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

TL;DR

This paper investigates finite-size effects and phase transitions in a disordered scalar field model, revealing critical lengths where second-order phase transitions occur with long-range correlations.

Contribution

It provides a detailed analysis of finite-size effects in a disordered ${ m f oldsymbol{ extlambda}oldsymbol{phi}^{4}}$ model, including one-loop and gap equation approaches, highlighting critical lengths for phase transitions.

Findings

Critical length exists where a second-order phase transition occurs.

Long-range correlations with power-law decay are observed at the transition.

Finite-size effects influence the phase behavior in disordered scalar field models.

Abstract

We discuss finite-size effects in one disordered ${\lambda}{\phi}^{4}$ 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.