General criterion for the quantum Mpemba effect in symmetry restoration
In plain words
In some quantum spin chains, a state that breaks a symmetry more strongly at the start regains that symmetry faster in a small region than a weakly broken one. The task is a general rule, valid beyond special solvable models, for when this happens.
Precise statement
Chain of $L$ qubits with conserved charge $Q = (1/2) \sum_i \sigma^{z}_{i}$, initial tilted ferromagnet with tilt angle $\theta$ that breaks the $U(1)$ symmetry, unitary $U(1)$-symmetric dynamics (Hamiltonian or random circuit). For a subsystem of length $l << L$, the entanglement asymmetry $A_{l}(t)$ = S(rho_A,Q) - S(rho_A), with rho_A,Q the reduced density matrix projected onto charge sectors and S the von Neumann entropy. Find necessary and sufficient conditions on the initial-state charge statistics and on the dynamics (integrable or chaotic, closed or open) for A_l(t) of a larger-$\theta$ state to cross below that of a smaller-$\theta$ state at finite $t$; the analogous question for a non-Abelian group $G$ with charges $Q_{a}$ uses the corresponding $G$-asymmetry.
What would settle it
A proved criterion for the crossing of $A_l(t)$ in generic chaotic symmetric dynamics, tested against exact numerics and quantum-simulator data.
Status in the literature
Unverified note
Criteria exist for integrable chains (Rylands et al., PRL 2024) and U(1) random circuits (Liu et al., PRL 133, 140405, 2024; Turkeshi, Calabrese and De Luca 2024), with a trapped-ion observation (Joshi et al., PRL 133, 010402, 2024); thermal (energy) analogues in chaotic chains are treated hydrodynamically (Muller, Pappalardi and Fazio, arXiv:2604.11876, 2026); no general criterion for symmetry restoration in chaotic dynamics is established.