Landau-Ginzburg Theory and Functional Integrals

Derive the Landau-Ginzburg free energy functional from the Ising model partition function via the Hubbard-Stratonovich transformation.

Show that the saddle-point approximation of the functional integral reproduces the mean-field results of Landau theory.

Analyze Gaussian fluctuations around the saddle point to derive the Ornstein-Zernike form of the correlation function (propagator).

Derive the Ginzburg criterion by comparing the magnitude of Gaussian fluctuation corrections to the mean-field order parameter.

Define the generating functional for correlation functions by introducing a source term into the functional integral.

Apply Wick's theorem for Gaussian functional integrals to evaluate multi-point correlation functions.

Represent the first-order perturbative correction from the quartic term diagrammatically using propagators and vertex factors.