Aging in Spherical Spin Glass Dynamics

Authors’ note

With the exception of this note, this entire document is machine written. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. However, the resulting document remains lacking in exposition and overall coherence. We are nonetheless releasing this manuscript on Hexagon to disseminate the result as quickly as possible, because we believe it may be of interest to the mathematics and physics communities. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

We thank Luca Leuzzi for many helpful comments on the theory of spherical spin glasses and his encouragement to explore their dynamics rigorously.

P.L. was partially supported by NSF grant DMS-2450004.

Introduction

Spin glass dynamics and aging

The static theory of mean-field spin glasses describes the Gibbs measure of a random Hamiltonian in high dimension. For the spherical $p$-spin model studied in this paper, the limiting free energy is given by the Crisanti–Sommers formula, a variational problem over probability measures on $[0,1]$ whose minimizer is the Parisi measure [46]. This formula was proved by Talagrand for even models [120] and by Chen in general [42]. For the pure model, the Parisi measure is a point mass at zero at high temperature, and it has one step of replica symmetry breaking at every temperature below the static transition [6, Section 5]; the two-point support property follows from [120, Proposition 2.2]. These results concern the equilibrium measure. In this paper, we study the dynamics.

The natural dynamics is the Langevin diffusion on the sphere, in which the configuration moves in the direction of increasing Hamiltonian, perturbed by Brownian noise; the inverse temperature sets the strength of the drift relative to the noise. This diffusion is reversible with respect to the Gibbs measure, and for each fixed dimension $N$, it converges to the Gibbs measure as time tends to infinity. At sufficiently low temperature, however, the spectral gap of its generator is exponentially small in $N$, with probability tending to one over the disorder [77], so the slowest relaxation modes have relaxation times that grow exponentially with $N$. On time scales that do not grow with $N$, the dynamics can stay far from equilibrium. In this paper, we take the limit $N \to\infty$ first, on every fixed time interval, and only then study the long-time behavior of the limiting dynamics. In this order of limits, the Gibbs measure and its Parisi measure need not describe the behavior at long times. The physical theory predicts instead that the system ages.

Aging is observed in disordered materials that do not reach equilibrium on any accessible time scale. After a rapid cooling to low temperature, called a quench, the response of such a material to a small perturbation depends on the time that has elapsed since the quench, and it becomes slower as the sample gets older. In spin glasses, Lundgren, Svedlindh, Nordblad, and Beckman found that magnetic relaxation in CuMn depends on the time spent at low temperature before the measurement [109]; the experimental picture is reviewed in [121]. In a trap model, the system jumps between traps and spends a random time in each of them. Bouchaud introduced weak ergodicity breaking through a trap model whose mean trapping time diverges: the system keeps aging, although the phase space does not split into mutually inaccessible regions [33, p. 1707]. Here the correlation measures the overlap of the configurations at two times, and the response measures the linear effect of a small force applied at the earlier time; we define both precisely in Section 1.2. In the formulation used for mean-field aging, the correlation with every fixed earlier configuration eventually vanishes, while relaxation becomes slower as the system gets older [52, arXiv version 1, Section 3].

Mean-field spin glasses with $p$-spin interactions are a setting in which aging can be derived from a microscopic dynamics. Derrida introduced the random energy model, in which the energies of different configurations are independent, as a limit of a family of $p$-spin models [62], and the Ising $p$-spin models themselves were studied by Gross and Mézard [85] and by Gardner [74]. Sompolinsky and Zippelius developed the dynamical mean-field theory of spin glass relaxation [115, 116]. This theory replaces the interaction of one coordinate with all the others by a random force and a memory of its own past, both determined self-consistently, and it describes the dynamics through closed equations for the correlation and response functions. Kirkpatrick and Thirumalai connected the dynamics of $p$-spin models with the glass transition of supercooled liquids, which are liquids cooled below their freezing temperature without crystallizing [97, 98]. For $p \ge3$, the dynamical equations resemble those of the mode-coupling theory of supercooled liquids, a self-consistent theory in which correlation functions satisfy nonlinear equations with memory and, at low temperature, develop a plateau, that is, a positive limit at large time differences (see [83] for a review). In these equations, the correlation of the equilibrium dynamics stops decaying to zero below a dynamical transition temperature. This temperature is higher than the static transition temperature, at which the Parisi measure changes. Kirkpatrick, Thirumalai, and Wolynes built on this correspondence a scaling theory of glasses in finite dimension, in which relaxation below the dynamical transition proceeds by thermally activated rearrangements, where the noise carries the system across barriers of the energy landscape [99]. The spherical $p$-spin model was solved at equilibrium by Crisanti and Sommers [46]. Crisanti, Horner, and Sommers studied its dynamics in [48], including the dynamical transition above the static one and the power-law relaxation near it. Cugliandolo and Kurchan constructed an asymptotic solution of the nonequilibrium equations at long times [51]. We describe their predictions in Section 1.3, after we introduce the model and the equations that they analyzed.

