I've recently worked on a lattice model relevant to condensed matter physics. The model features an ordered phase with a broken $U(1)$ symmetry, so I anticipated the presence of a Goldstone mode. One is indeed present, but it has a dispersion unlike anything I've seen before.

The system is two-dimensional, so we have a dispersion relation $\omega(k_{x},k_{y})$ with a zero at $\Gamma$ (i.e., $k_{x}=k_{y}=0)$. The strange thing is that a series expansion of $\omega$ about $\Gamma$ is linear in $k_{x}$ but quadratic in $k_{y}$:

$$\omega(0,0) \approx c_{0} + c_{x}k_{x} + c_{y}k_{y}^{2} + \cdots $$

I'm used to the usual story about Goldstone modes requiring symmetric dispersions with well-defined power laws, so that $\omega$ typically scales like $|k|^{z}$ near the node. I'm curious to learn if such a symmetry property is in fact required, and if it is not why we so often encounter Goldstone modes with this symmetry.


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