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