Statistical Mechanics - Physics at Oregon State University

212 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
Γj =
1
Z(T, h = 0, N)
σ0,···,σN−1
N−1
βJ
σ0σje i=0 σiσi+1 (9.102)
where we use periodic boundary conditions and have σ0 = σN .
In order to evaluate this expression we use a simple trick. First we make the
problem more complicated by assuming that all bonds can be different. This
means that the coupling constant between spins i and i+1 is Ji. We treat these
coupling constants as independent parameters. In the end we need, of course,
Ji = J. We define a function S by
S(J0, .., JN−1) =
σ0,···,σN−1
e β Jiσiσi+1 (9.103)
This is similar to a partition function, and we have that Z = S(J, J, J, · · ·).
Next we take partial derivatives with respect to J0 to Jj−1. This gives us
∂ ∂
· · ·
∂J0 ∂J1
Since σ 2 i
This leads to
∂
∂Jj−1
S =
σ0,···,σN−1
= 1 this simplifies to
∂ ∂
· · ·
∂J0 ∂J1
∂
∂Jj−1
S = β j
gj = 1
β j S
e β Jiσiσi+1 [βσ0σ1][βσ1σ2] · · · [βσj−1σj]
σ0,···,σN−1
∂ ∂
· · ·
∂J0 ∂J1
(9.104)
e β Jiσiσi+1 σ0σj (9.105)
∂
∂Jj−1
S
Ji=J
(9.106)
Therefore we would like to evaluate S. This is done using transfer matrices
like we did in the previous section. We define
T (i)
βJi e e
=
−βJi
(9.107)
e −βJi e βJi
or T (i)
′
σ,σ ′ = eβJiσσ . The definition of S has a sum in the exponent and we write
this exponent of a sum as the product of exponents in the form
S =
e βJ0σ0σ1 βJ1σ1σ2 βσN−1σN−1σ0 e · · · e (9.108)
From this we see that
σ0,···,σN−1
S = Tr T (0) T (1) · · · T (N−1)
(9.109)
If we need to calculate the trace or determinant of a matrix, it is always
very useful to know the eigenvalues and eigenvectors. The eigenvectors of the
matrices T (i) are easy, they are e± = 1 √ (1, ±1) independent of the index i.
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