Quantum Mpemba effects from symmetry perspectives

A new review article published in the AAPPS Bulletin on July 2, 2025, examines the quantum Mpemba effect through the specific lens of symmetry breaking and restoration in closed quantum many-body systems. Authored by Hui Yu, Shuo Liu, and Shi-Xin Zhang, the study provides a concise overview of this counterintuitive phenomenon, where systems initially exhibiting stronger symmetry breaking can restore symmetry faster than those with weaker initial asymmetry. The review distinguishes this closed-system manifestation from open quantum systems, where environmental dissipation typically dominates relaxation dynamics. Instead, the quantum Mpemba effect in isolated systems is driven by intrinsic quantum fluctuations following a global quantum quench, where the system is prepared in a pure, non-equilibrium state and evolves under unitary dynamics.

The classical Mpemba effect, where a hotter system cools faster than a colder one, has been debated for centuries and observed in various contexts such as supercooling and granular fluids. However, a universally accepted explanation for the classical version remains absent, with proposed mechanisms ranging from differences in thermal conductivity to complex non-equilibrium dynamics. The quantum version of this effect has recently expanded into the realm of quantum mechanics, manifesting in both open systems interacting with an environment and isolated systems evolving under unitary dynamics. The review focuses specifically on the isolated system case, highlighting how the effect emerges when a system is quenched from a highly asymmetric initial state under a U(1)-symmetric Hamiltonian.

To quantify the distance of a system from equilibrium, the authors introduce two complementary diagnostic measures: entanglement asymmetry and charge variance. Entanglement asymmetry is defined as the difference between the Rényi entropy of the reduced density matrix of a subsystem and its symmetry-projected counterpart. This measure originates from the study of entanglement properties in quantum many-body systems and has been widely employed to quantify symmetry breaking. The entanglement asymmetry satisfies two key properties: it is non-negative and equals zero if and only if the reduced density matrix is equal to its symmetry-projected version, thereby quantifying the degree of symmetry breaking in the subsystem.

Charge variance serves as a parallel metric, defined as the difference between the expectation value of the square of the total charge operator and the square of the expectation value of the charge operator. If the state resides entirely within a single charge sector, the charge variance is zero; conversely, if the charge distribution spans multiple sectors, the variance becomes nonzero. Unlike entanglement asymmetry, charge variance can be measured directly on the computational basis, offering a simpler experimental protocol. These two metrics provide distinct insights into equilibration, with entanglement asymmetry acting as a practical proxy for the distance to an equilibrium state in the context of symmetry restoration.

The review analyzes the early- and late-time dynamics of these quantities under Hamiltonian evolution and random unitary circuits. In the case of U(1)-asymmetric initial states evolving under U(1)-symmetric random circuits, the quantum Mpemba effect emerges in the early-time dynamics of entanglement asymmetry. States with stronger initial asymmetry, characterized by larger entanglement asymmetry, decay faster than those with weaker asymmetry. This behavior originates from distinct thermalization rates across different charge sectors, where states with greater overlap in the Q_A=0 sector thermalize more slowly due to the restricted Hilbert space dimension. The effect is observed for tilted ferromagnetic and domain wall states but is absent for tilted antiferromagnetic states, which display no quantum Mpemba signature.

When the circuit comprises only random Haar gates without any symmetry, the quantum Mpemba effect vanishes because no well-defined charge sectors exist. The absence of the effect persists in Z_2-symmetric circuits due to the equal Hilbert space dimensions associated with the two parity sectors. However, the effect re-emerges in SU(2)-symmetric circuits due to the varying sizes of symmetric sectors. This highlights the critical role of symmetry structure in determining the presence and nature of the quantum Mpemba effect. The review further investigates the robustness of the effect when the subsystem’s symmetry remains broken at late times, revealing that the quantum Mpemba effect in entanglement asymmetry remains robust against weak symmetry-breaking perturbations.

In Hamiltonian dynamics, the quantum Mpemba effect is demonstrated for both integrable and non-integrable systems. For integrable quantum systems, the effect is explained by the propagation of quasiparticle excitations. The initial state acts as a source of quasiparticle pairs emitted homogeneously, propagating at fixed velocities. The quantum Mpemba effect emerges because more asymmetric initial states contain more symmetry-breaking quasiparticle pairs, and these pairs propagate at state-dependent velocities. When the dominant symmetry-breaking pairs in a highly asymmetric state happen to be faster, they exit the subsystem more quickly, causing faster decay of entanglement asymmetry. This mechanism links the effect to the initial quasiparticle density and their velocities, which are computable via Bethe Ansatz for generic integrable systems.

