Quantum Mpemba Effect

• Quantum Mpemba Effect is a phenomenon where a state with greater initial symmetry breaking relaxes faster than one closer to equilibrium.
• It originates from charge fluctuations and quasiparticle dynamics that accelerate symmetry restoration through entanglement asymmetry decay.
• Analytical and numerical studies in models like free fermions and quantum cellular automata validate QME, offering insights for rapid equilibration in quantum systems.

The Quantum Mpemba Effect (QME) is a class of anomalous relaxation phenomena in quantum systems, characterized by the counter-intuitive result that a system initially further from equilibrium—often quantified by a larger amount of symmetry breaking or greater deviation in a relevant non-equilibrium observable—can restore symmetry or relax to equilibrium faster than another system prepared closer to the stationary state. In quantum integrable systems with a U(1)-conserved charge, this effect arises during the unitary relaxation of initially symmetry-broken states under Hamiltonian dynamics that preserve the symmetry. A microscopic criterion for the occurrence of QME can be constructed using the entanglement asymmetry, a measure quantifying how much a subsystem’s reduced density matrix deviates from the block structure imposed by the symmetry.

1. Conceptual Foundation and Definition

In classical systems, the Mpemba effect refers to the observation that, under certain conditions, hot water can freeze faster than cold water. The quantum analogue, as formulated for integrable many-body systems with a global U(1) charge, considers a relaxation scenario in which an initial state $\rho$ breaks the symmetry ($[\rho, Q] \neq 0$) and evolves under a symmetry-preserving, integrable Hamiltonian $H$ ($[H, Q] = 0$). The relaxation is studied through the time evolution of the reduced density matrix $\rho_A(t)$ of a subsystem $A$, which, by interacting with the rest of the system under unitary evolution, tends toward a symmetric stationary state.

The central diagnostic is the entanglement asymmetry,

$$\Delta S_A(t) = \mathrm{Tr}\left[\rho_A(t) (\log \rho_A(t) - \log \rho_{A,Q}(t))\right],$$

where $\rho_{A,Q}(t)$ is the U(1)-symmetrized reduced density matrix, obtained via projection onto the charge eigenspaces of the subsystem-restricted charge operator $Q_A$.

The QME is said to occur when, for two initial states with

$\Delta S_{A, 1}(0) > \Delta S_{A, 2}(0),$

one observes at some later time $t_M$ that

$\Delta S_{A, 1}(t_M) < \Delta S_{A, 2}(t_M),$

i.e., the more symmetry-broken (further-from-equilibrium) state overtakes the less broken (closer-to-equilibrium) state in the approach to symmetry restoration.

2. Microscopic Mechanism and Predictive Criteria

The microscopic origin of the QME is established through the joint consideration of charge fluctuations and their relaxation via quasiparticle transport. For a large subsystem of length $\ell$, the charge sector decomposition introduces coefficients

$$J_k(t) = \int_{-\pi}^\pi \frac{d\alpha}{2\pi} e^{ik\alpha} \exp\left\{ \ell \int d\lambda\, x_\zeta(\lambda) f_\alpha(\lambda) \right\},$$

where $x_\zeta(\lambda) = \max[1 - 2|v(\lambda)|\zeta, 0]$ with $\zeta = t/\ell$ (selecting slow modes), and $f_\alpha$ encodes mode-specific contributions. The initial charge fluctuation width $\sigma_0^2$ determines the probability distribution $J_k(0)$ (e.g., for large $\ell$,

$$J_k(0) \simeq \frac{1}{\sqrt{\pi \sigma_0^2}} \exp\left[ -\frac{(k - q_0)^2}{2\sigma_0^2} \right],$$

with $q_0$ the mean charge). A more asymmetric state has a broader distribution, smaller $J_{q_0}(0)$, and hence larger $\Delta S_A(0)$.

The relaxation of asymmetry is dominated by the ability of slowest quasiparticle modes to transport and "discharge" the symmetry-breaking fluctuations from the subsystem, reducing the weight $J_0(t)$. The QME occurs when a state with broader initial charge fluctuations (more asymmetry, larger $\sigma_0^2$) is associated with faster decay of the slow modes' contribution,

$$(i')\ J_{q_{0,1}}(0) < J_{q_{0,2}}(0), \quad (ii')\ J_{0,1}(t) > J_{0,2}(t),$$

so that the more asymmetric state overtakes the less asymmetric one in symmetry restoration.

