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To overcome the phase redundancies inherent in electronic wavefunctions, the authors construct a $U(1)$ gauge-equivariant autoencoder that compresses plane-wave density functional theory (DFT) Bloch states into a compact latent representation. Enforcing gauge equivariance naturally regularizes the latent manifold to preserve physical smoothness and underlying topological invariants. Evaluated on two-dimensional insulators, the model achieves a 0.970 average reconstruction fidelity in a 64-dimensional latent space and enables accurate downstream prediction of off-diagonal $GW$ self-energy elements.
Quantum wavefunctions suffer from arbitrary phase redundancies that derail standard neural representations, but enforcing exact $U(1)$ gauge equivariance compresses DFT Bloch states into 64 dimensions with 97% reconstruction fidelity while inherently preserving topological invariants.
Representation learning, or featurization, of mean-field Bloch states provides an important route for incorporating electronic information into machine-learning models of first-principles condensed-phase systems. A challenge in representing Bloch states is the gauge redundancy of each state. In this work, we propose to address this issue by explicitly incorporating gauge equivariance in the machine-learning framework. We develop a $U(1)$-equivariant autoencoder that compresses the plane-wave-basis Bloch states obtained from density functional theory into a low-dimensional latent representation. We achieve an average overlap of 0.970 between original wavefunctions and the reconstructions from a 64-dimensional latent space using a dataset of two-dimensional insulators. We further show that $U(1)$ equivariance endows the latent space with physically meaningful structures, including smoothness and topological information. Finally, we present a proof-of-principle application in which the latent representations are used to predict the off-diagonal elements of the $GW$ self energy matrix, and demonstrate the importance of enforcing gauge equivariance in the prediction.
