Hot-Ham: High-order Tensor Hamiltonians

• High-order Tensor Hamiltonian (Hot-Ham) is an E(3)-equivariant framework using symmetry-aware MPNNs and tensor contraction techniques to predict electronic structure Hamiltonians.
• By applying local coordinate transformation and the Gaunt tensor product, Hot-Ham reduces computational complexity from O(L^6) to near-linear scaling for efficient large-scale simulations.
• Hot-Ham achieves state-of-the-art accuracy with minimal parameters, as evidenced by low MAEs in systems like graphene, MoS₂, and twisted multilayer structures.

High-order Tensor Machine-learning Hamiltonian (Hot-Ham) is an E(3)-equivariant machine learning framework for accurate and efficient prediction of electronic structure Hamiltonians, particularly tailored for large-scale materials systems. Hot-Ham leverages symmetry-aware message passing neural networks (MPNNs) and novel tensor contraction strategies to reconstruct Hamiltonian blocks from density functional theory (DFT) data, conforming to the transformation laws prescribed by the Wigner–Eckart theorem. By directly modeling high-order tensor features representing atomic orbitals and coupling them efficiently, Hot-Ham addresses the prohibitive computational complexity of standard equivariant MPNNs and sets new accuracy and efficiency standards for data-driven ab initio electronic structure calculations (Liang et al., 5 Sep 2025).

1. Symmetry Principle and E(3)-Equivariant Modeling

Hot-Ham is fundamentally constructed to respect the symmetries of the Euclidean group in three dimensions, E(3), which includes translations, rotations, and inversion. The model’s target—blocks of the electronic Hamiltonian—are high-order tensors encoding orbital interactions and must transform covariantly under E(3). Hot-Ham achieves this by embedding both node and edge representations as high-order spherical harmonic tensors and explicitly designing all message passing and tensor operations to be equivariant, ensuring that predictions transform correctly under all relevant symmetries. This symmetry compliance is essential for transferability and reliability in both periodic crystals and complex disordered phases.

2. Computational Innovations: Local Coordinate Transformation and Gaunt Tensor Product

Hot-Ham introduces two key algorithmic advances to overcome the steep complexity of equivariant message passing found in earlier MPNN architectures:

a) Local Coordinate Transformation

High-order tensors associated with nodes and edges are transformed such that the principal axis aligns with the relevant interatomic direction (the “edge vector”). In this frame, spherical harmonic features become sparse, as only components with $m = 0$ remain nonzero (i.e., $Y_{\ell m}(r) = 0$ for $m \neq 0$), dramatically reducing the number of nontrivial summations involved in tensor products. The traditional Clebsch–Gordan tensor product (CGTP), whose complexity for maximum tensor order L is $\mathcal{O}(L^6)$, simplifies to a complexity of approximately $\mathcal{O}(L^3)$ via this transformation.

b) Gaunt Tensor Product (GTP)

Hot-Ham replaces the standard CGTP with the Gaunt tensor product, which computes products of spherical harmonics using the Gaunt coefficients: $$F(\mathbf{x}) = \int Y_{\ell_1 m_1}(\Omega) Y_{\ell_2 m_2}(\Omega) Y_{\ell_3 m_3}(\Omega) \, d\Omega \equiv G(\ell_1, \ell_2, \ell_3; m_1, m_2, m_3)$$ This allows the tensor contraction to be carried out in two alternative ways: - Using a 2D Fast Fourier Transform on spherical harmonic features (GTP(2D–FB)), yielding $\mathcal{O}(L^3)$ cost. - Using projection onto a spherical grid, enabling a further reduction to $\mathcal{O}(L^2 \log_2 L)$, where L is the maximal tensor order.

The GTP as implemented handles only symmetric components (even $\ell_1+\ell_2+\ell_3$), and Hot-Ham introduces antisymmetric tensors through a subsequent SO(2) convolutional layer, ensuring complete representation of physical interactions.

