A dynamical theory of the saturated-remanence regime of the Sherrington–Kirkpatrick spin glass at finite temperature

Parisi and Ritort (J. Phys. I France 3, 969, 1993) simulated the decay of the remanent magnetization after removal of a saturating field in infinite-range and short-range spin glasses and identified, as the central open task of that study, the construction of a real dynamical theory for this strongly off-equilibrium regime at finite temperature in the Sherrington–Kirkpatrick (SK) model. Three decades later the task remains open in sharpened form. The exact out-of-equilibrium solution of Cugliandolo and Kurchan covers only random (unmagnetized) initial conditions and rests on weak ergodicity breaking, which presumes rather than derives the vanishing of the remanence in the thermodynamic limit; the only exact analytic result for the saturated protocol is the critical-point decay exponent $\lambda(T_c)=5/4$. Large-scale numerics find a decay exponent $\delta(T)\simeq T/T_c$ and a residual magnetization scaling $m(\infty)\propto N^{-1/4}$ but cannot exclude a nonzero thermodynamic limit, while 2019–2025 work on mean-field aging reports strong ergodicity breaking — memory of the initial condition that never fades — undermining the assumption behind $m(\infty)\to 0$. The problem asks for closed, solvable dynamical mean-field equations of the SK model at $T<T_c$ with saturated initial data, and for the derivation from them of the decay law of $m(t)$ and of the excess energy, and of the thermodynamic-limit fate and finite-size behavior of $m(\infty)$.

Background

A spin glass is a magnet in which randomly placed, mutually competing (frustrated) interactions freeze the atomic moments at low temperature into a disordered, history-dependent configuration rather than an ordered magnet. The Sherrington–Kirkpatrick (SK) model is its exactly mean-field version: $N$ Ising spins $\sigma_i=\pm 1$, every pair coupled by a random interaction $J_{ij}$ of zero mean and variance $1/N$, with Hamiltonian $H=-\sum_{i<j}J_{ij}\sigma_i\sigma_j-h\sum_i\sigma_i$. Because all spins interact, its equilibrium thermodynamics is exact in the thermodynamic limit ($N\to\infty$): Parisi's replica-symmetry-breaking (RSB) solution describes the low-temperature phase ($T<T_c$, with $T_c=1$ in the standard normalization) as a hierarchy of many pure thermodynamic states rather than a single Gibbs state. The alternative droplet theory of finite-dimensional spin glasses (Fisher and Huse, 1988) instead predicts a two-state landscape with thermally activated dynamics; the competition between these pictures has organized much of the field.

If a magnetic field stronger than the saturation field is applied at low temperature and then suddenly removed, the magnetization does not relax promptly: a remanent magnetization $m(t)$ decays extremely slowly — in experiments over hours or days — while the internal energy relaxes toward its equilibrium value. Experiments fit this decay with the phenomenological law $m(t)\propto M(-T\ln(t/\tau)/T_f)$, where $T_f$ is the freezing temperature and $\tau$ a microscopic time; that is, the decay is controlled by the combination $T\ln t$, the signature of thermally activated crossing of energy barriers. When the field change is small, the decay additionally depends on the waiting time spent before the change — the aging phenomenon, one of the defining glassy behaviors.

In 1993 Parisi and Ritort measured by Monte Carlo simulation the decay of the remanent magnetization and of the excess energy after removal of a saturating field, in several infinite-range models (the SK model; random lattices of fixed coordination number $z=4$ and $z=6$ with up to $N=100\,000$ spins; a $D=15$ hypercubic cell with $N=32\,768$) and in short-range models in three and four dimensions, with SK sizes up to $N=2048$. Both classes agreed reasonably with the experimental $T\ln t$ law at low temperature and with power-law fits $m(t)=m(\infty)+A t^{-\alpha}$, $E(t)=E(\infty)+b t^{-\beta}$; but the fitted exponents drifted with temperature ($\alpha=0.29,0.48,0.745$ at $T=0.2,0.4,0.6$ for $N=2048$), with system size, and with the coupling distribution ($\alpha\simeq 0.38$ for Gaussian versus $0.58$ for binary couplings at $T=0.4$, $N=512$), and the fitted residual magnetization $m(\infty)\sim 10^{-2}$ was compatible with a finite-size suppression of order $N^{-1/2}$, as Kinzel had argued for any finite system relaxing from the all-aligned state. Since mean-field theory is exact in the SK model, they concluded that explaining remanence within mean-field theory was an extremely interesting and still open task, the key step toward it being to solve the dynamical equations of the SK model in the off-equilibrium regime at finite temperature. At that time the analytic footholds were equilibrium and near-equilibrium (linear-response) dynamics and the zero-temperature solution for parallel dynamics of Eissfeller and Opper; the weak-field (aging) regime was explicitly deferred to future work.

