# Extended Falicov-Kimball model: Exact solution for finite temperatures

## TL;DR

This paper provides an exact finite-temperature solution for the extended Falicov-Kimball model in large dimensions, revealing phase transitions, phase diagrams, and the effects of onsite and intersite interactions on electronic states.

## Contribution

It offers the first exact finite-temperature analysis of the extended Falicov-Kimball model including $U$ and $V$, detailing phase diagrams and insulator-metal transitions.

## Key findings

- Identification of temperature-dependent density of states (DOS)
- Discovery of insulator-metal transition temperature $T_{MI}$
- Detection of eight different ordered phases and their transitions

## Abstract

The extended Falicov-Kimball model is analyzed exactly for finite temperatures ($T\geq0$) in the limit of large dimensions. Onsite and intersite density-density interactions $U$ and $V$ are included in the model. Using the dynamical mean field theory formalism on the Bethe lattice we find rigorously the temperature dependent density of states (DOS) at half-filling. At $T=0$ the system is ordered to form the checkerboard pattern and the DOS has the gap $\Delta(\varepsilon_F) > 0$ at the Fermi level, if only $U\neq 0$ or $V\neq 0$. If $U <0$ or $U > 2V$, two additional subbands develop inside the principal energy gap. They become wider with increasing $T$ and at a certain $U$- and $V$-dependent temperature $T_{MI}$ they join with each other at $\varepsilon_F$. Since above $T_{MI}$ the DOS is positive at $\varepsilon_F$, we interpret $T_{MI}$ as the transformation temperature from insulator to metal. Moreover, we show that if $V\lesssim 0.54$ then $T_{MI}=0$ at two quasi-quantum critical points $U_{cr}^{\pm}$ (one positive and the other negative), whereas for $V\gtrsim 0.54$ there is only one negative $U^-_{cr}$. Having calculated the temperature dependent DOS we study thermodynamic properties of the system starting from its free energy and then we construct the phase diagrams in the variables $T$ and $U$ for a few values of $V$. Our calculations give that inclusion of the intersite coupling $V$ causes the finite temperature phase diagrams to become asymmetric with respect to a change of sign of $U$. On these phase diagrams we detected stability regions of eight different kinds of ordered phases, where both charge-order and antiferromagnetism coexists (five of them are insulating and three are conducting) and three different nonordered phases (two of them are insulating and one is conducting). Moreover, both continuous and discontinuous transitions between various phases were found.