Fig. 3: Example studies on twisted bilayer graphene (TBG).
From: A deep equivariant neural network approach for efficient hybrid density functional calculations
a Schematic workflow of the DeepH-hybrid method, which applies deep neural networks to learn from training datasets of non-twisted bilayer graphene and then generalizes to study TBG systems with varying twist angle θ. b–d Band structures of b (2, 1) TBG (θ ≈ 21.79°, 28 atoms/cell), c (3, 2) TBG (θ ≈ 13.17°, 76 atoms/cell), and d (17, 16) TBG (θ ≈ 2.00°, 3,268 atoms/cell) computed by hybrid density functional theory (DFT-hybrid) and DeepH-hybrid. e Computation time for predicting hybrid-functional Hamiltonians of TBGs by DFT-hybrid using linear-scaling algorithms versus by DeepH-hybrid using neural-network inference. The dashed black line represents the time cost trend as a function of system size for an algorithm with O(N) scaling. f Top-view atomic structure of magic-angle TBG (θ ≈ 1.08°, 11,164 atoms/cell). The Moiré supercell is highlighted by a red dashed frame. g,h Band structures of magic-angle TBG computed by g DeepH-PBE and h DeepH-hybrid. DeepH-PBE denotes the deep-learning DFT Hamiltonian of the PBE functional. The relaxed atomic structure of magic-angle TBG adapted from ref. 63 is used. The results of DeepH-PBE are adapted from ref. 28. Source data are provided as a Source Data file.
