1,720,991 research outputs found
Analytic solution of the random Ising model in one dimension
No full textAn analytic expression is derived for the Lyapunov exponents of the product of random transfer matrices related to the Ising model with quenched disorder in one dimension. We find a deterministic map which transforms the original system into a new one with zero external field and constant coupling. The free energy and the rate of correlation decay are thus obtained in terms of an exponentially convergent series. Our results can be generalized to the product of random matrices with nonzero entries
Beyond the mean field approximation for spin glasses
No full textWe study the d-dimensional random Ising model using a suitable type of Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model can be mapped into a single Ising model with a random Gaussian field, whose strength (related to the effective coupling between two replicas) is determined via a self-consistency equation. This allows us to obtain analytic estimates of the internal energy and of the critical temperature in d dimensions
Gibbs thermodynamic potentials for disordered systems
No full textWe propose a new method to estimate the quenched free energy in disordered systems. It uses generalized thermodynamic potentials, given by annealed averages of the partition function with appropriate constraints realized by the aid of Lagrange multipliers. The method is applied to Ising models with random magnetic fields. In that particular case, the constraints correspond to averaging only over the disorder configurations where the sum of the random variables has the correct mean value and variance
Bethe-Peierls approximation for the 2D random Ising model
No full textThe partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method
Rigorous bound of the Lyapunov exponents of the one-dimensional random Ising-model
No full textWe find analytic upper and lower bounds of the Lyapunov exponents of the product of random matrices related to the one-dimensional disordered Ising model, using a deterministic map which transforms the original system into a new one with smaller average couplings and magnetic fields. The iteration of the map gives bounds which estimate the Lyapunov exponents with increasing accuracy. We prove, in fact, that both the upper and the lower bounds converge to the Lyapunov exponents in the limit of infinite iterations of the map. A formal expression of the Lyapunov exponents is thus obtained in terms of the limit of a sequence. Our results allow us to introduce a new numerical procedure for the computation of the Lyapunov exponents which has a precision higher than Monte Carlo simulations
Constrained annealing for Spin-Glasses
No full textThe quenched free energy of spin glasses can be estimated by means of annealed averages where the frustration or other self-averaging variables of disorder are constrained to their average value. We discuss the case of d-dimensional Ising models with random nearest neighbour coupling, and for +/-J spin glasses we introduce a new method to obtain constrained annealed averages without recurring to Lagrange multipliers. It requires to perform quenched averages either on small volumes in an analytic way, or on finite size strips via products of random transfer matrices. We thus give a sequence of converging lower bounds for the quenched free energy of 2d spin glasses
Mean-field solution of the random Ising model on the dual lattice
No full textWe perform a duality transformation that allows one to express the partition function of the d-dimensional Ising model with random nearest neighbor coupling in terms of spin variables defined on the square plaquettes of the lattice. The dual model is solved in the mean-field approximation
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