1549380323-Statistical Mechanics Theory and Molecular Simulation

Spin correlations 633

r= 6

r= 6 r= 8

r= (^8) r= 8
Fig. 16.12Examples of graphs that contribute to the partition function of the two-dimen-
sional Ising model forr= 6 andr= 8.
and the number of nearest neighbors on the periodic lattice isν=N. Putting these
facts together gives the partition function as
Q(N,T) = 2N[cosh(K)]N

[

1 +vN

]

, (16.7.20)

which simplifies to

Q(N,T) = 2N

[

coshN(K) + sinhN(K)

]

(16.7.21)

in agreement with eqn. (16.7.19).

16.8 Spin correlations and their critical exponents

In Section 4.6.1, we considered spatial correlation functions in a liquid. Interestingly, it
is possible to define an analogous quantity for the Ising model. Consider the following
spin-spin correlation function at zero field:

〈σiσj〉=

1

Q(N,T)

∑

σ 1

···

∑

σN

σiσje−βH, (16.8.1)

whereQ(N,T) is the canonical partition function. Ifσiandσj occupy lattice sites
at positionsriandrj, respectively, then at large spatial separationr=|ri−rj|, the