Bounds in partition functions of the continuous random field Ising model

Abstract

We investigate the critical properties of continuous random field Ising model (RFIM). Using the distributional zeta-function method, we obtain a series representation for the quenched free energy. It is possible to show that for each moment of the partition function, the multiplet of $k$-fields the Gaussian contribution has one field with the contribution of the disorder and $(k-1)$-fields with the usual propagator. Although the non-gaussian contribution is non-perturbative we are able to show that the model is confined between two ${\mathbb{Z}}_2\times \mathcal{O}(k-1)$-symmetric models. Using arguments of lower critical dimension alongside with monotone operators, we show that the phase of the continuous RFIM can be restricted by an ${\mathbb{Z}}_2\times \mathcal{O}(k-1)\to \mathcal{O}(k-2)$ phase transition.

6 pages