Charge-Order on the Triangular Lattice: A Mean-Field Study for the Lattice $S=1/2$ Fermionic Gas

TL;DR

This study investigates charge ordering phenomena in a fermionic lattice gas model on a triangular lattice, revealing various phases and phase separation behaviors influenced by interactions and temperature.

Contribution

It extends the extended Hubbard model to the atomic limit on a triangular lattice, analyzing phase diagrams and charge orderings using a variational mean-field approach.

Findings

Multiple charge-ordered phases identified at zero and finite temperatures.

Phase separation occurs at specific interaction parameters and fillings.

Distinct behaviors compared to hypercubic lattice models.

Abstract

The adsorbed atoms exhibit tendency to occupy a triangular lattice formed by periodic potential of the underlying crystal surface. Such a lattice is formed by, e.g., a single layer of graphane or the graphite surfaces as well as (111) surface of face-cubic center crystals. In the present work, an extension of the lattice gas model to $S=1/2$ fermionic particles on the two-dimensional triangular (hexagonal) lattice is analyzed. In such a model, each lattice site can be occupied not by only one particle, but by two particles, which interact with each other by onsite $U$ and intersite $W_{1}$ and $W_{2}$ (nearest and next-nearest-neighbor, respectively) density-density interaction. The investigated hamiltonian has a form of the extended Hubbard model in the atomic limit (i.e., the zero-bandwidth limit). In the analysis of the phase diagrams and thermodynamic properties of this model with repulsive $W_{1}>0$, the variational approach is used, which treats the onsite interaction term exactly and the intersite interactions within the mean-field approximation. The ground state ($T=0$) diagram for $W_{2}\leq0$ as well as finite temperature ($T>0$) phase diagrams for $W_{2}=0$ are presented. Two different types of charge order within $\sqrt{3} \times \sqrt{3}$ unit cell can occur. At $T=0$, for $W_{2}=0$ phase separated states are degenerated with homogeneous phases (but $T>0$ removes this degeneration), whereas attractive $W_2<0$ stabilizes phase separation at incommensurate fillings. For $U/W_{1}<0$ and $U/W_{1}>1/2$ only the phase with two different concentrations occurs (together with two different phase separated states occurring), whereas for small repulsive $0<U/W_{1}<1/2$ the other ordered phase also appears (with tree different concentrations in sublattices). The qualitative differences with the model considered on hypercubic lattices are also discussed.