Statistical Mechanics - Physics at Oregon State University

9.6. SPIN-CORRELATION FUNCTION FOR THE ISING CHAIN. 211
1.0
0.75
0.5
0.25
0.0
0.0
0.25
0.5
T
0.75
1.0
−30
Figure 9.7: Magnetization of the one dimensional Ising chain, as a function
of log(h).
−20
−10
Γi = 〈(σ0 − m)(σi − m)〉 = gi − m 2
h
0
(9.100)
This is often a more useful quantity to study. In a true mean field theory
all fluctuations on different sites would be uncorrelated, and we would have
Γi = δi0(1 − m 2 ). That would imply χ = βΓ0 and this quantity does not
diverge at Tc. That is wrong, we looked at that before, so the mean field theory
makes another approximation.
For the average energy we have
〈H〉 = −JNqm 2 − hN − JNqΓnn
(9.101)
and mean field theory is obtained by requiring that the spin correlations between
neighboring sites are zero. We only need fluctuations on neighboring sites to be
uncorrelated. Further correlations can be non zero, and will have to be non-zero
because the susceptibility is diverging!
To solve the real problem we need the value of Γnn. We can find this function
by considering the exact behavior of a pair of atoms. But this pair is emerged
in the rest of the system. We now need to ask the question how large the value
of the spin on of of the sites is if the connections are all made to averages. That
requires the knowledge of both the nearest neighbor and the next nearest neighbor
spin correlation function. It is possible to build up a system of equations
where the equation for spin correlation functions at a given distance requires
knowledge of spin correlation functions one distance further apart. We need to
know all spin correlation functions to solve this system, or we need to have a
good approximation for how spin correlation functions at large distances decay.
Here we consider the one dimensional Ising chain again. We set the external
field equal to zero. Since without an external field there is no magnetization,
the spin-correlation function is given by