# Friedel oscillations of one-dimensional correlated fermions from perturbation theory and density functional theory

## TL;DR

This paper investigates the decay of Friedel oscillations in one-dimensional correlated fermions using perturbation theory and density functional theory, finding that interactions do not significantly alter the decay exponent in large systems.

## Contribution

It compares perturbation theory and density functional theory approaches to analyze Friedel oscillations in 1D fermionic systems, highlighting limitations and insights of each method.

## Key findings

- Perturbation theory shows a logarithmic divergence indicating power-law decay.
- Numerical results do not support a power-law decay from perturbation theory.
- Density functional theory results suggest the decay exponent remains close to the noninteracting case.

## Abstract

We study the asymptotic decay of the Friedel density oscillations induced by an open boundary in a one-dimensional chain of lattice fermions with a short-range two-particle interaction. From Tomonaga-Luttinger liquid theory it is known that the decay follows a power law, with an interaction dependent exponent, which, for repulsive interactions, is larger than the noninteracting value $-1$. We first investigate if this behavior can be captured by many-body perturbation theory for either the Green function or the self-energy in lowest order in the two-particle interaction. The analytic results of the former show a logarithmic divergence indicative of the power law. One might hope that the resummation of higher order terms inherent to the Dyson equation then leads to a power law in the perturbation theory for the self-energy. However, the numerical results do not support this. Next we use density functional theory within the local-density approximation and an exchange-correlation functional derived from the exact Bethe ansatz solution of the translational invariant model. While the numerical results are consistent with power-law scaling if systems of $10^4$ or more lattice sites are considered, the extracted exponent is very close to the noninteracting value even for sizeable interactions.