Statistical Mechanics - Physics at Oregon State University

8.4. DENSITY-MATRIX APPROACH (BRAGG-WILLIAMS APPROXIMATION.173
A = log(B). We first look at the logarithm of Rn, Tn = log(Rn). The operators
Rn have a simple form, they are Hermitian, and the eigenvectors are simple.
The only states that can be mixed are states with different values of σn. But
vectors that have different values of σi with i = n are not mixed, and all eigenvectors
of Rn have values for σi either +1 or −1. Since the eigenvectors of Tn
are identical to the eigenvectors of Rn, the same is true for Tn. This implies
that we can write
< σ1, · · · , σN|Tn|σ ′ 1, · · · , σ ′ N >= δσ1,σ ′ 1 δσ2,σ ′ 2 · · · τ(σn, σn) · · · (8.75)
The only part of the space that is mixed corresponds to σn, but since the
eigenvectors of both operators are teh same, we need to have
τ = log(ρ) (8.76)
Finally, because [Ri, Rj] = 0 we have log(RiRj) = log(Ri) log(Rj). This
gives us that log(ρ) =
i log(Ti). Hence
and
< σ1, · · · , σN| log(ρ)|σ ′ 1, · · · , σ ′ N >=
Tr ρ log(ρ) =
N
n=1
N
< σ1|ρ|σ1 >
< σ2|ρ|σ2 > · · ·
n=1
σ1
σ2
δσ1,σ ′ 1 δσ2,σ ′ 2 · · · 〈log(ρ)〉 (σn, σn) · · ·
σn,σ ′ n
(8.77)
< σn|ρ|σ ′ n >< σ ′ n| log(ρ)|σn > · · ·
(8.78)
The trace of the operators is equal to one, so all those sums in the equation are
replaced by one. This leaves
Tr ρ log(ρ) =
and we have
N
n=1 σn,σ ′ n
< σn|ρ|σ ′ n >< σ ′ n| log(ρ)|σn >= NTr ρ log(ρ) (8.79)
S = −kBNTr ρ log(ρ) (8.80)
as expected, since all sites are equivalent and contributions from different sites
are not mixed. Nevertheless, it was important to follow the mathematical derivation
to make sure that our intuitive expectation was indeed correct.
Next we need to minimize the free energy
G(h, T ) min
m,a
− 1
2 JNqm2
− hNm + NkBT Tr ρ log(ρ)
(8.81)