Machine-learning semi-local density functional theory for many-body lattice models at zero and finite temperature

TL;DR

This paper develops a machine-learning density functional theory for the 1D spinless Hubbard model, enabling efficient computation of ground-state and finite-temperature properties in the thermodynamic limit using neural networks.

Contribution

It introduces semi-local neural network functionals for the Hubbard model at zero and finite temperature, independent of system size, and capable of predicting thermodynamic quantities.

Findings

Functional accurately reproduces ground-state energies.

Neural networks predict finite-temperature thermodynamic properties.

Method enables analysis of many-body systems in the thermodynamic limit.

Abstract

We introduce a machine-learning density-functional-theory formalism for the spinless Hubbard model in one dimension at both zero and finite temperature. In the zero-temperature case this establishes a one-to-one relation between the site occupation and the total energy, which is then minimised at the ground-state occupation. In contrast, at finite temperature the same relation is defined between the Helmholtz free energy and the equilibrium site occupation. Most importantly, both functionals are semi-local, so that they are independent from the size of the system under investigation and can be constructed over exact data for small systems. These 'exact' functionals are numerically defined by neural networks. We also define additional neural networks for finite-temperature thermodynamical quantities, such as the entropy and heat capacity. These can be either a functional of the ground-state site occupation or of the finite-temperature equilibrium site occupation. In the first case their equilibrium value does not correspond to an extremal point of the functional, while it does in the second case. Our work gives us access to finite-temperature properties of many-body systems in the thermodynamic limit.