The obstruction is that the saturated-remanence protocol drives the system far from equilibrium, where the fluctuation-dissipation theorem (FDT — the equilibrium identity relating the response to a small perturbation to the correlation of spontaneous fluctuations) does not hold. Mean-field dynamics then requires solving closed equations for two-time functions, the autocorrelation $C(t,t')$ and response $R(t,t')$, without an equilibrium reference state. Cugliandolo and Kurchan derived and solved these equations asymptotically for the SK model — the achievement that founded the modern theory of mean-field aging — but only for a random (unmagnetized) initial configuration, and under the assumption of weak ergodicity breaking: the postulate that the system eventually decorrelates from any fixed earlier configuration. As they state, this assumption itself “implies that the remanent magnetization $C(t,0)$ tends to zero in the absence of a magnetic field” — the vanishing of the saturated remanence in the thermodynamic limit is thus assumed, not derived. Within this framework the deferred weak-field aging regime has since been worked out in depth (fluctuation-dissipation ratios, the off-equilibrium generalization of the FDT relating response and correlation; the quasi-equilibrium interpretation of the thermoremanent decay — the weak-field counterpart of the protocol, in which a small rather than a saturating field is removed — by Franz and Virasoro; experimental field-scaling support by Zotev, Kenning and Orbach, and the experimental analysis of initial-condition effects in aging by Zotev, Rodriguez, Kenning, Orbach, Vincent and Hammann), but the saturated regime — the actual remanence protocol, with initial magnetization of order one — has remained without an analytic solution at $T<T_c$. The only analytic inroad is at the critical point, where the remanence decay exponent $\lambda(T_c)=5/4$ was computed exactly for soft spins (continuous spin variables replacing the Ising $\pm 1$ degrees of freedom) from a uniform initial condition by Parisi, Ranieri, Ricci-Tersenghi and Ruiz-Lorenzo and confirmed numerically; below $T_c$, the dedicated large-scale simulations of Marinari, Parisi and Rossetti ($N$ up to 18432, $T=0.25$–$0.75$, Gaussian couplings, fully magnetized start) found a decay exponent $\delta(T)\simeq T/T_c$ and a residual magnetization $m(\infty)\sim N^{-1/4}$ — compatible with vanishing in the thermodynamic limit, though in their words “a constant behavior, with a non-zero residual magnetization, cannot be excluded”.

The question has recently become more urgent rather than less. Very large GPU simulations of mean-field spin glasses (Bernaschi, Billoire, Maiorano, Parisi and Ricci-Tersenghi, 2020, of order $2^{64}$ spin updates on SK and diluted SK models, the finite-connectivity variant) report that two-time correlations during isothermal aging do not decay to zero — strong ergodicity breaking: the aging dynamics remains confined to a part of configuration space determined by the initial condition and the early dynamics, contradicting the weak-ergodicity-breaking hypothesis underlying the Cugliandolo–Kurchan solution. Analogous restricted, initial-condition-dependent dynamics was found in the spherical mixed $p$-spin model — a mean-field glass with continuous spins confined to a sphere and random interactions acting on $p$ spins at a time — by Folena, Franz and Ricci-Tersenghi, and Citro and Ricci-Tersenghi (2025) integrated the dynamical mean-field equations of the spherical 3+4-spin model to times of order $10^6$ and report solid evidence that “the initial condition is never forgotten”. If strong ergodicity breaking holds for the SK model as well, the expectation that the saturated remanence vanishes as $N\to\infty$ loses its theoretical basis, and the sign of the answer to the 1993 question is again at stake. A dynamical theory of the saturated-remanence regime at finite temperature would have to settle this regime from first principles, deciding the decay law, the fate of $m(\infty)$, and the status of the decay exponents.

Problem Statement

Construct the dynamical theory of the saturated-remanence regime of the Sherrington–Kirkpatrick (SK) spin glass at finite temperature, and settle its quantitative content.

Set-up. Consider the SK model, $N$ Ising spins with random couplings $J_{ij}$ of zero mean and variance $1/N$ (Gaussian, or binary $\pm 1/\sqrt{N}$), in the thermodynamic limit, evolved with relaxational single-spin dynamics (sequential Monte Carlo or Langevin). At $t=0$ the system is prepared in a saturated configuration — equilibrium in a field above the saturation field, i.e. all spins aligned, $m(0)=1$ — and the field is removed. The temperature is finite and in the spin-glass phase, $T<T_c$ with $T_c=1$ in the standard normalization; this is the regime in which the fluctuation-dissipation theorem does not hold.

The objective is single: a first-principles dynamical mean-field account of this protocol, replacing the phenomenological fits of Parisi–Ritort and their successors. Establishing it involves, first:

1. A closed and solvable set of dynamical equations (for the two-time correlation and response together with the one-time magnetization) carrying the saturated initial data, and their long-time solution, in which the ergodicity-breaking scenario (weak versus strong) and the fluctuation-dissipation structure are derived rather than assumed.

The following are expected content and open sub-questions of that single objective, not separate acceptance conditions:

2. The resulting long-time decay law of the remanent magnetization $m(t)$: its functional form (power law $m(\infty)+A t^{-\delta(T)}$ versus the phenomenological $T\ln(t/\tau)$ scaling) and the temperature dependence of the exponent. Whether the decay parameters also depend on the coupling distribution (Gaussian versus binary) is an open issue, suggested by the source's observation that agreement with the phenomenological law is best in the case of continuously distributed exchange interactions.

3. The fate of the residual magnetization $m(\infty)$ in the thermodynamic limit — zero or strictly positive — together with its finite-size behavior. Here $m(\infty)$ denotes the residual in the fit $m(t)=m(\infty)+A t^{-\delta}$: at any fixed finite $N$ in zero field the true $t\to\infty$ magnetization vanishes, so $m(\infty)$ is a fitted residual (a long-lived plateau), and the order of limits must be stated. If a finite-size scaling description $m(\infty)\sim N^{-\omega}$ holds, the theory should predict the exponent $\omega$.

4. The decay law of the excess internal energy $E(t)-E(\infty)$, where $E(\infty)$ is its long-time limiting value.

The generalization of the same question to other mean-field spin glasses (fixed-coordination random lattices, mixed $p$-spin models) is the natural extension of the objective, not a separate one.

References

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