Simulations of Critical Properties of Classical Spin Models

Abstract

Two methods of simulations are described: an exact one -transfer matrix technique and a statistical one -Monte Carlo method. Both of them are applied to investigate critical properties of classical spin models. To do this we also exploit finite size scaling and the critical point ratio of the square of the second moment of the order parameter to its fourth moment. General definition of a classical spin model as well as particular definitions of models are presented. Results of both methods are in good agreement and, moreover, they are consistent with numerical results provided by literature. 1Classical spin models A spin is a quantity hard to understand on the grounds of classical physics. A notion of a spin comes from experimental results and as a physical quantity it is described by a spin operator in the formalism of quantum mechanics. As yet only in this formalism a definition of a spin is reasonable. Thus, when we talk about spin in the context of classical models we do not mean a quantum spin operator. A name "spin" is used to call a set of numbers labeling states of an atom or a molecule, or in general, a site of a lattice. The name comes from the fact that these numbers are eigenvalues of the z component of a real quantum spin operator. So, defining a quantum model (e.g. of a solid located at each site i of a lattice, but when we define a classical spin model we use variables S i located at each site i of a lattice, where S i is an eigenvalue of a z component of an operator Classical spin models may be sometimes regarded as approximation of quantum spin models, but not always. Sometimes a spin variable is used to write 39 state) we use operators

Visualization of Monte Carlo Spin Clusters for the Ising Model in 2 Dimension

“…The Monte Carlo method for sampling of the Gibbs distribution involves an element of chance [6,7]. Random (or, more strictly, pseudorandom) numbers are used to select the configurations of the system.…”