2026-08-29
SALTED trained only on 3×3 shifted bilayers; long-range LOVV keeps 1000-atom moiré band errors near 5 meV, while SOAP’s 0.09% MoS2 density error still yields 330 meV band error.
Once a 2D bilayer is twisted by a few degrees, the moiré cell runs to thousands of atoms. Direct DFT becomes impractical. Continuum and tight-binding models cut the cost, but their parameters move with twist angle, stacking, and material, so a new physical question often means a new fit.
Machine-learned electronic structure splits two ways: predict Hamiltonian matrix elements, or predict the electron density and derive ground-state properties from it. Hamiltonian models usually lean on nearsightedness and report sub-meV matrix-element errors. That is reasonable for nonpolar graphene. Polar bilayers such as hBN and the transition-metal dichalcogenides rearrange charge between layers, build long-range electrostatic fields, and make a local cutoff miss the moiré-scale potential. Density models have their own failure modes. Overcomplete density-fitting bases make the overlap matrix ill-conditioned on heavy elements. A small global density error also does not guarantee that bands derived from that density are right.
A team at the MPI for the Structure and Dynamics of Matter, with York and Cambridge, extends SALTED (Symmetry-Adapted Learning of Three-dimensional Electron Densities) to twisted bilayers. SALTED expands the density in atom-centered auxiliary functions and learns the coefficients with equivariant-kernel Gaussian process regression.
Training stays small on purpose: 512 aligned 3×3 bilayers per material, with random interlayer shifts, spacings, and in-plane rattling, split 400/112. The actual test set is twisted. Graphene and hBN run from 21.79° to 3.15° (up to 1324 atoms); the TMDCs from 21.78° to 3.89° (up to 1302 atoms). Reference calculations use PBE in FHI-aims.
The descriptor is the lever that matters. SOAP sees only the local atomic density inside a 6 Å cutoff. LODE tensors that density with an electrostatic-like representation and stays sensitive to about 13 Å. LOVV tensors two electrostatic representations, stays sensitive to about 17 Å, and can correlate two sites that both sit outside a local cutoff. On the TMDCs the density-fitting overlap matrices have condition numbers from 10^{9.8} to 10^{12.5}. Above an empirical threshold of 10^{9.5} the authors truncate singular values: 10^{-6} for TiS2, 10^{-2} for ZrS2 and MoS2, dropping near-null fitting noise.
The pipeline is density, then a Kohn-Sham Hamiltonian, then diagonalization. Spin-orbit coupling is added as a non-self-consistent perturbation on the same density, with no retraining. A separate MACE potential relaxes graphene and hBN geometries before SALTED sees them.
On the aligned validation set, with hyperparameters tuned for density, the numbers already disagree with the usual story:
| Material | Density error SOAP / LOVV | Band error SOAP / LOVV |
| Graphene | 0.48% / 3.63% | 1.86 meV / 7.1 meV |
| hBN | 0.65% / 2.12% | 1.51 meV / 4.0 meV |
| TiS2 | 1.71% / 2.12% | 49 meV / 9.6 meV |
| ZrS2 | 0.40% / 0.45% | 27 meV / 6.6 meV |
| MoS2 | 0.09% / 0.04% | 330 meV / 10.2 meV |
For graphene and hBN, better density tracks better bands. For the TMDCs the ranking flips. SOAP often wins on density and loses on bands. On ZrS2 it has the lowest density error and the highest band error. On MoS2, LOVV and LODE both sit at 0.04% density error, yet the band errors differ by a factor of five (10.2 meV vs 52 meV).
Extrapolation to small twist angles is sharper. All three descriptors keep graphene low-energy band error below 5 meV. SOAP exceeds 40 meV on TiS2 and 100 meV on MoS2 (off-scale in the figure). Only LOVV stays at ≲5 meV across all five materials, cells above 1000 atoms, and superlattices around 50 Å.
Bandwidth scans reach 4564 atoms in hBN and 3768 in the TMDCs; DFT checks stop near 1300 atoms. The hBN highest valence band narrows to about 5 meV; TMDC conduction groups approach 1 meV. The hBN gap stays near 4 eV and shrinks by about 0.5 eV from 21° to 1.8°. Below about 2°, the TiS2 model diverges: near a metal-insulator transition, tiny gaps feel every density mistake.
For magic-angle twisted bilayer graphene (1.05°, 11908 atoms) a SOAP model sits about 15 meV MAE from full DFT, with a density error of about 10^{-4} e per electron. Residuals pile up in AA domains, so the flat bands look hole-doped; normalizing the total charge does not fix the dispersion. On relaxed graphene at about 2.0° (3268 atoms), SALTED's low-energy band error is 1.5 meV against DeepH's 1.3 meV, both under 0.3% of a 600 meV window. On MoS2, SALTED beats a DeepH result published a few years ago by about an order of magnitude. The in-plane field at TB-hBN domain walls saturates near 9 mV/nm at a 2 nm probe height for twists below 1°, with a wall width of 4.23±0.01 nm. The current implementation is 10 to 100 times faster than converged DFT. LOVV prediction scales quadratically with system size; SOAP stays near-linear.
The working rule for anyone training density models on moiré materials is simple. A scalar density error is not an acceptance test. Where the residuals sit, and how they couple to the observable, decides whether bands and fields come out right. On polar bilayers, locality is not the default.
The practical gain is that training never leaves cheap 3×3 shifted bilayers. Small-angle supercells stay out of the training set. One density then feeds SOC, the Hartree potential, and real-space fields without a new model per property. Hamiltonian learners still look strong on graphene, where locality holds; DeepH's magic-angle flat-band width sits closer to the expected profile. Published Hamiltonian models have not yet shown equally stable bands for hBN or TMDCs below 5°.
This is an incremental step with a clear direction: move density learning from a pretty fit to a usable downstream electronic structure, and treat long-range geometric encoding as required physics on polar moiré stacks, not an optional extra.
The authors name the binding constraint: SALTED is a Gaussian process, so training-set size is capped, and so are accuracy and coverage. They want a neural-network version; the same density-representation instabilities would travel with it.
Physically, TiS2 extrapolation falls apart near gap closure. Spatial density errors at magic-angle graphene twist the flat bands into a doped-looking dispersion that charge normalization does not repair. All references are PBE, so band gaps are systematically too small, a GGA habit; hybrids are left for later. LOVV still scales quadratically. Thousand-atom cells beat DFT today; larger ones will need multipoles and GPUs. One model per material is a GPR-scaling compromise; a multi-material model needs more data. The DeepH numbers come from papers a few years old. How that line moves once long-range interactions are added is not measured here.