Statistical Mechanics - Physics at Oregon State University

170 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
and since
it follows that
1 = Tr ρ =
〈f(σk)〉 =
i
σi=±1
ρi(σi, σi)
σk=±1 f(σk)ρk(σk, σk)
σk=±1 ρk(σk, σk)
(8.57)
(8.58)
This result is independent of the state of the spins on all other sites. There
are no correlations between the sites for expectation values.
We still have to minimize the Gibbs energy and that procedure will determine
the form of ρk. Hence the nature of the system will only enter via the
thermodynamic average for the energy and entropy, and hence the density matrix
in this uncorrelated model depends only on these average quantities. We
expect therefore that this will give the same results as the mean field or average
field approximation. At this level the previous formulation of mean field theory
is often easier to apply, but the present formulation is an important starting
point for improving the model by including correlations in the density matrix.
For a general system the local density matrices ρi depend on the atomic site.
For an infinite solid without surfaces all sites are equivalent and all functions ρi
are identical to a simple function ρ. The matrix elements ρ(σ, σ ′ ) have to obey
certain requirements, though.
First of all, the density matrix has trace one, Tr ρ = 1, which gives
ρ(σ1, σ1) · · · ρ(σN, σN ) = 1 (8.59)
σ1,···,σN
or [Tr ρ] N = 1. The density matrix is Hermitian and if we consider one off
diagonal element only we have
< σ1, · · · , σ ′ k, · · · , σN |ρ|σ1, · · · , σk, · · · , σN >=
(< σ1, · · · , σk, · · · , σN|ρ|σ1, · · · , σ ′ k, · · · , σN >) ∗
(8.60)
We write the density matrix as a direct product again, and sum over all values
σi with i = k. This gives
[Tr ρ] N−1 < σ ′ k|ρ|σk >=
Now we use [Tr ρ] N = 1 and get
< σ ′ k |ρ|σk >
Tr ρ
[Tr ρ] N−1 < σk|ρ|σ ′ ∗ k >
< σk|ρ|σ
=
′ k >
∗ Tr ρ
(8.61)
(8.62)
ρ
The last equation means that the operator Tr ρ is Hermitian with trace one,
and therefore has real eigenvalues. Call the eigenvalues λ1 and λ2 and the