Problem: Consider a Landau-Ginzburg theory involving a single real scalar field \(\phi(\mathbf x)\) for \(\mathbf x\in\mathbf R^n\) governed by the canonically normalized free energy density:
\[\mathcal F(\phi,\partial\phi/\partial\mathbf x,…)=\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2+\frac{\phi^2}{2\xi^2}+…\]
Explain what the \(+…\) means, explain which terms have temperature \(T\)-dependence, and explain for which such terms does that \(T\)-dependence actually matter?
Solution: The \(+…\) includes any terms (each with their own coupling constants) compatible with the golden trinity of constraints: locality, analyticity, and symmetry (e.g. a quartic \(g\phi^4\) coupling). The part of the free energy density \(\mathcal F\) before the \(+…\) should compared to the Lagrangian density \(\mathcal L\) of Klein-Gordon field theory:
\[\mathcal L=\frac{1}{2c^2}\frac{\partial^2\phi}{\partial t^2}-\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2-\frac{\phi^2}{2\lambda^2}\]
with \(\lambda=h/mc\) the Compton wavelength playing a role analogous to the correlation length \(\xi\sim 1/\sqrt{|T-T_c|}\); indeed, this \(T\)-dependence in \(\xi=\xi(T)\) is the only \(T\)-dependence that matters, even though generically all the other coupling constants will also have \(T\)-dependence.