Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption

Abstract

A convenient way to study phase transitions of finite spins systems of linear size $L$ is to fix boundary conditions that impose the presence of a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension $d=2$ within Migdal-Kadanoff real space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of non-boundary states. Our main conclusion is that all singularities involve the correlation length $\xi_{av}(T) \sim (T_c-T)^{-\nu}$ appearing in the average free-energy $\bar{F} \sim (L/\xi_{av}(T))^{d_s}$ of the interface of dimension $d_s=d-1$, except for the free-energy width $\Delta F \sim (L/\xi_{var}(T))^{\theta}$ that involves the droplet exponent $\theta$ and another correlation length $\xi_{var}(T)$ which diverges more rapidly than $\xi_{av}(T)$. We compare with the spin-glass transition in $d=3$, where $\xi_{var}(T)$ is the 'true' correlation length, and where the interface energy presents unconventional scaling with a chaos critical exponent $\zeta_c>1/\nu$ [Nifle and Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale $L_{ch}(T)$ associated with the chaotic nature of the low-temperature phase, diverges more slowly than the correlation length.