1549380323-Statistical Mechanics Theory and Molecular Simulation

624 Critical phenomena

16.6 Ising model in one dimension

Solving the Ising model in one dimension is a relatively straightforwardexercise. As
we will show, however, the one-dimensional Ising model shows no ordered phases.
Why study it then? First, there are classes of problems that can bemapped onto
one-dimensional Ising-like models, such as the conformational equilibria of a linear
polymer (see Problems 10.9 and 16.5). Second, the mathematical techniques employed
to solve the problem are applicable to other types of problems. Third, even the one-
dimensional spin system must become ordered atT= 0, and therefore, understanding
the behavior of the system asT→0 will be important in our treatment of spin systems
via renormalization group methods.
From eqn. (16.5.2), the Hamiltonian for the one-dimensional Ising model is

H=−J

∑N

i=1

σiσi+1−h

∑N

i=1

σi. (16.6.1)

In order to complete the specification of the model, a boundary condition is also
needed. Since the variableσN+1appears in eqn. (16.6.1), it is convenient to impose
periodic boundary conditions, which leads to the conditionσN+1 =σ 1. The one-

...

σ 1 σ 2 σ 3 σN σN+1

Fig. 16.8 One-dimensional Ising system subject to periodic boundaryconditions.

dimensional periodic chain is illustrated in Fig. 16.8. Because of the periodicity, the
Hamiltonian can be written in a more symmetric manner as

H=−J

∑N

i=1

σiσi+1−

h

2

∑N

i=1

(σi+σi+1). (16.6.2)

The partition function corresponding to the Hamiltonian in eqn. (16.6.2) is

∆(N,h,T) =

∑

σ 1 =± 1

···

∑

σN=± 1

exp

[

βJ

∑N

i=1

σiσi+1+

βh

2

∑N

i=1

(σi+σi+1)

]

. (16.6.3)

Since each spin sum has two terms, the total number of terms represented by the spin
sums is 2N. A powerful method for evaluating the partition function is referred to as
thetransfer matrixmethod, first introduced by Kramers and Wannier (1941a, 1941b).
This method recognizes that the partition function can be expressed as a large product
of matrices. Consider the matrix P, whose elements are given by

〈σ|P|σ′〉= eβJσσ

′+βh(σ+σ′)/ 2

. (16.6.4)