The model and the Cugliandolo–Kurchan equations

We fix an integer $p \ge3$ and $\beta> 0$, and write $\xi(q) = q^p$. For $N \ge2$, let $S_N = \{x \in\mathbb{R}^N : \lVert x\rVert^2 = N\}$. The Hamiltonian $H_N$ is a centered Gaussian process on $\mathbb{R}^N$ with covariance

realized as in (3.1). This is the pure spherical $p$-spin model in the normalization of the rigorous static theory. Its Gibbs measure at inverse temperature $\beta$ has density proportional to $e^{\beta H_N}$ with respect to the uniform probability measure on $S_N$. Let $X_t$ be the Langevin dynamics on $S_N$ with generator

where $\Delta_S$ and $\nabla_S$ are the Laplace–Beltrami operator and gradient on $S_N$; equivalently, $X_t$ solves (3.3), whose noise has amplitude $\sqrt{2}$. The Gibbs measure is reversible for this diffusion. The initial point $X_0$ is uniform on $S_N$ and independent of the disorder and of the Brownian motion. Since the uniform measure is the Gibbs measure at $\beta=0$, this initial condition models a quench from infinite temperature. The drift points in the direction of increasing $H_N$, so the limiting energies and the threshold energy in this paper are positive; physics works usually minimize the Hamiltonian, which reverses these signs.

Since the uniform initial law is not invariant for the dynamics, its natural observables depend on two times. The correlation

is the overlap of the configurations at times $s$ and $t$. The response describes how the configuration at a time $t$ reacts to a small force applied at an earlier time $s$. Precisely, for a deterministic locally square-integrable force $h(u)$ projected onto the tangent space of $S_N$, the response is the zero-initial strong solution $V_t$ of the forced linearized equation along $X$, with the drift and noise derivatives evaluated on $X$. It satisfies $V_t=\int_0^t U_N(t,u)h(u)\,\mathrm{d}u$ for the random response matrix $U_N(t,u)$ (§ 4.1.1). We write

for its normalized trace, averaged over the disorder, the initial point, and the noise. Thus $\bar r_N(t,s)$ is the mean linear response of a coordinate at time $t$ to a unit impulse of force on the same coordinate at time $s$, that is, to a force concentrated near time $s$ with unit time integral. We measure the response to a force added to the drift without a factor $\beta$. The finite-dimensional response has diagonal value $\bar r_N(t,t)=\kappa_N=(N-1)/N$, because the force is projected onto the $(N-1)$-dimensional tangent space, and its limit below has diagonal value one (Remark 3.1).

As $N\to\infty$, these functions have deterministic limits on fixed time horizons. For every fixed $0\leq T<\infty$, the correlation $C_N(t,s)$ converges in probability to a function $C(t,s)$, uniformly on $[0,T]^2$, and the averaged response $\bar r_N(t,s)$ converges to a function $R(t,s)$, uniformly on $0 \le s \le t \le T$. Together with a function $\lambda(t)$, which keeps $C(t,t) = 1$, the limits form the unique solution of the system of integro-differential equations

for $t \ge s \ge0$, with $C(t,t) = R(t,t) = 1$, where $C$ is extended to $s > t$ by symmetry. We call these the Cugliandolo–Kurchan (CK) equations; they are stated with their conventions in (3.9)–(3.12). Existence and uniqueness of the solution were proved by Dembo, Guionnet, and Mazza [60, Proposition 1.1], and the solution satisfies $0 \le C \le1$ and $R \ge0$ (see Proposition 3.2). Sellke proved fixed-horizon convergence of $C_N$ and of the Brownian observable representing the integrated response [112, Theorem 4.3]. The force-response identification in Lemma 3.4 equates the annealed mean of its rescaled version $\Gamma_N(t,s) = X_t \cdot B_s/(\sqrt{2}N)$ with $\int_0^s \bar r_N(t,u)\,\mathrm{d}u$, where $B$ drives the diffusion. He built on the theorem of Ben Arous, Dembo, and Guionnet for the soft-spherical dynamics [24, Theorem 1.2], in which the constraint $\lVert X_t\rVert^2 = N$ is replaced by a confining potential. We state these results in our normalization in Proposition 3.3 and Lemma 3.4, and we deduce the uniform convergence of $\bar r_N$ in Remark 3.5. Throughout the paper, the CK solution is this solution $(C,R,\lambda)$, which describes the Langevin dynamics started from the uniform point.

The CK equations involve only $C$, $R$, and $\lambda$; the disorder no longer appears. The function $\lambda$ is a Lagrange multiplier for the spherical constraint, and the integrals are memory terms. In the physical derivation through dynamical mean-field theory, each coordinate of $X_t$ behaves in the limit like the solution $x_t$ of the linear equation (see [49, arXiv version 2, Section 9.5])

Here $x_0$, $b$, and $\eta$ are independent; $x_0$ is a standard Gaussian variable, $b$ is a standard Brownian motion, and $\eta$ is a centered Gaussian process with covariance $\mathbb{E}\eta_t\eta_u = \beta^2\xi'(C(t,u))$. The interaction of one coordinate with all the others produces the memory kernel $\beta^2\xi''(C(t,u))R(t,u)$, which weights the effect of the past values of $x$ on its present drift, and the noise $\eta$, which is called colored because its values at different times are correlated. The CK equations express the self-consistency conditions that $C(t,s) = \mathbb{E}x_t x_s$, that $R(t,s)$ is the response of $x_t$ to a force applied at time $s$, and that $\mathbb{E}x_t^2 = 1$, which determines $\lambda$. We use this effective process only as a guide. Its variation-of-constants formula writes $C(t,s)$ as the sum of the contributions of the initial condition, of the Brownian noise, and of the colored noise, each propagated by the response. This is the Green covariance identity of Proposition 4.2, which we prove directly from the CK equations. We write