= Mean-field tricritical exponent calculation
{title2=$\alpha=1/2,\quad\beta=1/4,\quad\gamma=1,\quad\delta=5$}
For the sextic <Landau free energy> $V=rM^2/2+vM^6/6-hM$, $v>0$, with $r$ linear in reduced temperature, the equation of state is $h=rM+vM^5$. Its ordered zero-field minimum satisfies $M^4=-r/v$, the inverse <magnetic susceptibility> is $4|r|$ below the transition and $r$ above, and the critical isotherm satisfies $h=vM^5$. The minimized singular <free-energy density> is $-|r|^{3/2}/(3\sqrt v)$ below and zero above. These facts give the displayed <mean-field critical exponents>, obeying both the <Rushbrooke scaling relation> and <Widom scaling relation>. Both the quadratic and quartic coefficients must be tuned; a generic path with a positive quartic coefficient instead has ordinary critical behavior. Fluctuations change the result below the tricritical <upper critical dimension> three, and marginal corrections can add logarithms at three.
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