= Solution
These are <mean-field critical exponents> at a <tricritical point>. Hold the other control at its tricritical value and take the coefficients to be <real analytic functions> of <temperature> along a thermal approach with $\mathcal A_2=a(T-T_c)+o(T-T_c)$, $a>0$, $\mathcal A_6\to s>0$, and $\mathcal A_4=0$ or $\mathcal A_4=o(|\mathcal A_2|^{1/2})$. At fixed tricritical $g$, a generic <real analytic> <temperature> path with nonzero thermal slope has $\mathcal A_4=O(T-T_c)$ and meets the last condition. Paths tangent to a scaling direction can have different apparent <temperature> powers.
At zero field on the ordered side with $\mathcal A_4=0$, stationarity gives $m^4=-\mathcal A_2/s$; under the more general approach $\mathcal A_4=o(|\mathcal A_2|^{1/2})$, this relation holds to leading order. Thus the <order-parameter critical exponent> is
$$
\boxed{m\sim\left(\frac{a(T_c-T)}s\right)^{1/4},\qquad \beta=\frac14.}
$$
For the zero-quartic representative, putting this minimum into the <Landau free energy> gives the singular order-parameter contribution (and the leading asymptotic contribution under the stated negligible-quartic approach)
$$
f_{\rm s}=-\frac{|\mathcal A_2|^{3/2}}{3\sqrt s}.
$$
Using $C_{\rm s}=-T\,\partial_T^2f_{\rm s}$ at fixed other control yields $C_{\rm s}\sim(T_c-T)^{-1/2}$, specifically $Ta^{3/2}/(4\sqrt{s(T_c-T)})$ to leading order. Hence
$$
\boxed{\alpha=\frac12.}
$$
This <heat-capacity critical exponent> refers to the nonzero ordered-side singular amplitude; in this <mean-field approximation> to <Landau theory>, the disordered <global minimum> contributes zero singular <free energy>, while any regular background remains. Fluctuation corrections lie outside the present approximation.
In the disordered phase, differentiating the equation of state at $m=h=0$ gives the <magnetic susceptibility> $\chi=1/\mathcal A_2$. On the ordered branch its curvature is $\mathcal A_2+5sm^4=4|\mathcal A_2|$, so $\chi=1/(4|\mathcal A_2|)$. Both give the <magnetic-susceptibility critical exponent>
$$
\boxed{\gamma=1.}
$$
At the tricritical <temperature> with both lower coefficients zero, $h=sm^5$, so $m=\operatorname{sgn}(h)(|h|/s)^{1/5}$ and the <critical-isotherm exponent> is
$$
\boxed{\delta=5,\qquad (\alpha,\beta,\gamma,\delta)=\left(\frac12,\frac14,1,5\right).}
$$
The <Widom scaling relation> $\gamma=\beta(\delta-1)$ and <Rushbrooke scaling relation> $\alpha+2\beta+\gamma=2$ both check these powers. Their validity here is a mean-field conclusion, not a claim that fluctuations in every spatial dimension preserve them.
Back to article page
