Statistical Mechanics - Physics at Oregon State University

172 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
but now restrict ourselves to density matrices ρ = (ρ) N , where the power means
a direct product and where we use the three parameters a and m as defined
before. This leads to
G(h, T ) min
m,a Tr (ρ) N H − (ρ) N hM + kBT (ρ) N log((ρ) N )
(8.70)
and this procedure gives an upper bound to the energy. Note that this does not
give any bounds on derivatives. But we know that derivatives are monotonous,
and hence we cannot have large oscillations in them. Therefore, derivatives have
errors similar to the errors in the free energy!
The energy U − Mh is calculated from Tr (H − hM)ρ, which becomes
U − Mh =
· · ·
⎡
ρ(σ1, σ1) · · · ρ(σN , σN) ⎣−J
σiσj − h
σ1
σN
i
σi
⎤
⎦ (8.71)
A given variable σj occurs only in a linear fashion in this summation, and
hence the sums over
the spin-variables either give a factor one or m, because
ρ(σ, σ) = 1 and σ σρ(σ, σ) = m. This leads to
σ
U − Mh = − 1
2 JNqm2 − hNm (8.72)
as expected. This expression is independent of the values of the complex number
a.
The harder task is to evaluate the entropy, because it has the logarithm of
a matrix in the expression.
The entropy can be obtained from
S = −kB
σ1,···,σN σ ′ 1 ,···,σ′ N
< σ1, · · · , σN|ρ|σ ′ 1, · · · , σ ′ N >< σ ′ 1, · · · , σ ′ N| log ρ|σ1, · · · , σN >
(8.73)
So how does one calculate the logarithm of a matrix? The final result is not
too surprising, it looks like log(xy) = log(x) + log(y), but here we work with
matrices in stead of numbers, so we have to be a little careful. In our case we
define N additional matrices related to ρ by
< σ1, · · · , σN|Rn|σ ′ 1, · · · , σ ′ N >= δσ1,σ ′ 1 δσ2,σ ′ 2 · · · ρ(σn, σn) · · · (8.74)
which is diagonal except for the n-th spin variable. Hence ρ = R1R2 · · · RN
where all the matrices are of rank 2N . These matrices commute, since the off
diagonal elements occur on different blocks that are not connected. So we have
[Ri, Rj] = 0.
The exponent of a matrix is defined via a power-series as usual, eA
=
n 1
n! An , and the logarithm is the inverse of the exponential, eA = B ⇒