Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 1 / b

Fourier expansion in volume $V$ diagonalizes the quadratic Landau-Ginzburg theory:whereEach independent real mode contributes a Gaussian integral proportional to $(T/K{)}^{1/2}$. Taking $-T\log Z$, dividing by $V$, and replacing the sum by an integral gives, up to field-independent conventions,

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For one mode put $f=-\frac{T}{2}\log (CT/K)$. Since $c=-T{\partial }_T^{2}f$,ThusAt $(T_c,0)$, $K_T=({\mu }^2{)}^{\prime }(T_c)\ne 0$ because ${\mu }^2$ has a simple zero, so $A(T_c,0)\ne 0$.

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Near the ordinary critical point, $K\sim at+\gamma k^2$. Rescaling $k=t^{1/2}q$ gives the Gaussian fluctuation correction near a critical pointThus ${\alpha }_{\mathrm{fl}}=2-d/2$. It becomes as important as the mean-field value $\alpha =0$ at $d=4$, so the ordinary upper critical dimension is $d_c=4$.

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The quadratic kernel still has $K\sim at+\gamma k^2$, so ${\alpha }_{\mathrm{fl}}=2-d/2$. The tricritical mean-field exponent is $\alpha =1/2$. Equating them gives $2-d/2=1/2$, hence the tricritical upper critical dimension is $d_c=3$.

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