# Compute the mean eld free energy and show that theory

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Unformatted text preview: onian 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 self-consistency 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 ) &lt; 0 and, for stability, c(T ) &gt; 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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