Full reduction of large finite random Ising systems by real space renormalization group.

Abstract

We describe how to evaluate approximately various physical interesting quantities in random Ising systems by direct renormalization of a finite system. The renormalization procedure is used to reduce the number of degrees of freedom to a number that is small enough, enabling direct summing over the surviving spins. This procedure can be used to obtain averages of functions of the surviving spins. We show how to evaluate averages that involve spins that do not survive the renormalization procedure. We show, for the random field Ising model, how to obtain Gamma(r)=<sigma(0)sigma(r)>-<sigma(0)><sigma(r)>, the "connected" correlation function, and S(r)=<sigma(0)sigma(r)>, the "disconnected" correlation function. Consequently, we show how to obtain the average susceptibility and the average energy. For an Ising system with random bonds and random fields, we show how to obtain the average specific heat. We conclude by presenting our numerical results for the average susceptibility and the function Gamma(r) along one of the principal axes. (In this work, the full three-dimensional (3D) correlation is calculated and not just parameters such nu or eta). The results for the average susceptibility are used to extract the critical temperature and critical exponents of the 3D random field Ising system.