Machine learning density functional theory for the Hubbard model

The solution of complex many-body lattice models can often be found by defining an energy functional of the relevant density of the problem. For instance, in the case of the Hubbard model the spin-resolved site occupation is enough to describe the system's total energy. Similarly to standard density functional theory, however, the exact functional is unknown, and suitable approximations need to be formulated. By using a deep-learning neural network trained on exact-diagonalization results, we demonstrate that one can construct an exact functional for the Hubbard model. In particular, we show that the neural network returns a ground-state energy numerically indistinguishable from that obtained by exact diagonalization and, most importantly, that the functional satisfies the two Hohenberg-Kohn theorems: for a given ground-state density it yields the external potential, and it is fully variational in the site occupation.