Statistical Mechanics - Physics at Oregon State University

194 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
As an example we will consider the result obtained in the Bethe approximation
for the one-dimensional Ising model without an external field (h = 0). The
effective field h ′ is a function of λ and is determined by
cosh β(λ + h ′ )
cosh β(λ − h ′ ) = eh′ 2β
(9.15)
But we already know that there are no phase transitions, independent of the
value of λ, and hence h ′ = 0. In the Bethe approximation the only interaction
term pertains to the bonds between the central site and its neighbors, and hence
the interaction energy of a cluster is
〈λV〉λ = 〈−λσ0
2
i=1
σi〉λ
(9.16)
Now we use the results from the previous chapter to calculate this average.
In order to find the potential energy we calculate the cluster energy from the
partition function. The cluster partition function was determined in the previous
chapter, and in this case is given by
and the cluster energy Ec follows from
Zc = 8 cosh 2 (βλ) (9.17)
Ec = − ∂
∂β log Zc = −2λ tanh βλ (9.18)
Since h = 0 we have H0 = 0, and all the internal energy is due to the interaction
term. The internal energy of the whole system of N particles is
U = 〈λV〉 = 1
2 NEc
(9.19)
The factor one-half is needed to avoid double counting of all bonds. A cluster
contains two bonds! As a result we find that
G(J) = G(0) − N
J
0
dλ tanh βλ = G(0) − NkBT log cosh βJ (9.20)
Next, we determine the free energy at λ = 0. There is only an entropy term,
since the internal energy is zero. Without a magnetic field and without interactions
all configurations are possible, thus S = NkB log 2, and we find after
combining the two logarithmic terms:
G(J) = −NkBT log(2 cosh βJ) (9.21)
The average of the interaction energy was obtained in the Bethe approximation.
It turns out, however, that the calculated free energy is exact! The reason
for that is the simplicity of the system. There is no phase transition, and the
correlation function in the cluster approximation is actually correct.