Dynamics within metastable states in a mean-field spin glass

In this letter we present a dynamical study of the structure of metastable states (corresponding to TAP solutions) in a mean-field spin–glass model. After reviewing known results of the statical approach, we use dynamics: starting from an initial condition thermalized at a temperature between the statical and the dynamical transition temperatures, we are able to study the relaxational dynamics within metastable states and we show that they are characterized by a true breaking of ergodicity and exponential relaxation. The recent developments in the theory of spin–glass dynamics [1] have made clearer the similarity of behaviour in spin glasses and in glasses [2, 3]. In this context it seems at the moment that a certain category of spin glasses, those which are described by a so-called one-step replica-symmetry breaking (RSB) transition [4], are good candidate models for a mean-field description of the glass phase [5, 6]. In these systems the presence of metastable states generates a purely dynamical transition (which is supposed to be rounded in finitedimensional systems [5, 6]) at a temperature Td higher than the one obtained within a theory of static equilibrium, Ts. The spherical p-spin spin glass introduced in [7, 8] is an interesting example of this category. It is a simple enough system in which the metastable states can be defined and studied by the TAP method [9]. In this paper we want to provide a better understanding of these metastable states, using a dynamical point of view. We shall show the existence of a true ergodicity breaking such that these metastable states, in spite of being excited states with a finite excitation free energy per spin, are actually dynamically stable even at temperatures above Td. Note that a connection between dynamics and TAP approach was made in [18], for a similar model, but not in the same spirit. The spherical p-spin spin glass describes N real spins si, i ∈ {1, . . . , N} which interact through the Hamiltonian H(s) = − ∑ 16i1<··· 2 case it shows an interesting spin–glass behaviour, simple enough to allow for detailed analytical treatment. † Unité propre du CNRS, associée à l’Ecole Normale Supérieure et à l’Université de Paris Sud, Paris, France. 0305-4470/96/050081+07$19.50 c © 1996 IOP Publishing Ltd L81 L82 Letter to the Editor In the static approach, one describes the properties of the Boltzmann probability distribution of this system. The replica method shows the existence of a static transition with a one-step RSB at temperature Ts [7]. This transition reflects the fact that, below Ts, the Boltzmann measure is dominated by a few pure states, a scenario which is well known from the random energy model [10]. Staying within a static framework, the TAP approach [11, 12] provides some more insight into the physical nature of this system. The TAP equations can be derived through a variational principle on the local magnetizations mi = 〈si〉, from a free energy f ({mi}) which is best written in terms of radial and angular variables, q and ŝi (with mi = √qŝi), in the form [11] f ({mi}) = qE ({ŝi}) − T 2 ln(1 − q) − 1 4T [(p − 1)q − pqp−1 + 1] (2) where the angular energy is E ({ŝi}) ≡ − ∑ 16i1<···<ip6N Ji1,...,ip ŝi1 . . . ŝip . (3) At zero temperature the TAP states are just unit vectors which minimize the angular energy E0. There actually exist such states for E0 ∈ [Emin, Ec = − √ 2(p − 1)/p]. Denoting by ŝ i one zero temperature state, of energy E 0 α , it gives rise at finite temperature T to one TAP state α given by