Statistical Mechanics - Physics at Oregon State University

206 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
G(T, N, h = 0) = −NkBT log(2 cosh(βJ) + kBT log cosh(βJ) (9.71)
In the thermodynamic limit the second term can be ignored, and the answer
is the same as we found by integrating over the coupling constant.
The previous paragraph gave the results for the Ising chain without a magnetic
field. What happens when a magnetic field is present? In that case the
calculation is more complicated if we use the same boundary conditions. The
main problem is that the term with the product of neighboring spins has fewer
terms than the term connecting to the magnetic field. It is hard treat both
terms at the same time.
It is possible, however, to change the boundary conditions, since we always
want to take the limit N → ∞. Therefore, we assume periodic boundary
conditions by connecting spin N back with spin 1. Hence in the calculations we
take σ0 = σN and σ1 = σN+1. The energy in the extra bond is small compared
to the total energy if N is large, and disappears in the thermodynamic limit.
The energy of a configuration {σ1, · · · , σN} is
H(σ1, · · · , σN) = −J
N
i=1
σiσi+1 − h
2
N
[σi + σi+1] (9.72)
where we made the magnetic contribution symmetric, which is possible because
we are using periodic boundary conditions. This is written with a single summation
in the form
H(σ1, · · · , σN ) = −
i=1
N
f(σi, σi+1) (9.73)
where we defined f(σ, σ ′ ) = Jσσ ′ + h
2 (σ + σ′ ). The partition function is
Z(T, N) =
σ1,···,σN
i=1
e β i f(σi,σi+1) =
σ1,···,σN
e βf(σi,σi+1)
i
(9.74)
This looks like the product of a set of two by two matrices. Hence we define
the matrix T by
T(σ, σ ′ ) = e βJf(σ,σ′ )
(9.75)
This matrix T is often called a transfer matrix. It is a two by two matrix
looking like
β(J+h) e e
T =
−βJ
e −βJ e β(J−h)
The partition function in terms of T is
(9.76)