Z. Phys, B - Condensed Matter 84, 95-114 (1991) Condensed Zeitschrift Matter for Physik B Springer-Verlag 1991 Dynamics of the finite SK spin-glass Harald Kinzelbach and Heinz Horner Institut fiir Theoretische Physik der Universit/it, Philosophenweg 19, W-6900 Heidelberg, Federal Republic of Germany Received January 31, 1991 The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of re- sponse and correlation function on time scales related to those free energy barriers which diverge with system size N. For the largest of these barriers we find a behav- ior proportional to N ~ with 2 = 1/3. 1. Introduction A large class of questions ranging from the field of com- binatorial optimization [-1, 2] to models for protein dy- namics [3], neural networks [1, 4], ecosystems [5], and prebiotic evolution [6] can be related to the properties of the low temperature phase of spin glass models. (For general reviews on spin glasses see [7, 8].) All those disordered systems are typically character- ized by a low temperature state with a large number of high free energy barriers separating different regions in phase space. Transitions between these regions are characterized by long relaxation times. The appropriate mean field model for spin glasses was introduced by Sherrington and Kirkpatrick (the SK-model) [9]. In this approach in the low temperature phase the heights of some of the barriers diverge in the thermodynamic limit (i.e. number of spins N ~ oo) [2]. At any temperature below a well defined critical temperature T~ the phase space is divided into a large number of components which are closed in a dynamical sense: the time evolution of the system is confined to one such component ('pure phase' or 'ergodic component') on all finite time scales. This means that ergodicity is broken in the thermody- namic limit [10]. The static SK-model for an infinite number of spins is solved by Parisi's replica symmetry breaking ansatz which produces a stable mean field solution [-2]. In a dynamical context a solution for the infinite sys- tem has been derived by Sompolinsky and Zippelius [12, 13]. In their treatment ill defined products of distribu- tions show up resulting from transitions between ergodic components on infinite time scales. In order to regularize them Sompolinsky considers a large but finite system of N spins and assumes the existence of a broad hierarch- ical spectrum of very long relaxation times diverging with N~ or. Taking the thermodynamic limit again leads to a well defined solution for the infinite system. It is essentially equivalent to the static Parisi treatment [12]. The structure of the spectrum of the diverging time- scales and their N-dependence as well as the dynamic behavior related to these times are questions which are investigated in the following by analytical means. The paper is organized as follows. In Sect. 2 the SK spin-glass model for N discrete clas- sical Ising spins is defined. The connection between breaking of ergodicity, free energy barriers and the oc- curence of time scales diverging in the limit N ~ co is discussed in a general dynamical framework for the SK model. Within this framework the time scales can be related to the structure of the phase space. The smallest diverging time scale, denoted by T1 in the following, gives the typical time the system needs to sample one of the regions in phase space which become ergodic compo- nents in the thermodynamic limit. The largest one, To, is connected with restauration of ergodicity in the finite system, i.e. with the time the system needs to sample the whole system. Additionally one identifies another exceptional scale T* related to the global up-down sym- metry of the model. Most of the results given in literature concerning the N-dependence of these scales are derived from numerical simulations. They are shortly summarized in the final part of this section.