3. Realizations in Integrable Many-Body Systems

The analytic criterion is exemplified in several models:

• Free Fermion Model: Squeezed initial states with non-trivial mode occupations yield charge variances $\sigma_0^2$ tied to $\theta(\lambda)$. The QME emerges if the state with larger charge susceptibility ($\mathcal{X}_1 > \mathcal{X}_2$) has a flatter second derivative of the mode occupation at zero ($\theta''_2(0) > \theta''_1(0)$), leading to weaker excitation of slow modes.
• Rule 54 Quantum Cellular Automaton: Exact calculation of $J_k$ from combinatorics reveals that higher variance in the charge sector population ($\theta(1-\theta)$ for parameter $\theta$) together with higher effective velocity $v_\theta$ (i.e., faster-moving slow modes) predicts QME, observable as crossings in time-dependent entanglement asymmetry.
• Lieb–Liniger Model: For coherent state quenches, analytic techniques using mode-occupation functions and local density of states determine that higher-density initial states (and thus broader distributions of charge) induce faster suppression of the slow-mode weights $J_0(t)$, yielding QME in the relaxation of entanglement asymmetry.

4. Analytical and Numerical Methods

The analysis relies on:

• Replica Trick and Fourier Methods: The entanglement asymmetry can be expressed through Rényi asymmetries and Fourier-projected charge sectors,

$$\Delta S_A^{(n)} = \frac{1}{1-n} \left[\log \mathrm{tr}(\rho_{A,Q}^n) - \log \mathrm{tr}(\rho_A^n)\right]$$

with charge projectors $\Pi_q$ constructed via Fourier integrals.

• Quasiparticle and Saddle-Point Approximations: In the thermodynamic limit, saddle-point evaluation of $J_k(0)$ quantifies the initial spread in the charge distribution, while time evolution under integrable dynamics isolates the role of quasi-local conserved charges in transmitting symmetry-breaking information.
• Numerical Simulations: For free models, the diagonalization of correlation matrices verifies the analytic predictions, plotting $\Delta S_A(t)$ over time and identifying the crossover points diagnostic of QME.

5. Implications, Extensions, and Open Directions

The emergence of the QME in integrable quantum systems points to a general scenario where anomalous relaxation is governed by the interplay between initial state fluctuations and the transport properties of the slowest excitations. This mechanism, while established in integrable settings, naturally raises questions regarding its persistence under weak integrability breaking, extension to fully chaotic dynamics, or relevance in open systems where coupling to the environment introduces decoherence and dissipation.

Potential future research avenues include:

• Probing QME in systems with weakly broken integrability or disorder to test the robustness of the effect and identify possible universality or breakdown.
• Extending the criterion to open-system dynamics, possibly connecting quantum and classical versions of the Mpemba effect.
• Exploiting insights from QME for engineered state preparation and rapid symmetry restoration in quantum technologies, where accelerated relaxation is advantageous.

6. Key Mathematical Expressions

Concept	Formula	Description
Entanglement Asymmetry	$$\Delta S_A(t) = \mathrm{Tr}[\rho_A(t)(\log \rho_A(t) - \log \rho_{A,Q}(t))]$$	Relative entropy between $\rho_A$ and its symmetry-projected form
Rényi Entanglement Asymmetry	$$\Delta S_A^{(n)} = \frac{1}{1-n} [\log \mathrm{Tr}(\rho_{A,Q}^n) - \log \mathrm{Tr}(\rho_A^n)]$$	Rényi generalization, $n\to 1$ gives von Neumann case
Charge Sector Projector	$$\Pi_q = \int_{-\pi}^{\pi} \frac{d\alpha}{2\pi} e^{i\alpha(Q_A - q)}$$	Projector onto subsystem charge sector $q$
Charged Moment Evolution	$$J_k(t) = \int_{-\pi}^{\pi} \frac{d\alpha}{2\pi} e^{ik\alpha} \exp\left\{ \ell \int d\lambda\, x_\zeta(\lambda) f_\alpha(\lambda) \right\}$$	Time-dependent weight for charge sector $k$
Asymmetry Time Evolution	$$\Delta S_A(t) = -\sum_k \mathrm{Re} [ J_k(t) \log J_k(t) ]$$	General analytic continued formula for asymmetry decay

7. Significance and Relation to Broader Research

The quantum Mpemba effect demonstrates that relaxation pathways in quantum many-body systems are determined not only by their proximity to equilibrium but by detailed properties of the initial non-equilibrium ensemble, specifically the distribution and transport of conserved quantities. The appearance of QME reveals the importance of symmetry sector fluctuations and quasiparticle diffusion as organizing principles for understanding, predicting, and potentially controlling anomalously fast or slow relaxation in quantum systems. The results in integrable systems provide a rigorous foundation and serve as a benchmark for exploring analogous phenomena in more complex or open quantum systems (Rylands et al., 2023).

This insight offers practical guidance for experimental protocols seeking rapid equilibration, as well as a theoretical framework connecting the structure of quantum states to macroscopic relaxation behavior.