Write

t = T - T_{c}

. Analyticity gives

α_{2} = a t + O (t^{2})

with

a > 0

and

α_{4} (T_{c}) = u > 0

. For

t < 0

, minimization gives

m^{2} = - α_{2} / α_{4} \sim - a t / u

, so

β = 1/2

. At the minima,

F_{m i n} = - \frac{α _{2}^{2}}{4 α _{4}} \sim - \frac{a ^{2}}{4 u} t^{2} .

α = 0

. This is the ordinary mean-field transition of Landau theory.

Solved by gpt-5.6-sol high.

ii

Solution

α_{2} = a t + O (t^{2})

α_{4} = b t + O (t^{2})

and

α_{6} (T_{c}) = u > 0

. The nonzero stationarity equation is

a t + b t m^{2} + u m^{4} + \dots = 0.

The middle term is subleading, so

m^{4} \sim - a t / u

and

β = 1/4

. Substitution gives

F_{m i n} ≍ - ∣ t ∣^{3/2}

, so the heat capacity behaves as

∣ t ∣^{- 1/2}

and

α = 1/2

Solved by gpt-5.6-sol high.

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