3. Model Structure and Performance Benchmarks

Hot-Ham is benchmarked against several public datasets and material systems:

System	Reported MAE (meV)	Parameters (millions)
Monolayer Graphene	~0.08	0.9
Monolayer MoS₂	~0.12	0.9
Bilayer Graphene	~0.15	0.9
(Baseline equivariant MPNN)	>0.15	4.5

The model achieves these accuracies with substantially fewer parameters compared to state-of-the-art alternatives, while maintaining linear or near-linear scaling with system size. On multilayer twisted MoS₂, Hot-Ham trained solely on bilayer and trilayer data (without four-layer examples) predicts Hamiltonians for twisted structures with MAE within a few meV and reproduces DFT band structures with high fidelity. For large graphene/h-BN heterostructures (up to 1022 atoms), band MAEs remain in the few-meV range, and Hamiltonian element MAEs as low as 0.49 meV are reported.

4. Generalization and Application Scope

Hot-Ham demonstrates strong generalization and computational efficiency across diverse applications:

• Twisted Multilayer Systems: Accurate predictions for moiré superlattices and double bilayer stacks, despite absent configurations in the training data.
• Heterostructures: Transferability across mismatch and twisting in graphene/h-BN interfaces.
• Bulk Allotropes: Robust reproduction of electronic structures in complex phosphorus allotropes (e.g., violet phosphorus with 84-atom unit cells), with MAEs below 1 meV on Hamiltonian elements and key band structure features preserved.

This generalization is attributed to strict symmetry enforcement, efficient high-order tensor contraction, and the model’s ability to extrapolate from lower-complexity training data.

5. Computational Cost, Scaling, and Efficiency

The combination of local coordinate transformation and GTP enables a significant reduction in tensor contraction cost. For a tensor order L, the cost per contraction scales as $\mathcal{O}(L^3)$ for the GTP(2D–FB) and as $\mathcal{O}(L^2\log_2 L)$ for the sphere grid implementation, compared to $\mathcal{O}(L^6)$ for CGTP. This allows the model to handle Hamiltonians with higher angular momentum channels (large L) and larger system sizes with modest computational resources. Empirical scaling studies on multilayer MoS₂ show near-linear growth of computation time with the number of atoms, consistent with the theoretical reduction in complexity.

6. Prospective Extensions and Future Applications

Hot-Ham’s architecture positions it as a platform for a range of advanced applications in ab initio materials modeling:

• Force Prediction and Molecular Dynamics: Differentiating the NN-predicted Hamiltonian enables force calculations for atomistic dynamics, potentially integrating electronic structure effects with large-scale dynamics.
• Electron–Phonon Interactions: The ability to efficiently differentiate Hamiltonians supports calculation of electron–phonon matrix elements, opening pathways to paper superconductivity or inelastic scattering phenomena.
• Quantum Transport: The prediction of orthogonal-base Hamiltonians facilitates coupling to linear scaling quantum transport methods, enabling electronic structure analysis of million-atom-scale devices.
• Integration with DFT and Learning Pipelines: The symmetry-aware architecture of Hot-Ham naturally adapts to further integration as a drop-in DFT surrogate for nano- and mesoscale materials simulations.

7. Broader Context and Methodological Relationships

The technological strategies used in Hot-Ham reflect the broader trend toward symmetry-enforced, high-order tensor machine learning frameworks in ab initio materials simulation (Yu et al., 11 Jun 2025, Yin et al., 1 Jan 2024). Hot-Ham’s use of local frames and efficient convolutional contractions aligns with efforts to reduce the overhead of equivariant neural networks and makes it amenable to continued scaling. Furthermore, the design is compatible with ongoing work in multi-stage neural protocols, tensor decomposition, and operator inference for learning both parametric and nonparametric Hamiltonian models.

In summary, Hot-Ham introduces an efficient E(3)-equivariant MPNN built upon local coordinate transformations and Gaunt tensor products, profoundly reducing the cost of high-order tensor interactions in electronic structure Hamiltonian learning. Its strong performance across standard and challenging materials datasets, efficient scaling, and symmetry-consistent architecture make it a promising foundation for next-generation machine-learned ab initio electronic structure methods (Liang et al., 5 Sep 2025).