1549380323-Statistical Mechanics Theory and Molecular Simulation

Ising model 613

system). We are interested in the transition from a disordered to an ordered state that
occurs either by the application of an external field or spontaneously as the temper-
ature is reduced. The models that are often invoked to describe this process are of a
type known aslattice models. Since we know that magnetization involves the collective
alignment of the quantum spins of particles along a particular spatialaxis, the basic
lattice model consists ofNinteracting particles having spinsSˆ 1 ,...ˆSNplaced at the
points of a regular lattice, as illustrated in Fig. 16.3. For simplicity, we consider the

Fig. 16.3 Small part of a cubic spin lattice.

particles to be spin-1/2, so thatSˆi= ( ̄h/2)σˆi, whereσˆis the vector of Pauli matrices
σˆ= (ˆσx,ˆσy,ˆσz) (see Problem 16.3). When diagonalized, each of the Pauli matrices
has eigenvalues±1. We consider only pairwise interactions among the spins described
by a general spin-spin coupling tensorJijso that the Hamiltonian for the system in
the absence of an external magnetic field is

Hˆ 0 =−^1

2

∑

i,j

σˆi·Jij·σˆj. (16.3.1)

The model in eqn. (16.3.1) is known as theHeisenberg model. The ‘0’ subscript reminds
us that eqn. (16.3.1) prescribes a field-free Hamiltonian. When a magnetic fieldBis
applied to the system, the total Hamiltonian takes the formHˆ=Hˆ 0 +Hˆ 1 , whereHˆ 1
is a perturbation due to the field and is given by

Hˆ 1 =−

∑

i

γB·ˆSi=−

∑

i

h·σˆi, (16.3.2)

whereγis the spin gyromagnetic ratio andh=γ ̄hB/2 (see Problems 9.1 and 10.2).
As we will discuss in the next section, the Heisenberg and liquid–gas models do
not belong to the same universality class. Therefore, we need to consider a simplified
version of the model in eqns. (16.3.1) and (16.3.2), which ultimately renders it easier
to solve. The simplification we will make is to restrict the spins to lie alongonly
one spatial axis, specifically, thez-axis. With this choice, the spin matrices become
σˆi= (0, 0 ,σˆzi). In addition, since ˆσzis diagonal, we can reduce each spin matrix ˆσzito