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

A convenient way to study phase transitions is by considering finite spin systems of linear size L with boundary conditions that produce 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 nonboundary states. Our main conclusion is that all singularities involve the correlation length xi(av)(T)similar to(T-c-T)(-nu) appearing in the average free-energy (F) over bar similar to[L/xi(av)(T)](d)(s) of the interface of dimension d(s)=d-1, except for the free-energy width Delta F similar to[L/xi(var)(T)](theta) that involves the droplet exponent theta and another correlation length xi(var)(T) that 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 [M. Nifle and H. J. 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.