Statistical Mechanics - Physics at Oregon State University

9.6. SPIN-CORRELATION FUNCTION FOR THE ISING CHAIN. 215
ξ ≈ 1
2 e2βJ
(9.124)
which diverges faster that any power law. Normally we would expect that
ξ ∝ |T − Tc| −ν , but that is not the case here.
Before we draw any conclusions we need to check one thing. The thermodynamic
limit should always be taken last. So we need to take the limit to zero
temperature first! We should not take the limit T → 0 in the expression above.
As usual, the limit T → 0 has to be taken before the limit N → ∞. Before the
thermodynamic limit we have
ξ = −j
log
tanh j (βJ) + tanh N−j (βJ)
1 + tanh N (βJ)
−1
(9.125)
Using the exact expression (no approximation) tanh(βJ) = 1 − x, with |x| ≪ 1
we have
j N−j
(1 − x) + (1 − x)
ξ = −j log
1 + (1 − x) N
If we take linear terms in the powers only, we see that
(1 − x) j + (1 − x) N−j
1 + (1 − x) N
≈
−1
(1 − jx) + (1 − (N − j)x
1 + (1 − Nx)
Therefore, we need second order terms. We have
(1 − x) j + (1 − x) N−j
1 + (1 − x) N
This is equal to
≈
1 + 1
2−Nx
which can be approximated by
(1 +
1 j(j − 1)
x
2 − Nx 2
2 +
which is approximated by
1 +
1 j(j − 1)
x
2 − Nx 2
2 +
2 − Nx + j(j−1)
2
j(j−1)
2
1 + 1
2−Nx
x2 + (N−j)(N−j−1)
2 x2 2 − Nx + N(N−1)
2
x 2
x2 + (N−j)(N−j−1)
2 x2
N(N−1)
2 x2 (N − j)(N − j − 1)
x
2
2
)(1 −
(N − j)(N − j − 1)
x
2
2
−
(9.126)
= 1 (9.127)
1
2 − Nx
1
2 − Nx
(9.128)
(9.129)
N(N − 1)
x
2
2 )
(9.130)
N(N − 1)
x
2
2
(9.131)