relaxation phenomena.
Finite-temperature spin transport in integrable isotropic spin chains (i.e., spin chains with continuous non-Abelian symmetries) is known to be superdiffusive, with anomalous transport properties displaying remarkable robustness to isotropic integrability-breaking perturbations. Using a discrete-time classical model, we numerically study the crossover to conventional diffusion resulting from both noisy and Floquet isotropic perturbations of strength $\ensuremath{\lambda}$. We identify an anomalously-long crossover timescale ${t}_{★}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}6$ in both cases. We discuss our results in terms of a kinetic theory of transport that characterizes the lifetimes of large solitons responsible for superdiffusion.
Superdiffusion is surprisingly easily observed even in systems without the integrability underpinning this phenomenon. Indeed, the classical Heisenberg chain-one of the simplest many-body systems, and firmly believed to be nonintegrable-evinces a long-lived regime of anomalous, superdiffusive spin dynamics at finite temperature. Similarly, superdiffusion persists for long timescales, even at high temperature, for small perturbations around a related integrable model. Eventually, however, ordinary diffusion is believed to be asymptotically restored. We examine the timescales governing the lifetime of the superdiffusive regime, and argue that it diverges algebraically fast-both in deviation from the integrable limit, and at low temperature, where we find t^{*}∼T^{-ζ} with an exponent possibly as large as ζ=8. This can render the crossover to ordinary diffusion practically inaccessible.
Abstract Entanglement asymmetry is a quantity recently introduced to measure how much a symmetry is broken in a part of an extended quantum system. It has been employed to analyze the non-equilibrium dynamics of a broken symmetry after a global quantum quench with a Hamiltonian that preserves it. In this work, we carry out a comprehensive analysis of the entanglement asymmetry at equilibrium taking the ground state of the XY spin chain, which breaks theU(1) particle number symmetry, and provide a physical interpretation of it in terms of superconducting Cooper pairs. We also consider quenches from this ground state to the XX spin chain, which preserves theU(1) symmetry. In this case, the entanglement asymmetry reveals that the more the symmetry is initially broken, the faster it may be restored in a subsystem, a surprising and counter-intuitive phenomenon that is a type of a quantum Mpemba effect. We obtain a quasi-particle picture for the entanglement asymmetry in terms of Cooper pairs, from which we derive the microscopic conditions to observe the quantum Mpemba effect in this system, giving further support to the criteria recently proposed for arbitrary integrable quantum systems. In addition, we find that the power law governing symmetry restoration depends discontinuously on whether the initial state is critical or not, leading to new forms of strong and weak Mpemba effects.
Abstract Local relaxation after a quench in 1D quantum many-body systems is a well-known and very active problem with rich phenomenology. Except in pathological cases, the local relaxation is accompanied by the local restoration of the symmetries broken by the initial state that are preserved by unitary evolution. Recently, the entanglement asymmetry has been introduced as a probe to study the interplay between symmetry breaking and relaxation in an extended quantum system. In particular, using the entanglement asymmetry, it has been shown that the more a symmetry is initially broken, the faster it may be restored. This surprising effect, which has also been observed in trapped-ion experiments, can be seen as a quantum version of the Mpemba effect, and is manifested by the crossing at a finite time of the entanglement asymmetry curves of two different initial symmetry-breaking configurations. In this paper we show that, by tuning the initial state, the symmetry dynamics in free fermionic systems can display much richer behavior than seen previously. In particular, for certain classes of initial states, including the ground states of free fermionic models with long-range couplings, the entanglement asymmetry can exhibit multiple crossings. This illustrates that the existence of the quantum Mpemba effect can only be inferred by examining the late-time behavior of the entanglement asymmetry.