Numerical results for the random field Ising model (invited)

Abstract

We have performed numerical calculations for the random field (H) ferromagnetic Ising model in two dimensions. To study equilibrium properties, we have used the transfer matrix technique, thus bypassing serious equilibration problems which would arise in a Monte Carlo calculation for random systems. We have calculated the structure factor S vs H. Our results are consistent with S∼exp(c/H2) as predicted by theories which yield a lower critical dimension dc=2, whereas the expected behavior for dc=3, S∼H-4, is inconsistent with our results. To study dynamic properties we have used Monte Carlo simulations to determine the equilibration of Ising systems in random fields at low temperatures T following a quench from high T. The rate at which domains grow with time is determined as a function of the random field strength H, the linear dimension of the system L and temperature T. Domains are found to grow logarithmically with time. For small systems, L≤L*=(4J/H)2, the exponents a and b of the exponential equilibration time τ∼exp[(H/T)aLb] are found to be a≅1.0 and b≅0.5, in agreement with recent calculations based on approximate interface models. We tested the L and H/T dependence of τ in 3D and found a≅1.0, b≅0.5 also in 3D.