Statistical Mechanics - Physics at Oregon State University

192 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
9.2 Integration over the coupling constant.
Since exact answers are hard to get for most realistic models, it is useful to know
a variety of tricks to obtain good approximations. The procedure of integration
over the coupling constant is one of those tricks. We will not investigate the
relations between this method and others, but simply remark that this method
is another way of obtaining a correlated density matrix. We use the Ising model
again for illustration. We will break up the Hamiltonian into two parts:
V = −
{σ1,···,σN }
H0 = −h
{σ1,···,σN }
and study the general Hamiltonian
|σ1, · · · , σN >
σiσj < σ1, · · · , σN | (9.1)
|σ1, · · · , σN >
σi < σ1, · · · , σN| (9.2)
i
H = H0 + λV (9.3)
In this division the term H0 contains all the easy terms. In other systems it
would include the kinetic energy in many cases. The important point is that
the operator H0 can be diagonalized, we can find eigenvalues and eigenvectors.
The operator V contains all the hard terms, the ones that we do not know how
to deal with, and have approximated before. We are obviously interested in the
case λ = J. We will, however, treat λ as a variable parameter. The division
is made in such a way that the problem is simple for λ = 0. Again, we always
make sure that for H0 the solutions are known.
The free energy as a function of λ is
and the derivative with respect to λ is
G(λ) = −kBT log Tr e −βH0−βλV
(9.4)
dG kBT d
= −
Tr
dλ Tr e−βH0−βλV dλ e−βH0−βλV
(9.5)
One always has to be careful when taking derivatives of operators. In this
case H0 and V commute and the exponent of the sum is the product of the
single exponents.
[H0, V] = 0 ⇒ e −βH0−βλV = e −βH0 e −βλV
The derivative with respect to λ affects the second term only, and we have
d
dλ e−βλV = d
dλ
which leads to
∞
n=0
1
n! (−βλ)n V n =
∞
n=1
(9.6)
1
(n − 1)! λn−1 (−β) n V n = −βVe −βλV
(9.7)