Qi and Zheng (arXiv:2512.15223; published as Phys. Rev. B 113, 214311 (2026)) showed numerically that in the (1+1)-dimensional lattice Schwinger model the gauge symmetry of a subsystem, initially broken by preparing a superposition of gauge superselection sectors, is dynamically restored under gauge-invariant evolution for any finite Maxwell coupling, and that the restoration rate of each sector grows with the gauge-charge difference between sectors — a mechanism they exploited to engineer a quantum Mpemba effect (states with stronger initial symmetry breaking restoring faster). The supporting arguments combine a dephasing/spectral-broadening heuristic, perturbation theory, an unproven assumption of order-one local-operator matrix elements between eigenstates in different gauge sectors, and finite-size exact diagonalization. Two gaps remain open: (i) an analytic derivation of the gauge-sector-resolved relaxation rates and their dependence on the gauge-charge difference, with scaling in the hopping, mass, Maxwell coupling, and theta angle — the dependence the authors themselves flag as awaiting rigorous demonstration; and (ii) a rigorous proof of subsystem gauge-symmetry restoration in the thermodynamic limit for every finite Maxwell coupling, replacing the dephasing/spectral-broadening heuristics and the unproven matrix-element assumption on which the claimed result rests.

Background

The classical Mpemba effect is the counterintuitive observation that hot water can freeze faster than cold water. Its quantum analogue, the quantum Mpemba effect (QME), concerns relaxation of isolated many-body systems: after a quench, a state with stronger initial symmetry breaking can restore the symmetry of a subsystem faster than a more symmetric one. Symmetry breaking is quantified by the entanglement asymmetry (EA), introduced by Ares, Murciano, and Calabrese: for a subsystem AA with reduced density matrix ρA(t)\rho_A(t) and ρA,G(t)\rho_{A,G}(t) its group-twisted (symmetrized) counterpart, the EA is ΔSA(t)=S[ρA,G(t)]−S[ρA(t)]\Delta S_A(t)=S[\rho_{A,G}(t)]-S[\rho_A(t)], where SS is the von Neumann entropy; it vanishes exactly when ρA\rho_A is symmetric. The QME has been observed in trapped-ion quantum simulations (Joshi et al., 2024) and analyzed mostly in spin chains with global Abelian or non-Abelian symmetries, and is reviewed by Ares, Calabrese, and Murciano (2025).

Qi and Zheng (2025, published in Physical Review B in 2026) extended this framework to local gauge symmetries. They studied the lattice Schwinger model — the (1+1)-dimensional lattice version of quantum electrodynamics — governed by the Kogut–Susskind staggered-fermion Hamiltonian

H^=−w∑n(ψ^n†U^nψ^n+1+h.c.)+m∑n(−1)nψ^n†ψ^n+J∑n(L^n+θ2π)2,\hat{H}=-w\sum_{n}(\hat{\psi}^\dagger_n\hat{U}_n\hat{\psi}_{n+1}+\mathrm{h.c.})+m\sum_n(-1)^n\hat{\psi}^\dagger_n\hat{\psi}_n+J\sum_n\left(\hat{L}_n+\frac{\theta}{2\pi}\right)^2,

where ψ^n\hat{\psi}_n are fermion operators, U^n\hat{U}_n are gauge link operators, L^n\hat{L}_n are integer-valued electric fields, ww is the hopping, mm the staggered mass, JJ the Maxwell (electric-field energy) coupling, and θ\theta the topological angle. The Gauss-law operators G^n=L^n−L^n−1−ψ^n†ψ^n+12[1−(−1)n]\hat{G}_n=\hat{L}_n-\hat{L}_{n-1}-\hat{\psi}^\dagger_n\hat{\psi}_n+\frac{1}{2}[1-(-1)^n] commute with H^\hat{H} and generate a local U(1)U(1) gauge symmetry; their eigenvalues gn∈Zg_n\in\mathbb{Z} label gauge superselection sectors, the physical sector being gn=0g_n=0 for all nn. In quantum simulators, gauge charges are not restricted to the physical sector, and initial states may be superpositions of different sectors — that is, gauge-breaking states. Qi and Zheng showed that the block structure of ρA(t)\rho_A(t) in gauge sectors is fixed by the initial state, and studied whether gauge symmetry is dynamically restored. Their central findings: at J=0J=0 the gauge field is static and an emergent local conservation law makes the spectrum degenerate across sectors, so restoration fails; for any finite JJ, spectral broadening of the sector-transition frequency distribution z(ω)=∑αvα δωα,ωz(\omega)=\sum_\alpha v_\alpha\,\delta_{\omega_\alpha,\omega} — whose weights vα=⟨Egn∣ρini∣Eg′n′⟩⟨Eg′n′∣O^A∣Egn⟩v_\alpha=\langle E_{\boldsymbol{g}n}|\rho_{\mathrm{ini}}|E_{\boldsymbol{g}'n'}\rangle\langle E_{\boldsymbol{g}'n'}|\hat{O}_A|E_{\boldsymbol{g}n}\rangle are products of an initial-state overlap factor and a matrix element of a local subsystem operator O^A\hat{O}_A between energy eigenstates belonging to different gauge sectors, occurring at energy differences ωα\omega_\alpha — leads to dephasing and restoration in the thermodynamic limit, consistent with an intuitive appeal to Elitzur's theorem (the impossibility of spontaneously breaking a local symmetry at equilibrium). They then constructed families of initial states — superpositions of two gauge sectors with charges {0,0}\{0,0\} and {q,−q}\{q,-q\}, tuned so that states with larger initial EA decay faster — realizing the QME, and observed numerically that the sector relaxation rate increases with the gauge-charge difference ∣gA−gA′∣|g_A-g'_A|. Because the Schwinger model has an infinite-dimensional link Hilbert space, they also worked out the QME in the quantum link model, a finite-dimensional truncation of the electric field realized in trapped-ion and cold-atom experiments, and proposed a measurable order parameter O^=12∑i∈A(S^i++S^i+S^i+/2+h.c.)\hat{O}=\frac{1}{2}\sum_{i\in A}(\hat{S}^+_i+\hat{S}^+_i\hat{S}^+_i/\sqrt{2}+\mathrm{h.c.}) whose expectation value is nonzero only when gauge symmetry is broken and which reproduces the QME crossing behavior (the quantum link truncation and the spin operators S^i+\hat{S}^+_i entering O^\hat{O} are defined in Sec. IV of arXiv:2512.15223).

