Machine learning semilocal exchange-correlation functionals for Kohn-Sham density functional theory of the Hubbard model

The Hubbard model provides a test bed to investigate the complex behavior arising from electron-electron interaction in strongly correlated systems and naturally emerges as the foundation model for lattice density functional theory (DFT). Similarly to conventional DFT, lattice DFT computes the ground-state energy of a given lattice model, by minimizing an energy functional of the on-site occupations. The energy then comprises a contribution that depends on the external potential and a universal contribution describing the kinetic and the electron-electron interaction energy. Here we use machine learning to construct a class of scalable “semilocal” exchange-correlation functionals with an arbitrary degree of nonlocality for the one-dimensional spinfull Hubbard model. Then, by functional derivative we construct an associated Kohn-Sham potential, that is used to solve the associated Kohn-Sham equations. This effectively forms a close workflow for machine-learning Kohn-Sham lattice DFT. After having investigated how the accuracy of the semilocal approximation depends on the degree of nonlocality, we use our Kohn-Sham scheme to compute the polarizability of linear chains, either homogeneous or disordered, approaching the thermodynamic limit.