Lorentz Violation in Condensed Matter Systems

Question

I have heard from a few different sources that certain condensed matter systems like topological superconductors and insulators violate Lorentz invariance. Can someone explain how that happens?

Answer 1

When we study the electronic states of crystalline solids, we usually treat the crystal structure as a static background, and add dynamical effects like lattice vibrations as perturbations. This is the Born-Oppenheimer approximation, which essentially assumes that the electronic states and states of the lattice can be understood independently.

In that context, the presence of the (non-dynamical) background lattice manifestly breaks Lorentz symmetry in a fairly obvious way. The fact that electronic states are not shackled by Lorentz symmetry means that a vastly wider array of phenomena are possible than in the case of the Minkowski vacuum. This is partly why I find condensed matter so enjoyable.

From a fundamental perspective, yes - if you include the full dynamics of the lattice and treat everything relativistically, then Lorentz invariance would be recovered. However, what you'll find (if you haven't already) is that treating everything at that level is completely untenable even for systems of a few particles, to say nothing of an entire crystal. Sacrificing Lorentz invariance is often necessary to obtain models which you can actually get anything out of in the first place (and, as I say, I see this as a bonus, not an issue to overcome).

If you're interested in this viewpoint, it was very eloquently discussed by Laughlin in his book A Different Universe.

Answer 2

For context, you're definitely referring to your earlier post.

To clarify, nobody was saying that there are real Lorentz violations in condensed matter systems. We were saying that in some systems (like condensed matter), we don't bother accounting for relativistic effects. That's all there is to it. We're not saying that we're observing physics that displays Lorentz violations, but rather that we don't feel the need to account for relativity in our models for said systems.