Machine learning density functional theory for the Hubbard model

Abstract

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 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.

Opportunities and Challenges for Machine Learning in Materials Science

“…Bogojeski et al (119) found that one can efficiently learn the energy differences from DFT and coupled cluster simulations and use ML to provide a promising avenue to have coupled-cluster-level accuracy and DFT-level speed for physical situations where standard DFT is inadequate. Another strategy is to use ML to learn the computationally expensive portions of solving the Kohn-Sham equations in a DFT calculation, namely contributions to the exchangecorrelation energy (120)(121)(122). To this end, Snyder et al (120) modeled the kinetic energy of a onedimensional system of noninteracting electrons; this analysis was then extended to more general cases (121).…”

Perspective on integrating machine learning into computational chemistry and materials science

“…A universal density functional provided by an ML model could potentially eliminate the need for exhaustively comparing different types of functionals for a given chemical problem. So far, ML has been used to generate new DFT functionals or to adjust the energy functional, bypassing the need to solve the iterative Kohn-Sham equations and accelerating simulations for the ground state 104,107,[129][130][131][132][133][134] and excited states 135 significantly. These models further promise better transferability for different types of molecular systems.…”

Electron-electron versus electron-phonon interactions in lattice models: Screening effects described by a density functional theory approach

“…In addition to ab initio descriptions of matter, (TD)DFT has been applied to model Hamiltonians to explore conceptual and methodological aspects of the theory [39][40][41][42][43][44][45][46][47][48][49][50][51], as well as for specific applications to cold atoms [52][53][54][55], Kondo physics [56][57][58], quantum transport [59][60][61], quantum electrodynamics [33], and nonequilibrium thermodynamics [62], to mention a few [63]. We here consider a two-component DFT for the HH model, where the basic variables are given by the set (n, x) ≡ ({n i }, {x i }), with n i = n i↑ + n i↓ being the total electron density at site i, and the conjugated fields are (v, η) ≡ ({v i }, {η i }).…”