Gauge Equivariance in Wavefunction-based Machine Learning for Many-body Interactions

Simulation of many-body systems remains challenging because post-density-functional-theory (post-DFT) methods that account for many-body interactions are computationally expensive. Recently, machine learning (ML) approaches have been employed to accelerate post-DFT calculations and thereby enable simulations at larger scales. Motivated by the fact that many post-DFT formalisms take DFT Kohn-Sham states as their starting point, several ML models have been developed to use DFT states as input to predict many-body quantities. A remaining difficulty, however, is the uncontrolled random gauge of the DFT eigenstates. In this work, we introduce a U(1) gauge-equivariant ML approach for the GW plus Bethe-Salpeter equation (GW-BSE) formalism. Specifically, we develop a U(1)-equivariant variational autoencoder and a gauge-equivariant transformer that explicitly enforce gauge symmetry to predict many-body quantities, and we investigate the capabilities of this equivariant framework. Our approach provides a general framework for wavefunction-based ML models in materials study and design.