Dynamics of rough surfaces generated by two-dimensional lattice spin models

Abstract

We present an analysis of mapped surfaces obtained from configurations of two classical statistical-mechanical spin models in the square lattice: the $q$-state Potts model and the spin-1 Blume-Capel model. We carry out a study of the phase transitions in these models using the Monte Carlo method and a mapping of the spin configurations to a solid-on-solid growth model. The first- and second-order phase transitions and the tricritical point happen to be relevant in the kinetic roughening of the surface growth process. At the low and high temperature phases the roughness $W$ grows indefinitely with the time, with growth exponent ${\beta }_w\simeq 0.50(W\sim t^{{\beta }_w})$. At criticality the growth presents a crossover at a characteristic time $t_c$, from a correlated regime (with ${\beta }_w\ne 0.50$) to an uncorrelated one $({\beta }_w\simeq 0.50)$. We also calculate the Hurst exponent $H$ of the corresponding surfaces. At criticality, ${\beta }_w$ and $H$ have values characteristic of correlated growth, distinguishing second- from first-order phase transitions. It has also been shown that the Family-Vicsek relation for the growth exponents also holds for the noise-reduced roughness with an anomalous scaling.