Equivariant Deep Learning in Direct-Product Space for Electronic-Structure Hamiltonian Prediction

Deep learning provides a promising route to efficient prediction of electronic-structure Hamiltonians, enabling high-throughput screening across large collections of materials. However, most existing equivariant architectures propagate features in direct-sum spaces of irreducible spherical tensors, whereas the target Hamiltonian is naturally an operator block in a direct-product representation over local orbital and spin bases. This representation mismatch fragments contiguous Hamiltonian blocks, obscures direct manipulation of orbital couplings, and organizes features into irregular angular-momentum components that require numerous transformations and are poorly aligned with efficient hardware execution and memory management. To address both the representation mismatch and computational bottleneck, we propose DP-Learner, an equivariant learning framework that performs message passing directly on Hamiltonian-like operator blocks in the direct-product space. Theoretically, we propose Lie-generator-based directional embedding and invariantly weighted left–right products and contractions that preserve equivariance to $SO(3)$, $SU(2)$, and spatial inversions, without explicit Clebsch–Gordan decomposition or inverse reconstruction. Methodologically, we instantiate this principle as a Transformer architecture, in which Hamiltonian-like feature matrices or tensors evolve through equivariant operators in the direct-product space, while non-linear invariant scalar routing controls structured gating over matrix units, directional propagation, attention, and environment-embedded context interactions. We further construct MP-HAM-110K, a large-scale Hamiltonian dataset comprising more than 110,000 material structures covering 68 elements with both Hamiltonian and wavefunction data. Experiments on MP-HAM-110K and Materials-HAM-SOC demonstrate that DP-Learner dramatically improves Hamiltonian prediction accuracy and inference efficiency over the state-of-the-art method, with up to a $40.6$% reduction in prediction error, an $11.2\times$ speedup in neural inference, and a $95.2$% reduction in peak GPU memory consumption.