TheXYspin glass with slow dynamic couplings

Abstract

Abstract. We investigate an XY spin-glass model in which both spins and couplings evolve in time: the spins change rapidly according to Glauber-type rules, whereas the couplings evolve slowly with a dynamics involving spin correlations and Gaussian disorder. For large times the model can be solved using replica theory. In contrast to the XY-model with static disordered couplings, solving the present model requires two levels of replicas, one for the spins and one for the couplings. Relevant order parameters are defined and a phase diagram is obtained upon making the replica-symmetric Ansatz. The system exhibits two different spin-glass phases, with distinct de AlmeidaThouless lines, marking continuous replica-symmetry breaking: one describing freezing of the spins only, and one describing freezing of both spins and couplings.Recently various models with a coupled dynamics of fast Ising spins and slow couplings have been studied (see e.g. [1,5] and references therein). In addition to physical motivations, such as understanding the simultaneous learning and retrieval in recurrent neural networks or the influence of slow atomic diffusion processes in disordered magnetic systems, there is a more theoretical interest in such models in that they generate the replica formalism for a finite number of replicas n. Moreover, the replica number is found to have a physical meaning as the ratio of two temperatures (characterizing the stochasticity in the spin dynamics and the coupling dynamics, respectively). In this letter we extend the methods and results obtained for Ising spin models to a classical XY spin-glass with dynamic couplings, whose spin variables are physically more realistic than Ising ones. In addition, the XY model is closely related to neural network models of coupled oscillators, which provide a phenomenological description of neuronal firing synchronization in brain tissue. We solve our model upon making the replica-symmetric Ansatz, and calculate the de Almeida-Thouless (AT) lines [6] (of which here there are two types), where continuous transitions occur to phases with broken replica symmetry. In doing so we also improve the calculations of [3]. As §

Slowly evolving connectivity in recurrent neural networks: I. The extreme dilution regime

“…The fraction φ defined in (20) can be written as φ = φ ξ ξ , where the sublattice fractions φ ξ are expressed in terms of the replicated distributions (29) in the following way: for integer values of n between 1 and 5, and α = 0.1, α = 0.5 and α = 1 respectively, as a function of temperature. The degree of alignment of spins to their local fields increases with n outside the paramagnetic phase.…”

“…These areas have by now been investigated quite extensively. In contrast, only a modest number of studies involved coupled dynamical laws for both neurons and interactions [15,16,17,18,19,20,21,22], to reflect the complex dynamical interplay between synapses and neurons found in the real brain. The approach usually adopted in the latter studies, to obtain analytically solvable models, is the introduction of a hierarchy of adiabatically separated time-scales, such that the fast variables (taken to be the neurons) are in equilibrium on the time-scales where the slow variables (the interactions, taken to be symmetric) evolve.…”

“…Here the system is allowed to adapt its geometry autonomously. It should be emphasized that there is an important difference between having dynamic bonds with Hebbian type dynamical laws, as in [15,16,17,20,21,22], and the present situation of having dynamic geometry with fixed Hebbian values for active bonds. The former definitions imply irreversible modification or even elimination of stored information, whereas in the present paper, since the values assigned to the active bonds are not modified, the slow adaptation is fully reversible (one can always return to random dilution) and all stored information is retained.…”

Model glasses coupled to two different heat baths

“…This analogy between corresponding free energies was considered in the SK model with similar but more complicated techniques including two types of frozen variables [15]. Recently XY spin-glass model has been investigated by the same approach [16]. We shall show that a similar, perhaps more informative analogy can be found in the dynamics of the present model.…”

“…In this context Eq. (3.23) corresponds also to recently introduced models with two-level disorder: Coupling constants are also considered as frozen variables with respect to some other set of variables [15] [16].…”