Spin and charge density waves in the Lieb lattice

TL;DR

This paper investigates the mean-field phase diagram of the 2D Hubbard model on the Lieb lattice, demonstrating that allowing magnetization and charge density modulation aligns the mean-field results with Lieb's theorem across various interaction strengths.

Contribution

It introduces a more complex mean-field approach for bipartite lattices, ensuring the magnetization difference matches Lieb's theorem predictions at all interaction strengths.

Findings

Magnetization difference saturates at 1/2 for all U values.

Lieb's relation is approximately verified for large U in a specific filling range.

Magnetization correlates with filling of quasi-flat bands in the energy dispersion.

Abstract

We study the mean-field phase diagram of the two-dimensional (2D) Hubbard model in the Lieb lattice allowing for spin and charge density waves. Previous studies of this diagram have shown that the mean-field magnetization surprisingly deviates from the value predicted by Lieb's theorem \cite{Lieb1989} as the on-site repulsive Coulomb interaction ($U$) becomes smaller \cite{Gouveia2015}. Here, we show that in order for Lieb's theorem to be satisfied, a more complex mean-field approach should be followed in the case of bipartite lattices or other lattices whose unit cells contain more than two types of atoms. In the case of the Lieb lattice, we show that, by allowing the system to modulate the magnetization and charge density between sublattices, the difference in the absolute values of the magnetization of the sublattices, $m_{\text{Lieb}}$, at half-filling, saturates at the exact value $1/2$ for any value of $U$, as predicted by Lieb. Additionally, Lieb's relation, $m_{\text{Lieb}}=1/2$, is verified approximately for large $U$, in the $n \in [2/3,4/3]$ range. This range includes not only the ferromagnetic region of the phase diagram of the Lieb lattice (see Ref.~\onlinecite{Gouveia2015}), but also the adjacent spiral regions. In fact, in this lattice, below or at half-filling, $m_{\text{Lieb}}$ is simply the filling of the quasi-flat bands in the mean-field energy dispersion both for large and small $U$.