The arguments supporting these findings are, however, heuristic and numerical: the dephasing analysis assumes that matrix elements of local operators between eigenstates in different gauge sectors scale as O(1)O(1) with system size; the perturbative treatment is limited to small hopping; the numerics reach N≈16N\approx 16 sites; and the order parameter is only a sufficient indicator of gauge breaking, while the EA it replaces requires full state tomography. The authors explicitly flag, as an open challenge, the absence of a rigorous demonstration of the dependence of the sector relaxation rates on the gauge-charge difference; the remaining items are limitations they acknowledge in the heuristic and numerical arguments supporting their claimed all-finite-JJ restoration result.

Problem Statement

Establish the rigorous analytic foundations of the quantum Mpemba effect in subsystem gauge-symmetry restoration discovered numerically by Qi and Zheng in the lattice Schwinger model (and its finite-dimensional quantum link truncation). The overarching objective is a complete quantitative theory of gauge-sector relaxation in this setting, with two mutually supporting components. (1) Sector-resolved relaxation rates: derive analytically the decay rates of the off-diagonal gauge-sector blocks of the subsystem reduced density matrix ρA(t)\rho_A(t), including the dependence of these rates on each of the hopping ww, the mass mm, the Maxwell coupling JJ, the topological angle θ\theta, the subsystem size, and the total system size, and prove, or refute, the observed monotone dependence of the restoration rate on the gauge-charge difference ∣gA−gA′∣|g_A-g'_A| between the sectors in superposition — the mechanism underlying the Mpemba ordering. The parameter-resolved rate derivation and the monotone-dependence statement are each required parts of component (1). (2) Thermodynamic-limit restoration: replace the dephasing and spectral-broadening heuristics (in particular the unproven assumption that local-operator matrix elements between eigenstates in different gauge sectors are O(1)O(1) in system size) by a controlled proof that subsystem gauge symmetry is restored in the thermodynamic limit for every finite Maxwell coupling J>0J>0 and fails at J=0J=0. Component (1) supplies the mechanism — the dependence the authors explicitly flag as awaiting rigorous demonstration — and component (2) the limit in which the mechanism is exact; together they convert the heuristic and numerical picture into verified knowledge.

References

  1. Hao-Yue Qi and Wei Zheng, "Quantum Mpemba effect in Local Gauge Symmetry Restoration", arXiv:2512.15223 (2025), DOI: 10.48550/arXiv.2512.15223, https://arxiv.org/abs/2512.15223; published as "Quantum Mpemba effect in subsystem gauge symmetry restoration", Phys. Rev. B 113, 214311 (2026), DOI: 10.1103/b7ls-215n, https://doi.org/10.1103/b7ls-215n
  2. Filiberto Ares, Sara Murciano, and Pasquale Calabrese, "Entanglement asymmetry as a probe of symmetry breaking", Nat. Commun. 14, 2036 (2023), DOI: 10.1038/s41467-023-37747-8, https://doi.org/10.1038/s41467-023-37747-8
  3. Filiberto Ares, Pasquale Calabrese, and Sara Murciano, "The quantum Mpemba effects", Nat. Rev. Phys. 7, 451-460 (2025), DOI: 10.1038/s42254-025-00838-0, https://doi.org/10.1038/s42254-025-00838-0
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  5. Adrien Florio and Sara Murciano, "Entanglement asymmetry in gauge theories: Chiral anomaly in the finite temperature massless Schwinger model", arXiv:2511.01966 (2025), DOI: 10.48550/arXiv.2511.01966, https://arxiv.org/abs/2511.01966
  6. Shion Yamashika, Shimpei Endo, and Hiroyasu Tajima, "Quantum Fisher Information as a Measure of Symmetry Breaking in Quantum Many-Body Systems", arXiv:2509.07468 (2025), DOI: 10.48550/arXiv.2509.07468, https://arxiv.org/abs/2509.07468
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  9. Zhenyu Xiao, Hao-Kai Zhang, and Shuo Liu, "Nonstabilizerness Mpemba Effects", arXiv:2605.04155 (2026), DOI: 10.48550/arXiv.2605.04155, https://arxiv.org/abs/2605.04155