In weakly non-integrable isotropic classical spin chains, a recent letter (A. J. McRoberts, J. Phys. A: Math. Theor. 59, 24LT01 (2026)) identified a distinct, integrability-based mechanism for the symmetry-restoration Mpemba effect — the phenomenon in which an initially more symmetry-broken ensemble restores a Hamiltonian symmetry faster than a less-broken one, so that two equilibration curves cross. Near the integrable point, the parametrically long-lived superdiffusive spin-transport window makes colder chains equilibrate faster at late times, while hot chains equilibrate faster at early times; the letter's evidence for the observable criterion, the early-time equilibration rate, and the crossover scalings is numerical or imported from equilibrium-correlation results. This problem asks for the analytic completion of that framework within the same quench-and-relax protocol: (i) a rigorous proof or counterexample that restoration of the second moment of the one-spin distribution implies restoration of the full distribution, justifying δKz=0\delta K_z=0 as the isotropy criterion; (ii) closed-form or controlled-approximation evaluation of the early-time equilibration coefficient Γ(T,η)\Gamma(T,\eta) away from T=∞T=\infty; (iii) a systematic microscopic derivation of the crossover scalings τ∼δ−3\tau\sim\delta^{-3} and τ∼T−8\tau\sim T^{-8} for the relaxation of the anisotropy, with the regime of validity; and (iv) the resulting characterization of the degree of integrability breaking δ\delta that maximises the speed of the late-time crossing (equivalently, that minimises the late-crossing time t∗t^{*}).

Background

The Mpemba effect denotes the counterintuitive situation in which a system prepared further from equilibrium relaxes to its final state faster than one prepared closer, so that the two relaxation curves cross at some intermediate time. In Markovian (dissipative) systems this is understood through the eigenmode structure of the transition matrix (Z. Lu and O. Raz, PNAS 2017; I. Klich, O. Raz, O. Hirschberg, M. Vucelja, PRX 2019). In closed, Hamiltonian systems the analogous phenomenon is the symmetry-restoration Mpemba effect (often called the quantum Mpemba effect, though it is not intrinsically quantum): initial statistical ensembles are prepared to break a symmetry of the Hamiltonian — such as spin rotations about the zz axis — and the restoration of that symmetry is tracked through an asymmetry measure; if the initially more symmetry-broken ensemble restores the symmetry faster, the equilibration curves cross. This has been studied extensively in integrable quantum spin chains (C. Rylands, E. Vernier, P. Calabrese, J. Stat. Mech. 2024; S. Murciano, F. Ares, I. Klich, P. Calabrese, J. Stat. Mech. 2024; C. Rylands et al., PRL 2024), in random circuits, and in quantum-simulation experiments; a recent review is F. Ares, P. Calabrese, S. Murciano, Nat. Rev. Phys. 7, 451 (2025).

The source letter (A. J. McRoberts, J. Phys. A: Math. Theor. 59, 24LT01 (2026); arXiv:2603.10164) studies this effect in classical isotropic spin chains whose integrability — the property of possessing infinitely many conserved quantities that constrains the dynamics and underlies exact solutions — is weakly broken. The models are one-dimensional chains of classical unit-length spins with ferromagnetic nearest-neighbour couplings: the integrable Ishimori chain (Y. Ishimori, J. Phys. Soc. Jpn. 1982), the non-integrable classical Heisenberg chain, and the interpolating family

H=−2Jγ−1∑ilog⁡(1+γ Si⋅Si+1−12),\mathcal{H}=-2J\gamma^{-1}\sum_{i}\log\left(1+\gamma\,\frac{\mathbf{S}_{i}\cdot\mathbf{S}_{i+1}-1}{2}\right),

