Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 237
– J + J
Figure 5.1: Two nearest neighbor spins (in any dimension) have an interaction energy −J if they
are parallel and interaction energy +J if they are antiparallel.
In the Ising model the spin at every site is either up (+1) or down (−1). Unless otherwise
stated, the interaction is between nearest neighbors only and is given by −J if the spins are parallel
and +J if the spins are antiparallel. The total energy can be expressed in the form 4
E = −J
N
i,j=nn(i)
sisj −H
N
si (Ising model), (5.35)
i=1
where si = ±1 and J is known as the exchange constant. We will assume that J > 0 unless
otherwise stated and that the external magnetic field is in the up or positive z direction. In the
following we will refer to s as the spin. 5 The first sum in (5.35) is over all pairs of spins that are
nearest neighbors. The interaction between two nearest neighbor spins is counted only once. A
factor of µ has been incorporated into H, which we will refer to as the magnetic field. In the same
spirit the magnetization becomes the net number of positive spins, that is, the number of up spins
minus the number of down spins.
Because the number of spins is fixed, we will choose the canonical ensemble and evaluate the
partition function. In spite of the apparent simplicity of the Ising model it is possible to obtain
exact solutions only in one dimension and in two dimensions in the absence of a magnetic field. 6
In other cases we need to use various approximation methods and computer simulations. There is
no general recipe for how to perform the sums and integrals needed to calculate thermodynamic
quantities.
5.5 The Ising Chain
In the following we obtain an exact solution of the one-dimensional Ising model and introduce an
additional physical quantity of interest.
4If we interpret the spin as an operator, then the energy is really a Hamiltonian. The distinction is unimportant
here.
5Because the spin Sˆ is a quantum mechanical object, we might expect that the commutator of the spin operator
with the Hamiltonian is nonzero. However, because the Ising model retains only the component of the spin along
the direction of the magnetic field, the commutator of the spin ˆ S with the Hamiltonian is zero, and we can treat
the spins in the Ising model as if they were classical.
6It has been shown that the three-dimensional Ising model (and the two-dimensional Ising model with nearest
neighbor and next nearest neighbor interactions) is computationally intractable and falls into the same class as other
problems such as the traveling salesman problem. See
and . The Ising model is of interest to computer scientists in part for
this reason.