Critical exponents of the random field hierarchical model

Abstract

We study the one-dimensional Dyson hierarchical model in the presence of a random field. This is a long range model where the interaction scales with the distance in a power-law-like form, $J(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\rho}}$, and we can explore mean-field and non-mean-field behavior by changing $\ensuremath{\rho}$. We analyze the model at $T=0$ and we numerically compute the non-mean-field critical exponents for Gaussian disorder. We also compute an analytic expression for the critical exponent $\ensuremath{\delta}$, and give an interesting relation between the critical exponents of the disordered model and the ones of the pure model, which seems to break down in the non-mean-field region. We finally compare our results for the critical exponents with the expected ones in $D$-dimensional short range models and with the ones of the straightforward one-dimensional long range model.