Interacting spinless fermions in a diamond chain

TL;DR

This paper investigates the effects of Coulomb interactions on spinless fermions in a flux-threaded AB$_2$ chain, revealing localized states, phase separation, and metal-insulator transitions through exact and mean-field analyses.

Contribution

It provides the first detailed analysis of Coulomb interactions in the AB$_2$ chain, combining exact diagonalization and mean-field methods to explore localized states and phase transitions.

Findings

Localized states due to geometry and interactions identified.

Phase separation occurs at specific fillings leading to metal-insulator transition.

Spectrum near/above 2/9 filling resembles a mix of Luttinger liquids.

Abstract

We study spinless fermions in a flux threaded AB$_2$ chain taking into account nearest-neighbor Coulomb interactions. The exact diagonalization of the spinless AB$_2$ chain is presented in the limiting cases of infinite or zero nearest-neighbor Coulomb repulsion for any filling. Without interactions, the AB$_2$ chain has a flat band even in the presence of magnetic flux. We show that the respective localized states can be written in the most compact form as standing waves in one or two consecutive plaquettes. We show that this result is easily generalized to other frustrated lattices such as the Lieb lattice. A restricted Hartree-Fock study of the $V/t$ versus filling phase diagram of the AB$_2$ chain has also been carried out. The validity of the mean-field approach is discussed taking into account the exact results in the case of infinite repulsion. The ground-state energy as a function of filling and interaction $V$ is determined using the mean-field approach and exactly for infinite or zero $V$. In the strong-coupling limit, two kinds of localized states occur: one-particle localized states due to geometry and two-particle localized states due to interaction and geometry. These localized fermions create open boundary regions for itinerant carriers. At filling $\rho=2/9$ and in order to avoid the existence of itinerant fermions with positive kinetic energy, phase separation occurs between a high-density phase ($\rho=2/3$) and a low-density phase ($\rho=2/9$) leading to a metal-insulator transition. The ground-state energy reflects such phase separation by becoming linear on filling above 2/9. We argue that for filling near or larger than 2/9, the spectrum of the t-V AB$_2$ chain can be viewed as a mix of the spectra of Luttinger liquids (LL) with different fillings, boundary conditions, and LL velocities.