Past exam of the theoretical physics major option course of the University of Oxford / 2018 / 6 / e / Solution

Take $a=a_0(T-T_c)$ and hold $b,c$ fixed. The heat capacity comes from the minimized free energy, $C=-T{d}^2{F}_{\min }/d{T}^2$, with any smooth background added separately. For $b>0$ and $T<T_c$,Above $T_c$, $m=0$ and the polynomial contributes zero. There is a finite jump, rather than a divergence: the conventional ordinary mean-field critical exponents areThe upper branch has zero order-parameter contribution; a finite regular heat capacity has exponent zero.

Along the tuned tricritical path $b=0$, the ordered minimum givesUpper-side qualification: the vanishing disordered-side heat-capacity amplitude in Landau theory matters here: for $b=0$ and $a>0$, the minimizing solution is still exactly $m=0$, so this saddle-point polynomial gives $F_{\min }-F_{\mathrm{reg}}=0$ and $C_{\mathrm{sing},+}=0$. A zero function cannot determine a nonzero-amplitude power-law exponent. For example, adding the allowed smooth background $F_{\mathrm{reg}}=-C_{0}T\log (T/T_{\ast })$, with $C_0>0$, leaves the entire phase diagram unchanged but makes the upper-side heat capacity exactly $C_0$, not a $(T-T_c{)}^{-1/2}$ divergence.

One convention assigns the tricritical scaling exponent $\alpha =1/2$ to both sides, writing $C_{\mathrm{sing},\pm }=A_{\pm }|T-T_c{|}^{-1/2}$ with $A_{+}=0$ at this mean-field approximation saddle. If the source's literal $C\sim |T-T_c{|}^{-{\alpha }_{+}}$ is meant to have a nonzero leading coefficient, the given polynomial establishes no upper-side tricritical singular exponent; with a nonzero smooth background its leading upper-side exponent is zero. A nonzero upper-side singularity requires fluctuation information beyond this minimization. This distinction prevents claiming an upper-side divergence that the displayed Landau free energy does not produce.