1549380323-Statistical Mechanics Theory and Molecular Simulation

Two-dimensional Ising model 629

an exact expression for the magnetization at zero field and showedthat whenT < Tc,
whereTcis given by

2tanh^2 (2J/kTc) = 1, kTc≈ 2. 269185 J, (16.7.11)

the magnetization is nonzero, indicating that spontaneous magnetization occurs in two
dimensions. The magnetization is

m=

{

0 T > Tc

{

1 −[sinh(2βJ)]−^4

} 1 / 8

T < Tc

. (16.7.12)

In addition to the existence of a spontaneously ordered phase, the heat capacityCh
diverges asT→Tc. The expression for the heat capacity ath= 0 nearT=Tcis

Ch(T)

k

=

2

π

(

2 J

kTc

) 2 [

−ln

∣

∣

∣

∣^1 −

T

Tc

∣

∣

∣

∣+ ln

(

kTc

2 J

)

−

(

1 +

π

4

)]

, (16.7.13)

which diverges logarithmically. A graph ofChvs.Tis shown in Fig. 16.10. The log-
arithmic divergence emerges because the model is solved in two rather than three
dimensions; in the latter, we would expect a power-law divergence. The other critical
exponents can be derived for the two-dimensional Ising model andareα= 0 (logarith-
mic divergence),β= 1/8,γ= 7/4, andδ= 15. These are the exact exponents for the
d= 2,n= 1 universality class. To date, the three-dimensional Ising model remains

0

T

Ch

Tc

/k

Fig. 16.10Heat capacity of the two-dimensional Ising model (see eqn. (16.7.13)).