Statistical Mechanics - Physics at Oregon State University

214 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
Because of the periodic boundary conditions this result is symmetric under the
change j ↔ N − j. Also, we have ΓN = 1, which is to be expected because site
N is equal to site 0, and < σ0σN >=< σ 2 0 >= 1.
The expression is simplified when we divide the numerator and denominator
by the eigenvalue λ+ to the power N. Note that λ+ > λ− > 0. The ratio of the
two eigenvalues is simple, we have
This gives
λ−
λ+
= eβJ − e−βJ eβJ = tanh(βJ) (9.118)
+ e−βJ Γj = tanhj (βJ) + tanh N−j (βJ)
1 + tanh N (βJ)
(9.119)
This expression is complicated, but we have to remember that we always
need to invoke the thermodynamic limit after all calculations have been done.
That is the case here, we have our final result. If the temperature is non-zero,
tanh(βJ) < 1 and in the limit N ≫ j only the lowest order terms survives. Note
that we have given explicit meaning to the idea of the thermodynamic limit, we
require that N is large compared to j! In that case we can write
Γj = tanh j (βJ) (9.120)
This formula shows that as a function of the distance j the spin correlation
function decays as a power law. Hence it can be related to a correlation length
via
where the correlation length ξ is found by
j
−
Γj = e ξ (9.121)
ξ = − j
= −[log tanh(βJ)]−1
log(Γj)
(9.122)
As expected, we find that the spin correlation decays exponentially. If Γj is
close to one, spins tend to point in the same direction, and the magnetization
is strongly correlated. The spin correlation length measures the distance over
which spins are correlated and tend to point in the same direction. Note that
this common direction can fluctuate!
The correlation length increases with decreasing temperature. We see that
in the limit T → 0 the hyperbolic tangent approaches one and the logarithm
gives zero. We can find the behavior by
tanh(βJ) =
1 − e−2βJ
≈ 1 − 2e−2βJ
1 + e−2βJ (9.123)
and the logarithm of one plus a small term is approximated by the small term.
That gives