Statistical Mechanics - Physics at Oregon State University

204 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
or
χ =
Tr S0βMe −β(H−hM)
Tr e −β(H−hM)
This can be transformed to
−
Tr S0e −β(H−hM) TrβM e −β(H−hM)
Tr e −β(H−hM) 2
(9.60)
χ = β
{〈S0Si〉 − 〈S0〉〈Si〉} (9.61)
i
χ = β
〈S0 − m〉〈Si − m〉 (9.62)
i
which clearly shows that χ is directly related to the fluctuations in the spin
variables. Here we used 〈S0〉 = 〈Si〉 = m.
If fluctuations are uncorrelated we have < AB >=< A >< B > which would
seem to imply that for uncorrelated fluctuations χ = 0. That is not correct,
however. Fluctuations on the same site are always correlated, and in that case
we would get χ =< (S0 − m) 2 >, which is always positive indeed.
It is customary to define a spin-correlation function Γi by
Γi(T ) = 〈S0Si〉T − 〈S0〉T 〈Si〉T
(9.63)
If the spins at site 0 and site i are uncorrelated, Γi = 0. This spin-correlation
function can also be expressed in terms of real coordinates r, as long as we
understand that the value Γ(r, T ) is actually an average over a number of atomic
sites in a volume ∆V which is small compared to the total volume of the crystal.
In that case we find
χ = β d 3 rΓ(r, T ) (9.64)
If for large values of r the spin-correlation function is proportional to r −p the
integral is well-defined only if p > 3. Therefore, at Tc where the spin-correlation
function diverges we need to have p 3. At temperatures away from the critical
temperature the spin correlation function does not diverge. If the correlation
function would always be a power law, Γ ∝ r −p(T ) , this would imply p = 3 at the
critical temperature, since p(T ) > 3 away from the critical point. That is not
correct, however, because we also have an exponential factor, which disappears
at the critical temperature. In general we write Γ ∝ r −p e −α(T )r and we use
α(T ) > 0 away from the critical point and hence α(Tc) = 0. Since these are the
only two possibilities and since we know from experiment that the first one is
wrong, we find that correlations have to die out exponentially. The length scale
corresponding to this exponential decay is called the correlation length, and at
the critical point the correlation length diverges and becomes infinite.
If we have no correlations we find χ = β(< S 2 0 > −m 2 ) = β(1 − m 2 ). Here
we know < S 2 0 >= 0 because the value of the spin variable is ±1 and hence