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arXiv:2509.05597 · Journal of High Energy Physics

Entanglement asymmetry and quantum Mpemba effect for non-Abelian global symmetry

Harunobu Fujimura, Soichiro Shimamori

Published Mar 25, 2026• 2 citations
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Abstract

A bstract Entanglement asymmetry is a measure that quantifies the degree of symmetry breaking at the level of a subsystem. In this work, we investigate the entanglement asymmetry in $$ \hat{su}{(N)}_k $$ su ̂ N k Wess-Zumino-Witten model and discuss the quantum Mpemba effect for SU( N ) symmetry, the phenomenon that the more symmetry is initially broken, the faster it is restored. Due to the Coleman-Mermin-Wagner theorem, spontaneous breaking of continuous global symmetries is forbidden in 1 + 1 dimensions. To circumvent this no-go theorem, we consider excited initial states which explicitly break non-Abelian global symmetry. We particularly focus on the initial states built from primary operators in the fundamental and adjoint representations. In both cases, we study the real-time dynamics of the Rényi entanglement asymmetry and provide clear evidence of quantum Mpemba effect for SU( N ) symmetry. Furthermore, we find a new type of quantum Mpemba effect for the primary operator in the fundamental representation: increasing the rank N leads to stronger initial symmetry breaking but faster symmetry restoration. Also, increasing the level k leads to weaker initial symmetry breaking but slower symmetry restoration. On the other hand, no such behavior is observed for adjoint case, which may suggest that this new type of quantum Mpemba effect is not universal.

DOI
10.1007/jhep03(2026)244

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A bstract Entanglement asymmetry is a measure that quantifies the degree of symmetry breaking at the level of a subsystem. In this work, we investigate the entanglement asymmetry in su^(N)k\hat{su}{(N)}_ksu^(N)k​ su ̂ N k Wess-Zumino-Witten model and discuss the quantum Mpemba effect for SU( N ) symmetry, the phenomenon that the more symmetry is initially broken, the faster it is restored. Due to the Coleman-Mermin-Wagner theorem, spontaneous breaking of continuous global symmetries is forbidden in 1 + 1 dimensions. To circumvent this no-go theorem, we consider excited initial states which explicitly break non-Abelian global symmetry. We particularly focus on the initial states built from primary operators in the fundamental and adjoint representations. In both cases, we study the real-time dynamics of the Rényi entanglement asymmetry and provide clear evidence of quantum Mpemba effect for SU( N ) symmetry. Furthermore, we find a new type of quantum Mpemba effect for the primary operator in the fundamental representation: increasing the rank N leads to stronger initial symmetry breaking but faster symmetry restoration. Also, increasing the level k leads to weaker initial symmetry breaking but slower symmetry restoration. On the other hand, no such behavior is observed for adjoint case, which may suggest that this new type of quantum Mpemba effect is not universal.
Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.
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.
The quantum Mpemba effect is the counterintuitive nonequilibrium phenomenon wherein the dynamic restoration of a broken symmetry occurs more rapidly when the initial state exhibits a higher degree of symmetry breaking. The effect has been recently discovered theoretically and observed experimentally in the framework of global quantum quenches, but so far it has only been investigated in one-dimensional systems. Here we focus on a two-dimensional free-fermion lattice employing the entanglement asymmetry as a measure of symmetry breaking. Our investigation begins with the ground-state analysis of a system featuring nearest-neighbor hoppings and superconducting pairings, the latter breaking explicitly the U(1)\text{U}(1)U(1) particle-number symmetry. We compute analytically the entanglement asymmetry of a periodic strip using dimensional reduction, an approach that allows us to adjust the extent of the transverse size, achieving a smooth crossover between one and two dimensions. Further applying the same method, we study the time evolution of the entanglement asymmetry after a quench to a Hamiltonian with only nearest-neighbor hoppings, preserving the particle-number symmetry which is restored in the stationary state. We find that the quantum Mpemba effect is strongly affected by the size of the system in the transverse dimension, with the potential to either enhance or spoil the phenomenon depending on the initial states. We establish the conditions for its occurrence based on the properties of the initial configurations, extending the criteria found in the one-dimensional case.
The Kramers-Wannier duality is the prototypical example of a non-invertible symmetry, yet little is known about its fate away from criticality and out of equilibrium. We introduce the Kramers-Wannier entanglement asymmetry, a quantum-information measure that quantifies the breaking of this non-invertible symmetry in the transverse-field Ising chain. We first investigate its equilibrium properties, showing that it exhibits a striking crossover between the ordered and disordered phases together with a pronounced dip at the critical point that becomes increasingly sharp with subsystem size. We then study quantum quenches from both gapped phases to criticality and show that the Kramers-Wannier entanglement asymmetry decays to zero, signaling the dynamical restoration of the non-invertible symmetry. Remarkably, we uncover the emergence of a quantum Mpemba effect: under suitable conditions, states initially farther from equilibrium restore the Kramers-Wannier symmetry faster than states prepared closer to it. We provide both analytical and numerical evidence for this phenomenon and identify the mechanism responsible for its occurrence. Our work establishes entanglement asymmetry as a powerful probe of non-invertible symmetries beyond equilibrium and opens new perspectives on the dynamics of dualities in quantum many-body systems.
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.
Entanglement asymmetry, which serves as a diagnostic tool for symmetry breaking and a proxy for thermalization, has recently been proposed and studied in the context of symmetry restoration for quantum many-body systems undergoing a quench. In this Letter, we investigate symmetry restoration in various symmetric random quantum circuits, particularly focusing on the U(1) symmetry case. In contrast to nonsymmetric random circuits where the U(1) symmetry of a small subsystem can always be restored at late times, we reveal that symmetry restoration can fail in U(1)-symmetric circuits for certain weak symmetry-broken initial states in finite-size systems. In the early-time dynamics, we observe an intriguing quantum Mpemba effect implying that symmetry is restored faster when the initial state is more asymmetric. Furthermore, we also investigate the entanglement asymmetry dynamics for SU(2) and Z_{2} symmetric circuits and identify the presence and absence of the quantum Mpemba effect for the corresponding symmetries, respectively. A unified understanding of these results is provided through the lens of quantum thermalization with conserved charges.