which reduces to the Ishimori chain at γ=1\gamma=1 and to the Heisenberg chain as γ→0\gamma\to 0; δ=1−γ\delta=1-\gamma measures the degree of integrability breaking. Ferromagnetic isotropic chains exhibit spin superdiffusion — anomalous spin transport in the Kardar-Parisi-Zhang (KPZ) universality class, characterised by dynamical exponent 3/23/2 — which is exact in integrable chains (A. Das, M. Kulkarni, H. Spohn, A. Dhar, PRE 2019) and carried by long-lived solitons (localized nonlinear spin-wave excitations) in the non-integrable Heisenberg chain (A. J. McRoberts, T. Bilitewski, M. Haque, R. Moessner, PRB 2022). While non-integrable chains are asymptotically diffusive, the crossover away from superdiffusion is parametrically slow: within equilibrium spin dynamics the crossover time grows as τ∼δ−3\tau\sim\delta^{-3} with integrability breaking and τ∼T−8\tau\sim T^{-8} with temperature (A. J. McRoberts, R. Moessner, PRL 2024; C. McCarthy, S. Gopalakrishnan, R. Vasseur, PRB 2024), with parallel quantum results in J. De Nardis, S. Gopalakrishnan, R. Vasseur, B. Ware, PRL 2021 and K. Wang, J. E. Moore, PRB 2025.

The protocol of the letter is a quench-and-relax experiment: an ensemble of thermal states at temperature TT (sampled by heatbath Monte Carlo, a local Gibbs-sampling algorithm) is quenched by suppressing each spin's zz-component by a factor η\eta (with in-plane components rescaled so each spin stays unit length, and no net magnetisation induced), and then evolves under the same Hamiltonian that generated the thermal ensemble. The anisotropy is measured by δKz(t)=1−3⟨(Siz(t))2⟩\delta K_z(t)=1-3\langle (S_i^z(t))^2\rangle, where the average runs over sites and ensemble members; δKz=0\delta K_z=0 iff the state is isotropic. The Mpemba effect is said to occur for two ensembles when their δKz(t)\delta K_z(t) curves cross.

The letter establishes three things. First, early-time crossings — the classical analogue of the quantum Mpemba effect — occur in both integrable and non-integrable chains and are controlled by the initial equilibration coefficient Γ(T,η)=3(⟨z˙2⟩+⟨zz¨⟩)\Gamma(T,\eta)=3(\langle \dot z^2\rangle+\langle z\ddot z\rangle) obtained from a Maclaurin expansion of δKz(t)\delta K_z(t) in tt, where z˙\dot z and z¨\ddot z are initial velocities and accelerations evaluated on the quenched ensemble; Γ\Gamma is expressible through finite Boltzmann-weighted three- and five-site phase-space integrals that can be evaluated analytically only at T=∞T=\infty. Second, late-time crossings occur only in non-integrable chains — the integrability-breaking-induced Mpemba effect — because colder chains remain in the superdiffusive window for a parametrically longer time; inserting the crossover form D(t)∼D/(1+λt−1/3)D(t)\sim\mathcal{D}/(1+\lambda t^{-1/3}) for the time-dependent diffusion constant gives the hydrodynamic equilibration form δKz(t)∼κλt−4/3+t−1\delta K_z(t)\sim\kappa\sqrt{\lambda t^{-4/3}+t^{-1}} with crossover time τ∼λ3\tau\sim\lambda^3, and hence an explicit late-crossing time t∗=(κA2λA−κB2λBκB2−κA2)3t^{*}=\left(\frac{\kappa_A^2\lambda_A-\kappa_B^2\lambda_B}{\kappa_B^2-\kappa_A^2}\right)^3 for two ensembles A,BA,B. Third, depending on temperatures and quench parameters, two curves can show an early crossing, a late crossing, both, or neither; the value of δ\delta giving the fastest late crossing is non-universal. Crucially, the sufficiency of δKz=0\delta K_z=0 as an isotropy criterion rests on the numerically observed fact that the one-spin distribution ρt(z)\rho_t(z) of zz-components stays non-increasing in ∣z∣|z| (so all even moments remain below their equilibrium values and ⟨z2⟩=1/3\langle z^2\rangle=1/3 forces ρt\rho_t to be the equilibrium uniform distribution on [−1,1][-1,1]) — a statement supported only by simulation. What remains missing is the analytic layer: a rigorous criterion, a closed-form Γ\Gamma, a microscopic derivation of the crossover scalings for this nonequilibrium protocol, and the resulting optimal-δ\delta characterization.

