Finite-Size Effects in Disordered $\lambda {\varphi }^4$ Model

We discuss finite-size effects in one disordered $\lambda {\varphi }^4$ model defined in a $d$-dimensional Euclidean space. We consider that the scalar field satisfies periodic boundary conditions in one dimension and it is coupled with a quenched random field. In order to obtain the average value of the free energy of the system we use the replica method. We first discuss finite-size effects in the one-loop approximation in $d=3$ and $d=4$. We show that in both cases there is a critical length where the system develop a second-order phase transition, when the system presents long-range correlations with power-law decay. Next, we improve the above result studying the gap equation for the size- dependent squared mass, using the composite field operator method. We obtain again, that the system present a second order phase transition with long-range correlation with power-law decay.