Extended Falicov-Kimball model: Exact solution for finite temperatures

The extended Falicov-Kimball model is analyzed exactly for finite temperatures in the limit of large dimensions. The onsite, as well as the intersite, density-density interactions represented by the coupling constants $U$ and $V$, respectively, 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 zero temperature ($T=0$), the system is ordered to form the checkerboard pattern and the DOS has the gap $\mathrm{\ensuremath{\Delta}}({\ensuremath{\varepsilon}}_F)>0$ at the Fermi level, if only $U\ensuremath{\ne}0$ or $V\ensuremath{\ne}0$. With an increase of $T$, the DOS evolves in various ways that depend both on $U$ and $V$. 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 ${T}_{\text{MI}}$ they join with each other at ${\ensuremath{\varepsilon}}_F$. Since above ${T}_{\text{MI}}$ the DOS is positive at ${\ensuremath{\varepsilon}}_F$, we interpret ${T}_{\text{MI}}$ as the transformation temperature from insulator to metal. It appears that ${T}_{\text{MI}}$ approaches the order-disorder phase transition temperature ${T}_{\text{OD}}$ for $|U|=2$ and $0<U\ensuremath{\lesssim}2V$, but otherwise ${T}_{\text{MI}}$ is substantially lower than ${T}_{\text{OD}}$. Moreover, we show that if $V\ensuremath{\lesssim}0.54$, then ${T}_{\text{MI}}=0$ at two quasi-quantum-critical points ${U}_cr^{\ifmmode\pm\else\textpm\fi{}}$ (one positive and the other negative), whereas for $V\ensuremath{\gtrsim}0.54$ there is only one negative ${U}_cr^{\ensuremath{-}}$. Having calculated the temperature-dependent DOS, we study thermodynamic properties of the system starting from its free energy $F$ 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 coexist (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.