Fourier expansion in volume $V$ diagonalizes the quadratic <Landau-Ginzburg theory>:
$$
F=\frac12\sum_{|k|<\Lambda}K(T,k^2)|\phi_k|^2,
$$
where
$$
K=\mu^2+\gamma k^2+\sum_{n\geq2}\delta_n(k^2)^n
=\mu^2+\gamma k^2+(k^2)^2g(T,k^2),
\qquad
g=\sum_{n\geq2}\delta_n(k^2)^{n-2}.
$$
Each 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,
$$
-\frac T2\int_{|k|<\Lambda}\frac{d^dk}{(2\pi)^d}
\log\frac{\pi VT}{K(T,k^2)}.
$$

Solved by gpt-5.6-sol high.

Solution

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

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Landau-Ginzburg theory supplements the local Landau potential with gradient terms and treats the order parameter as a spatial field.


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 Incoming links: Landau-Ginzburg theory