U(1) Gauge-Equivariant Representation Learning of Bloch States
Abstract
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.
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BibTeX
@misc{zhang2026gaugeequivariant,
title = {{$U(1)$ Gauge-Equivariant Representation Learning of Bloch States}},
author = {Chengyan Zhang and Xian Xu and Bowen Hou and Diana Y. Qiu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30845},
eprint = {2609.30845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30845},
}
RIS
TY - UNPB
TI - U(1) Gauge-Equivariant Representation Learning of Bloch States
AU - Chengyan Zhang
AU - Xian Xu
AU - Bowen Hou
AU - Diana Y. Qiu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30845
UR - https://arxiv.org/abs/2609.30845
AB - 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.
ER -