Problem Statement

Develop the analytic theory of the symmetry-restoration equilibration curves δKz(t)\delta K_z(t) for the following pinned protocol, replacing the source letter's numerical evidence with derivation. The system is the interpolating classical spin-chain family H=−2Jγ−1∑ilog⁡(1+γ(Si⋅Si+1−1)/2)\mathcal{H}=-2J\gamma^{-1}\sum_i\log(1+\gamma(\mathbf{S}_i\cdot\mathbf{S}_{i+1}-1)/2) with degree of integrability breaking δ=1−γ\delta=1-\gamma (γ=1\gamma=1 is the integrable Ishimori chain, γ→0\gamma\to 0 the non-integrable Heisenberg chain); the initial ensemble is a thermal state at temperature TT with each spin's zz-component suppressed by a factor η\eta (in-plane components rescaled to preserve unit spin length, no net magnetisation), evolved under the same Hamiltonian. The four components of one common objective are:

  1. (Criterion) Prove, or give a counterexample to, the sufficiency of the second moment as an isotropy criterion: under this dynamics, if at some time tt the ensemble satisfies ⟨(Sz)2⟩t=1/3\langle (S^z)^2\rangle_t=1/3, must the full one-spin distribution ρt(z)\rho_t(z) equal the equilibrium uniform distribution on [−1,1][-1,1]? The numerically observed route is the property that ρt(z)\rho_t(z) is non-increasing in ∣z∣|z| at all times; a proof of that property (or of the implication directly) establishes δKz=0\delta K_z=0 as a rigorous criterion, while any single parameter set (γ,T,η)(\gamma,T,\eta) and time with ⟨(Sz)2⟩t=1/3\langle (S^z)^2\rangle_t=1/3 but ρt\rho_t non-uniform refutes it.

  2. (Early-time coefficient) Evaluate Γ(T,η)=3(⟨z˙2⟩+⟨zz¨⟩)\Gamma(T,\eta)=3(\langle \dot z^2\rangle+\langle z\ddot z\rangle) in closed form for finite temperature, or provide controlled approximations with explicit error bounds as functions of (T,η)(T,\eta); only the T=∞T=\infty limit is currently known analytically.

  3. (Crossover scalings) Starting from the microscopic dynamics of the interpolating family, derive the late-time hydrodynamic form δKz(t)∼κλt−4/3+t−1\delta K_z(t)\sim\kappa\sqrt{\lambda t^{-4/3}+t^{-1}} — in particular first-principles expressions for λ(δ,T)\lambda(\delta,T) and κ(T,η)\kappa(T,\eta) reproducing the crossover scalings τ∼λ3∼δ−3\tau\sim\lambda^3\sim\delta^{-3} and τ∼T−8\tau\sim T^{-8} for the anisotropy relaxation, and the regime of validity in (δ,T,η)(\delta,T,\eta) — rather than importing these scalings from equilibrium-correlation results.

  4. (Optimal integrability breaking) Using the results of component 3, characterize the value of δ\delta that maximises the speed of the late-time crossing (when one exists) — equivalently, that minimises the late-crossing time t∗t^{*} defined above — as a function of the two ensembles' temperatures and quench parameters, within component 3's range of validity. State the answer either as a closed form in (TA,ηA,TB,ηB)(T_A,\eta_A,T_B,\eta_B), or as the monotonicity of t∗t^{*} in δ\delta together with the resulting feasible range, or by showing that t∗t^{*} has no interior minimum in δ\delta.

These components jointly constitute a single objective: an analytic, rather than purely numerical, account of how weak integrability breaking controls the symmetry-restoration Mpemba effect in this protocol. Component 1 secures the observable in terms of which all crossings are stated; components 2 and 3 supply the analytic early-time and late-time branches of the equilibration curves; component 4 is their synthesis. All four components are required for a full resolution; a complete solution to any subset of them constitutes a partial result.

References

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