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