  The marriage of density functional theory (DFT) and deep learning methods has
the potential to revolutionize modern research of material science. Here we
study the crucial problem of representing DFT Hamiltonian for crystalline
materials of arbitrary configurations via deep neural network. A general
framework is proposed to deal with the infinite dimensionality and covariance
transformation of DFT Hamiltonian matrix in virtue of locality and use message
passing neural network together with graph representation for deep learning.
Our example study on graphene-based systems demonstrates that high accuracy
($\sim$meV) and good transferability can be obtained for DFT Hamiltonian,
ensuring accurate predictions of materials properties without DFT. The Deep
Hamiltonian method provides a solution to the accuracy-efficiency dilemma of
DFT and opens new opportunities to explore large-scale materials and physics.
