# Density Functional Theory of the Hubbard-Holstein Model

## TL;DR

This paper develops a density functional theory tailored for lattice models with both electron-electron and electron-phonon interactions, providing a new approach to study correlated systems with phonons.

## Contribution

It introduces a DFT framework for the Hubbard-Holstein model, deriving exchange-correlation potentials from dynamical mean field theory and analytical solutions.

## Key findings

- DFT captures linear conductance accurately.
- DFT describes real-time dynamics well.
- Potentials show density-dependent discontinuities.

## Abstract

We present a density functional theory (DFT) for lattice models with local electron-electron (e-e) and electron-phonon (e-ph) interactions. Exchange-correlation potentials are derived via dynamical mean field theory for the infinite-dimensional Bethe lattice, and analytically for an isolated Hubbard-Holstein site. These potentials exhibit discontinuities as a function of the density, which depend on the relative strength of the e-e and e-ph interactions. By comparing to exact benchmarks, we show that the DFT formalism gives a good description of the linear conductance and real-time dynamics.