Mean-field theory 619β. The location of the thin solid line is determined using a procedure known as the
Maxwell construction, which states that the areas enclosed above and below the thin
line and the isotherm must be equal. Once the isotherms forT < Tcare corrected in
this manner, then the exponentβcan be calculated (see Problem 16.3).
In order to apply the mean-field approximation to the Ising model, weassume that
the system is spatially isotropic. That is, for the spin-spin coupling∑ Jij, we assume
jJijis independent of the lattice locationi. Since the sum in eqn. (16.3.4) is per-
formed over nearest neighbors ofi, under the assumption of isotropy,
∑
jJij=zJ ̃,
whereJ ̃is a constant andzis the number of nearest neighbors of each spin (z= 2 in
one dimension,z= 4 on a two-dimensional square lattice,z= 6 on a three-dimensional
simple cubic lattice,z= 8 on a three-dimensional body-centered cubic lattice, etc.).
Absorbing the factorzinto the constantJ ̃, we defineJ=zJ ̃.
Next, we consider the Hamiltonian in the presence of an applied magnetic fieldh:
H=−
1
2
∑
<i,j>Jijσiσj−h∑
iσi. (16.5.2)The partition function is given by
∆(N,h,T) =∑
σ 1 =± 1∑
σ 2 =± 1···
∑
σN=± 1exp
β
^1
2
∑
<i,j>Jijσiσj+h∑
iσi
. (16.5.3)
To date, it has not been possible to obtain a closed-form expressionfor this sum in
three dimensions. Thus, to simplify the problem, we write the spin-spin productσiσjin
terms of the difference of each spin from the magnetization per spinm= (1/N)〈
∑
iσi〉:
σiσj= (σi−m+m)(σj−m+m)
=m^2 +m(σi−m) +m(σj−m) + (σi−m)(σj−m). (16.5.4)Sincem∼〈σ〉, the last term in eqn. (16.5.4) is a fluctuation term, which is neglected
in the mean-field approximation. If this term is dropped, then
1
2∑
<i,j>Jijσiσj≈1
2
∑
<i,j>Jij[
−m^2 +m(σi+σj)]
=−
1
2
m^2 NJ+Jm∑
iσi, (16.5.5)where the assumption of spatial isotropy has been used. Thus, the Hamiltonian reduces
to
H=−1
2
∑
<i,j>Jijσiσj−h∑
iσi≈1
2
NJm^2 −(Jm+h)∑
iσi, (16.5.6)and the partition function becomes