Discrete Spin Models Research Articles

A discrete vector spin model

A modification of the classical Heisenberg model is proposed in which the continuously rotating spin variables are replaced by spins which are only allowed an isotropically distributed discrete set of directions. The change is motivated by a need to simplify the calculations associated with Monte Carlo simulation of vector spin systems, a goal accomplished by the present model. An efficient table-based simulation technique is described and preliminary results for the critical region presented; various extensions of the model are discussed.

Decay of correlations and master equation in Kac' lattice model

We discuss on a simple discrete time spin model how a deterministic classical evolution may be transformed into a stochastic process. Simple averaging over a population of fixed scatterers yields a non-markovian microscopic evolution and is not compatible with a macroscopic description in terms of populations. Coupling each scatter-er to an independent reservoir leads to a Markov process in phase space, which can be projected on a macroscopic birth and death process. Some limiting cases (continuous time interpolation, ...) are considered.

Spin systems on hierarchical lattices. Introduction and thermodynamic limit

A number of exactly soluble models in statistical mechanics can be produced with the use of spins interacting with nearest neighbors on a hierarchical lattice. A general definition and several examples of such lattices are given, and the topological properties of one of these, the diamond lattice, are discussed in detail. It is shown that the free energy has a well-defined thermodynamic limit for a large class of discrete spin models on hierarchical lattices.

Cubic spin model with competing bilinear and higher-degree pair interactions on a linear chain in a magnetic field

Interesting ferrimagnetic structures occur in discrete spin models with competing interactions as well as in several rare-earth compounds with the rocksalt structure. We have determined the ground states of some one-dimensional models in a magnetic field. We present exact calculations using the transfer matrix approach of the magnetization curves of the cubic spin model on a linear chain with competing nearest and next nearest neighbor bilinear as well as biquadratic interactions. The implications of the structures obtained, to models on three dimensional face centered cubic lattices, are discussed.

Discrete and Abelian lattice gauge theories

Gauge-invariant discrete and Abelian ( X- Y) spin models with compact degrees of freedom are studied in ( D + 1) Euclidean (“space-time”) dimensions ( D ⩽ 3). They are formulated in terms of quantum Hamiltonians on a D-dimensional spatial lattice. In this approach the periodic Gaussian and Abelian models are identical. Duality transformations are obtained which relate the topological excitations of one model to the fundamental quantum spin excitations of another. These motivate the extension of the usual gauge-invariant structure to “gauge invariance of the third kind” which involves degrees of freedom associated with the surfaces of the plaquettes. The nature of the excited states and the behaviour of the order (and disorder) parameters are discussed in various limits of the coupling constants. For both discrete and Abelian theories the behaviour of the order parameter (the Wilson loop) depends crucially on whether the spins in the theory carry the fundamental unit of charge or a multiple of it.

Phase transitions in two-dimensional systems

Relations between two-dimensional physical systems, continuous-spin Hamiltonians, and discrete spin or Potts models are established by Landau symmetry and renormalization-group arguments. Experimental realizations of XY and Heisenberg models with cubic anisotropy as well as of different Potts models are reviewed. Then a unified model encompassing these classes of models is introduced. For the special case of a six-state discrete model a phase diagram is proposed based on duality and Migdal renormalization-group transformations.

An intermediate spherical model of a ferromagnet

A one-parameter family of partition functions is considered which for zero value of the parameter α reduces to the spherical model of a ferromagnet. The model for α> 0 is closer to the usual discrete lattice spin model of a ferromagnet than is the spherical model. The first four terms in α of the limiting value of the partition function are calculated above and below the critical temperature for arbitrary interactions using the saddle point method to calculate certain correlation functions for the spherical model. These calculations indicate that the critical temperature is independent of α for small α and certain interactions.

Magnetostatic modes of a gyrotropic cylinder and bar

The classically integrable discrete spin model with a logarithmic ferromagnetic nearest-neighbor interaction is semiclassically quantized. The resulting spectrum consists of linear magnons and nonlinear quantum solitons. The lowest soliton branch is identical with the magnon as S→∞; in practice, deviations amount to a maximum of 10% for S = 1. The early termination points in momentum space proposed by Haldane as a result of lattice discreteness are found. The singular case m = 2S is characterized by a logarithmic divergence of the energy at the cutoff wavevector.