Statistical Mechanics - Physics at Oregon State University

8.4. DENSITY-MATRIX APPROACH (BRAGG-WILLIAMS APPROXIMATION.169
−kBTr [ρ log(ρ)] − βkB [Tr(ρH) − U] + βkBh [Tr(ρM) − M] + kBλ [Trρ − 1]
(8.50)
with Tr ρ = 1 , ρ = ρ † , and ρ 2 ρ, the last inequality in an operator sense.
But we can rewrite the previous expression in the form:
−kBTr [ρ log(ρ)]−βkBTr(ρH)+βkBhTr(ρM)+kBλ [Trρ − 1]+kBβU −kBβhM
(8.51)
The operator for the Gibbs-like free energy for a magnetic problem is given by
G = H − T Sen − hM = Tr(ρH) + kBT Tr [ρ log(ρ)] − hTr(ρM) (8.52)
This means that we have to maximize the following expression for ρ:
− 1
T G + kBλ [Trρ − 1] + kBβU − kBβhM (8.53)
and then choose h and T to give the correct values of U and M. In other words
we have to minimize G!!! This change is equivalent to what we saw in thermodynamics,
where we derived minimum energy principles from the maximum
entropy principle.
Therefore, what we plan to do is to use a smaller set of density matrices and
minimize the operator form of the Gibbs energy within this set to get an upper
bound of the true thermodynamical Gibbs energy.
In our case, we will use the following approximation of the density matrix,
as defined via the matrix elements
< σ1, · · · , σN|ρ|σ ′ 1, · · · , σ ′ N >= ρ1(σ1, σ ′ 1)ρ2(σ2, σ ′ 2) · · · ρN (σN , σ ′ N) (8.54)
The functions ρi(σ, σ ′ ) represent two by two matrices. Essentially, we decompose
the density matrix as a direct product of N two by two matrices:
ρ = ρ1
ρ2
· · · ρN
(8.55)
Writing the density matrix as a direct product in this way is equivalent to
ignoring correlations between different sites. The correlation is not completely
gone, of course, since the energy still connects neighboring sites.
Why does this form ignore correlations between different sites? If we use this
form to calculate the thermodynamic average of a quantity which only depends
on a single spin variable σk we find
〈f(σk)〉 =
σk=±1
f(σk)ρk(σk, σk)
i=k
σi=±1
ρi(σi, σi)
(8.56)