Roughness exponent in two-dimensional percolation, Potts model, and clock model

Abstract

We present a numerical study of the self-affine profiles obtained from configurations of the q-state Potts (with $q=2,3,$ and 7) and $p=10\mathrm{}$ clock models as well as from the occupation states for site percolation on the square lattice. The first and second order static phase transitions of the Potts model are located by a sharp change in the value of the roughness exponent α characterizing those profiles. The low temperature phase of the Potts model corresponds to flat $\left(\alpha \simeq 1\right)$ profiles, whereas its high temperature phase is associated with rough $(\alpha \simeq 0.5)$ ones. For the $p=10\mathrm{}$ clock model, in addition to the flat (ferromagnetic) and rough (paramagnetic) profiles, an intermediate rough $(0.5<\alpha <1)$ phase—associated with a soft spin-wave one—is observed. Our results for the transition temperatures in the Potts and clock models are in agreement with the static values, showing that this approach is able to detect the phase transitions in these models directly from the spin configurations, without any reference to thermodynamical potentials, order parameters, or response functions. Finally, we show that the roughness exponent α is insensitive to geometric critical phenomena.