Cluster evolution and critical cluster sizes for the square and triangular lattice Ising models using lattice animals and Monte Carlo simulations

Abstract

Growth and decay of clusters at temperatures below T c have been studied for a two-dimensional Ising model for both square and triangular lattices using Monte Carlo (MC) simulations and the enumeration of lattice animals. For the lattice animals, all unique cluster configurations with their internal bonds were identified up to 25 spins for the triangular lattice and up to 29 spins for the square lattice. From these configurations, the critical cluster sizes for nucleation have been determined based on two (thermodynamic) definitions. From the Monte Carlo simulations, the critical cluster size is also obtained by studying the decay and growth of inserted, most compact clusters of different sizes. A good agreement is found between the results from the MC simulations and one of the definitions of critical size used for the lattice animals at temperatures T > ∼0.4 T c for the square lattice and T > ∼0.2 T c for the triangular lattice (for the range of external fields H considered). At low temperatures (T ≈ 0.2 T c for the square lattice and T ≈ 0.1 T c for the triangular lattice), magic numbers are found in the size distributions during the MC simulations. However, these numbers are not present in the critical cluster sizes based on the MC simulations, as they are present for the lattice animal data. In order to achieve these magic numbers in the critical cluster sizes based on the MC simulation, the temperature has to be reduced further to T ≈ 0.15 T c for the square lattice. The observed evolution of magic numbers as a function of temperature is rationalized in the present work.