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H = −J ij si · sj where si take values in the set
1
0 si ∈ , −1/2
√
3/2 , −1/2
√
− 3/2 By developing a mean ﬁeld theory for H determine the selfconsistency requirement
for the magnetisation m = si . Compute the mean ﬁeld free energy and show that
theory undergoes a ﬁrst order phase transition even in the absence of an external ﬁeld.
[Hint: This calculation will be simpler if you argue that you can focus on magnetisation vectors of the form m = (m, 0).]
8. Consider the free energy
F = a(T )m2 + b(T )m4 + c(T )m6
3 where b(T ) < 0 and, for stability, c(T ) > 0 for all T . Sketch the possible behaviours
of the free energy as a(T ) varies and, in each case, identify the ground state and
metastable states. Show that the system undergoes a ﬁrst order phase transition at
some temperature Tc . Determine the value a(Tc ) and the discontinuity in m at the
transition. 4...
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 Spring '14

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