Umklapp-equivariant Bloch-state representation learning

Establish equivariance of the Bloch-state autoencoder under reciprocal-lattice translations, so that the latent representations of lattice-periodic Bloch functions at crystal momenta related by an umklapp transformation obey the required position-dependent phase relation.

Background

The autoencoder encodes the lattice-periodic part of each Bloch state rather than the full Bloch function. Under a reciprocal-lattice translation, the crystal momentum transforms as $\bm{k}\mapsto\bm{k}+\bm{G}$, while the lattice-periodic function transforms through the position-dependent phase $$u_{n\bm{k}+\bm{G}}(\bm{r})=e^{-i\bm{G}\cdot\bm{r}}u_{n\bm{k}}(\bm{r})$$. This is not a global $U(1)$ phase and therefore is not captured by the autoencoder’s present global $U(1)$-equivariance constraint.