Analytic structure of the one-dimensional random-bond Ising model

Abstract

The quenched average of the one-dimensional random-bond (+or-J) Ising model in a magnetic field is studied using a technique based on the transformation of a product of random 2*2 matrices to an iterated conformal map. This approach allows for the underlying analytic structure of the random matrix product to appear as a natural consequence of the conformal mapping of the extended complex plane. The quenched average of the characteristic exponent (i.e. the free energy) is obtained by averaging over a specified probability distribution of nearest-neighbour coupling. The evaluation of the quenched average of the characteristic exponent is performed within the isoentropic approximation. The isoentropic approximation treats all bond configurations with the same number of antiferromagnetic domains having the same entropy and enables the enumeration of equivalent configurations. Combinatorial methods can then be used to express the average over all realizations of the random matrix product as a combinatorial sum. The combinatorial sum is evaluated using resummation methods and an explicit expression for the quenched average of the characteristic exponent is thereby obtained for the case of the random-bond Ising model. The explicit expression for the free energy is dependent on constants which are calculated numerically.