Transfer matrix approach to the disordered Ising model on hierarchical lattices

Abstract

A disordered short-range Ising model on the diamond hierarchical lattice, where the magnetic coupling constants Jij=±1 occur with probabilities p and 1−p, is investigated within a transfer-matrix based framework. Results are obtained after the evaluation of a large number of independent samples, where each individual coupling constant is randomly chosen according to the given probability distribution. The iteration of an exact set of discrete maps leads to the values of thermodynamic functions until a large but finite generation. An approximate scheme is developed to extend the results for each individual sample to the thermodynamic limit. Thermodynamic functions are evaluated and the (T,p) phase diagram is obtained on the basis of the behavior of the correlation length. The critical value pN for the emergence of magnetic ordering is found to lie on the Nishimori line, and a small reentrant effect is observed. The exponent ν is evaluated for several values of p. Exact log-periodic oscillations, found when p=1, vanish rapidly as p decreases.