1549380323-Statistical Mechanics Theory and Molecular Simulation

626 Critical phenomena

g(h,T) =−kTlnλ+

=−J−kTln

[

cosh(βh) +

√

sinh^2 (βh) + e−^4 βJ

]

. (16.6.11)

From eqn. (16.6.11), the magnetization per spin can be computed as

m=

(

∂g

∂h

)

=

sinh(βh) + sinh(βh)cosh(βh)/

√

sinh^2 (βh) + e−^4 βJ

cosh(βh) +

√

sinh^2 (βh) + e−^4 βJ

. (16.6.12)

Ash→0, the magnetization vanishes, since cosh(βh)→1 and sinh(βh)→0. Thus,
there is no magnetization at any finite temperature in one dimension,and hence,
no nontrivial critical point. Note, however, that asT →0 (β→ ∞), the factors
exp(− 4 βJ) vanish, andm→±1 ash→ 0 ±. This indicates that an ordered state does
exist at absolute zero of temperature. The fact thatmtends toward different limits
depending on the approach ofh→0 from the positive or negative side indicates that
T= 0 can be thought of as a critical point, albeit an unphysical one. Indeed, such
a result is expected since the entropy vanishes at absolute zero, and consequently an
ordered state must exist atT= 0. Though unphysical, we will find this critical point
useful for illustrative purposes later in our discussion of the renormalization group.

16.7 Ising model in two dimensions

In contrast to the one-dimensional Ising model, which can be solvedwith a few lines of
algebra, the two-dimensional Ising model is a highly nontrivial problem that was first
worked out exactly by Lars Onsager (1903–1976) in 1944 (Onsager, 1944). Extensive
discussions of the solution of the two-dimensional Ising model can be found in the
books by K. Huang (1963) and by R. K. Pathria (1972). Here, we shall give the basic
idea behind two approaches to the problem and then present the solution in its final
form.

Transfer matrix approach: The first method follows the transfer matrix approach em-
ployed in the previous section for the one-dimensional Ising model. Consider the simple
square lattice of spins depicted in Fig. 16.9, in which each row and eachcolumn con-
tainsnspins, so thatN=n^2. Ifiindexes the rows andjindexes the columns, then
the Hamiltonian, taking into account the restriction to nearest-neighbor interactions
only, can be written as

H=−J

∑n

i=1

∑n

j=1

[σi,jσi+1,j+σi,jσi,j+1]−h

∑n

i=1

∑n

j=1

σi,j. (16.7.1)

As in the one-dimensional case, we impose periodic boundary conditions on the square
lattice so that the spins satisfyσn+1,j=σ 1 ,jandσi,n+1=σi, 1. The partition function
can now be expressed as