Statistics of Ising Spin System

Alfred Levitas

T. Kaneyoshi

The theoretical framework for investigating the critical behaviour of an Ising multilayer system consisting of alternating spin-1/2 and spin- S ( S ½) magnetic layers is given within the cluster approximation introduced into the differential operator technique. It has the statistical accuracy corresponding to the Bethe-Peierls approximation. The critical temperatures T C of some multilayer systems are studied numerically. We find some characteristic features, including the behaviour of a critical transverse field in the transverse spin-1/2 ferromagnetic multilayer system and the different behaviour of the T C curve in an alternating spin-1/2 and spin- S ( S >½) Ising multilayer system, depending on whether S is an integer or a half-integer.

Harold Falk, Masuo Suzuki

A cumulant-like expansion for the entropy of an $N$-spin system is presented. The successive terms in the expansion relate to successively higher orders of statistical association among spins. It is proved that for any Ising system of general dimensionality with ferromagnetic interactions of arbitrary range, the first two terms in the entropy expansion provide a lower bound for the exact entropy. A corollary of the theorem is that the lower-bound property is also valid for any two-sublattice Ising system with antiferromagnetic interactions between sublattices. An example is given which illustrates the fact that the vanishing of the two-spin cumulant (correlation) does not necessarily imply that the spins are statistically independent. The sum of the first two terms in the expansion is compared numerically with the exact entropy of an $N$-spin chain (and also a ring) with nearest-neighbor ferromagnetic or antiferromagnetic Ising interactions. The comparison, which measures the validity of a Kirkwood-type truncation in this context, is favorable only at sufficiently high temperatures.

Masuo Suzuki

Masuo Suzuki

Belal Ehsan Baaquie

The simplest possible quantum degree of freedom has only two discrete possible values and is called an Ising spin. This could be a spin system having only two possible values, namely “up” or “down,” or this could be a quantum particle that can occupy only two possible positions. To simplify the problem to its bare essentials, a two-state system is studied in zero-dimension space and in Euclidean time; the two-state system occupies a single point and, as it evolves in time, it can point either up or down. Furthermore, time evolution is simplified by discretizing time into a finite time lattice. The two-state degree of freedom in zero-dimension space propagating on a time lattice is called the Ising model. The one-dimensional Ising model is a toy model that is studied to develop the essential ideas of quantum mechanics. The concepts of state space, Hamiltonian, evolution kernel, and path integration have a discrete and simple representation in the Ising model. The Ising model can be viewed as a statistical mechanical system in thermal equilibrium, with the Ising spins occupying a one-dimensional space lattice, and hence is called the one-dimensional Ising model. The Ising model can be defined on higher dimensional lattices and forms one of the bedrocks of theoretical statistical mechanics. In Sections 8.1 to 8.5 the degree of freedom, state space, Hamiltonian, path integral and correlator for the Ising model are discussed. In Section 8.6, the technique of spin decimation is illustrated using an Ising model with coupling depending on the lattice site.