What does the absence of these Goldstone boson interactions mean physically?

Question

I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative terms like $(\nabla\theta)^2$.

For instance, in the XY model, the free energy has terms like $$F\sim\frac\gamma2 \int (M_0^2+2M_0\delta M)(\nabla\theta)^2.$$ Or in the Heisenberg model, there are terms like $$F\sim\frac\gamma2 \int M_0^2[(\nabla\theta)^2+\sin^2\theta(\nabla\phi)^2],$$ for the two Goldstone modes $\phi$ and $\theta$.

My question is, what is the physical interpretation of the absence of $\theta^{2n}$ interactions, and only derivative ones?

Answer 1

When there's a single $\nabla \theta$ term, this tells you that the theory has a shift symmetry of $\theta \mapsto \theta + a$ which moves you between different vaccua (each with a different $U(1)$ charge). Expressions can become logner with additional Goldstones but the principle is the same. Your expression with $(\theta, \phi)$ is invariant under $SO(3)$ rotations which treat these fields as angles on $S^2$.

This structure appears because the potential in the original theory is a function on $\mathbb{R}^n$ and Goldstones parameterize the submanifold in $\mathbb{R}^n$ which minimizes it. The theory which describes their fluctuations must therefore preserve the interpretation that each Goldstone field is a co-ordinate on a target space. The way we do this is through the sigma model \begin{equation} S = \int dx \, G_{ij}(\phi) \nabla \phi^i \cdot \nabla \phi^j \end{equation} where $G_{ij}$ is the metric of the target space. You can check that the examples above have this form. If there were additional terms like $G_{ij} \phi^i \phi^j$ without derivatives, $S$ would no longer be similar to the action used to derive the geodesic equation and therefore not describe the "length of a path".