Charge-order on the triangular lattice: Effects of next-nearest-neighbor attraction in finite temperatures

TL;DR

This study explores how next-nearest-neighbor attraction influences charge order and phase separation in the extended Hubbard model on a triangular lattice at finite temperatures, revealing complex phase diagrams with multiple ordered and phase-separated states.

Contribution

It introduces a variational approach to analyze the extended Hubbard model with next-nearest-neighbor attraction, mapping out finite-temperature phase diagrams and identifying stable charge-ordered and phase-separated states.

Findings

Attractive next-nearest-neighbor interactions stabilize phase-separated states.

Multiple charge-ordered phases (DCO and TCO) are identified with specific stability regions.

Critical values of interaction ratios determine coexistence of ordered phases.

Abstract

The extended Hubbard model in the atomic limit, which is equivalent to lattice $S=1/2$ fermionic gas, is considered on the triangular lattice. The model includes onsite Hubbard $U$ interaction and both nearest-neighbor ($W_{1}$) and next-nearest-neighbor ($W_{2}$) density-density intersite interactions. The variational approach treating the $U$ term exactly and the $W_l$ terms in the mean-field approximation is used to investigate thermodynamics of the model and to find its finite temperature ($T>0$) phase diagrams (as a function of particle concentration) for $W_{1}>0$ and $W_{2}<0$. Two different types of charge-order (i.e., DCO and TCO phases) within $\sqrt{3} \times \sqrt{3}$ unit cells as well as the nonordered (NO) phase occur on the diagram. Moreover, several kinds of phase-separated (PS) states (NO/DCO, DCO/DCO, DCO/TCO, and TCO/TCO) are found to be stable for fixed concentration. Attractive $W_{2}<0$ stabilizes PS states at $T=0$ and it extends the regions of their occurrence at $T>0$. The evolution of the diagrams with increasing of $|W_{2}|/W_{1}$ is investigated. It is found that some of the PS states are stable only at $T>0$. Two different critical values of $|W_{2}|/W_{1}$ are determined for the PS states, in which two ordered phases of the same type (i.e., two domains of the DCO or TCO phase) coexist.