For non-integrable chaotic systems, the mechanism is similar to that in symmetric circuit cases, where relaxation speeds differ for symmetry sectors of different dimensions. The late-time entanglement asymmetry in chaotic systems vanishes in the thermodynamic limit, as the reduced density matrix of a small subsystem thermalizes to the equilibrium finite-temperature state. In contrast, for integrable systems, the steady-state entanglement asymmetry is not necessarily zero, as the late-time states are expected to be the generalized Gibbs ensemble. This distinction underscores the different thermalization properties of integrable and chaotic quantum systems and their impact on the persistence of the quantum Mpemba effect.

The review also explores the quantum Mpemba effect in many-body localized systems, which respect U(1) symmetry as particle conservation. Unlike integrable or chaotic systems, many-body localized phases exhibit the quantum Mpemba effect universally for any tilted product state. The timescale for the effect grows exponentially with subsystem size, reflecting the logarithmic lightcone nature of many-body localized dynamics. Remarkably, subsystem symmetry fully restores in the long-time limit, even though the steady state never thermalizes. These findings reveal a unique mechanism in many-body localized systems, distinct from thermal or integrable regimes, where symmetry restoration and the quantum Mpemba effect are present without thermal equilibrium.

The charge variance version of the quantum Mpemba effect emerges only in non-symmetric Hamiltonian dynamics. For tilted ferromagnetic states, the charge variance exhibits an early-time crossing, signifying the presence of the quantum Mpemba effect. This arises from the t^2 coefficient in the early-time expansion of the charge variance, which governs the initial dynamics. For small tilt angles, the charge variance initially grows concavely, while for larger angles, it exhibits a convex decay at early times. This qualitative difference in dynamical behavior underlies the emergence of the quantum Mpemba effect in the charge variance. The review concludes by discussing significant challenges and promising avenues for future research, including the potential for a unified description of classical and quantum anomalous thermalization.

Beyond theoretical models, the review details how entanglement asymmetry can be quantified experimentally using quantum hardware. One approach reconstructs the reduced density matrix of the subsystem through quantum state tomography supported by the classical shadow framework, projecting out the required block-diagonal components numerically. An alternative protocol prepares the block-diagonal state directly on quantum processors by applying symmetry-preserving local random unitary gates to decohere off-block-diagonal elements between charge sectors, followed by randomized measurements of the Rényi entropy. Although both approaches carry exponential measurement complexity with subsystem size, experimental implementations in trapped-ion quantum simulators have demonstrated that the quantum Mpemba effect remains resilient against noise and disorder. Crucially, these experiments revealed that when environmental decoherence alone drives relaxation, no crossings appear, proving that intrinsic quantum fluctuations are indispensable for generating the phenomenon.

The authors also examine non-symmetric dynamics where Hamiltonians or random circuits actively break symmetry. When symmetric initial states evolve under such dynamics, entanglement asymmetry displays early-time overshooting, surging to transient peaks that far exceed long-term values. In random circuit models doped with random Haar gates, this peak asymmetry follows a power-law scaling relative to the doping probability for probabilities below one tenth. Under anisotropic Hamiltonians with broken symmetry, late-time entanglement asymmetry oscillates around non-zero values rather than vanishing, because the subsystem thermalizes toward a local Hamiltonian that includes symmetry-breaking terms. Furthermore, the review demonstrates that the effect can withstand weak Hamiltonian symmetry breaking, persisting across specific anisotropy ranges such as 0.8 to 1 for ferromagnetic states and 0.4 to 1 for antiferromagnetic states before disappearing.

Looking toward broader applications, the review highlights how anomalous relaxation could aid quantum technology while framing unresolved theoretical puzzles. In quantum control and quantum simulation, the quantum Mpemba effect offers a non-adiabatic shortcut to target states, avoiding slow adiabatic cooling protocols by reaching symmetric configurations rapidly from highly non-equilibrium states. Future research directions outlined by the authors include evaluating whether analogous behavior persists under discrete or non-Abelian symmetries, boundary impurities, time-dependent perturbations, Floquet protocols, or long-range interactions in higher spatial dimensions. Clarifying potential relationships with dynamical phase transitions, prethermalization, the Kibble-Zurek mechanism, and discrete time crystals may ultimately establish whether anomalous equilibration reflects universal non-equilibrium physics or isolated model-dependent dynamics, potentially bridging classical and quantum thermalization.

Subject of Research: Quantum Mpemba effect and symmetry dynamics in closed quantum many-body systems

Article Title: Quantum Mpemba effects from symmetry perspectives

Article References: Yu, H., Liu, S., & Zhang, S.-X. (2025). Quantum Mpemba effects from symmetry perspectives. AAPPS Bulletin, 35(1), Article 17. https://doi.org/10.1007/s43673-025-00157-7

Image Credits: AI Generated

DOI: 10.1007/s43673-025-00157-7

Keywords: Quantum Mpemba effect, Symmetry breaking, Entanglement asymmetry, Charge variance, Quantum many-body systems, Non-equilibrium dynamics, Random unitary circuits, Hamiltonian evolution, Quantum, Mpemba, effects, symmetry