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 $\mathrm{\Delta }\left({\varepsilon }_F\right)>0$ at the Fermi level, if only $U\ne 0$ or $V\ne 0$. 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 ${\varepsilon }_F$. Since above $T_{\text{MI}}$ the DOS is positive at ${\varepsilon }_F$, 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 $\left|U\right|=2$ and $0<U\lesssim 2V$, but otherwise $T_{\text{MI}}$ is substantially lower than $T_{\text{OD}}$. Moreover, we show that if $V\lesssim 0.54$, then $T_{\text{MI}}=0$ 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 $F$ 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.

Figure 1

The itinerant electron densities of states ${\rho }^{+}$ (solid line) and ${\rho }^{-}$ (dotted line) in each sublattice (i) left column: for $T=0.15$ and $V=0.1$ in phases with $d_1>0$: ${\mathrm{COI}}_1$ ($U=-2.0,d=0.949,d_1=0.801$), ${\mathrm{COM}}_1$ ($U=-1.5,d=0.698,d_1=-0.630$), and ${\mathrm{COI}}_2$ ($U=-1.0,d=0.724,d_1=0.502$), from the top, respectively; (ii) right column: for $T=0.05$ and $V=0.1$ in phases with $d_1<0$: ${\mathrm{AFI}}_1$ ($U=1.6,d=0.994,d_1=-0.701$), AFM ($U=1.3,d=0.987,d_1=-0.630$), and ${\mathrm{AFI}}_2$ ($U=1.0,d=0.947,d_1=-0.524$), from the top, respectively. The Fermi level is located at $\varepsilon ={\varepsilon }_F=0$. The gray shadows indicate schematically the principal gap between the main (lower and upper) bands, where the subbands appear at $T>0$. In all insulators ${\rho }^{\pm }\left({\varepsilon }_F\right)=0$ and $\mathrm{\Delta }\left({\varepsilon }_F\right)>0$.

Figure 2

The itinerant electron densities of states ${\rho }^{+}$ (solid line) and ${\rho }^{-}$ (dotted line) in each sublattice for $U=0.16$ and $V=0.1$ for phases with $d_1>0$: ${\mathrm{COI}}_3$ ($T=0.040,d=0.799,d_1=0.051$; left panel) and ${\mathrm{COM}}_2$ ($T=0.047,d=0.234,d_1=0.013$; right panel). The Fermi level is located at $\varepsilon ={\varepsilon }_F=0$. ${\rho }^{\pm }\left({\varepsilon }_F\right)=0$ and $\mathrm{\Delta }\left({\varepsilon }_F\right)>0$ in the ${\mathrm{COI}}_3$.

Figure 3

(a) Energy gap $\mathrm{\Delta }\left({\varepsilon }_F\right)$ at the Fermi level as a function of $U$ at $T=0$ [i.e., $\mathrm{\Delta }{\left({\varepsilon }_F\right)}_{T=0}$, dashed lines] and in the limit $T\to 0^{+}$ [i.e., $\mathrm{\Delta }{\left({\varepsilon }_F\right)}_{T\to 0^{+}}$, solid lines] for $V=0.1,V=0.4$, and $V=0.8$ (as labeled). (b) Difference $D_{\mathrm{\Delta }}=\mathrm{\Delta }{\left({\varepsilon }_F\right)}_{T\to 0^{+}}-\mathrm{\Delta }{\left({\varepsilon }_F\right)}_{T=0}$ for the same values of $V$ is shown. The vertical dotted and dashed-dotted lines correspond to location (at $T=0$) of the quasicritical points ($U_{cr}^{\pm }$) and of the discontinuous transition for $U=2V$, respectively (for each value of $V$ from the figure, correspondingly).

Figure 4

Temperature $T_{\text{OD}}$ of the order-disorder transition for different $V$ (from 0.0 to 1.6 with a step of 0.1). Only second-order lines are determined. The dotted line connects points at the boundaries, where parameter $d_1$ change its sign in the ordered phase below the boundary ($d_1>0$ on the left, $d_1<0$ on the right). The dashed line denotes $k_{B}T=1/\left(2U\right)$ dependence for large $U$. Note that for large $V$ (larger that $V\approx 0.7$) the order-disorder transition is first-order one for some range of $U$ (not shown in this figure).

Figure 5

The finite-temperature phase diagram for $V=0.1$. Solid and dashed lines denote second-order (continuous) and first-order (discontinuous) transitions, respectively. Each region is labeled by name of a phase, which is stable in the particular region (details in text of Sec. 3b). To determine the diagram free energies of all solutions found were compared. Dashed-dotted lines denote the metal-insulator transformations, which are continuous but they are not phase transitions in the usual sense (details in text of Sec. 3b4).

Figure 6

The diagram for $V=0.5$. The figure presents details of the phase diagram in the neighborhood of first-order transition associated to a change of sign of $d_1$ parameter. The vertical dashed-dotted line indicates the location of the bicritical point. The features of the diagram do not change in a range of $0<V<0.54$. Other denotations as in Fig. 5.

Figure 7

The diagram for $V=0.6$ and $U>0$ (cf. Fig. 10). The region of the ${\mathrm{AFI}}_2$ disappeared. The region of the ${\mathrm{COM}}_2$ is very thin. The vertical dashed-dotted line indicates the location of the bicritical point. Other denotations as in Fig. 5.

Figure 8

The diagram for $V=1.0$. The order-disorder ${\mathrm{COI}}_3$-NOM transition (omitting the metallic phase) is discontinuous. The dashed line denotes the second-order boundary from Fig. 4, which is not a boundary between the phases with the lowest free energies. Other denotations as in Fig. 5.

Figure 9

The diagram for $V=1.6$. The region of the AFM disappeared. ${\mathrm{COI}}_3-{\mathrm{NOI}}_{\mathrm{B}}$ transition is present on the diagram. The dashed line denotes the second-order boundary from Fig. 4, which is not a boundary between the phases with the lowest free energies. Other denotations as in Fig. 5.

Figure 10

The diagram for $V=0.6$ and $U<0$ (cf. Fig. 7). All boundaries are continuous. Other denotations as in Fig. 5.

Figure 11

Dependencies of parameters $d$ (solid line), $|d_1|$ (dashed line), ${\mathrm{\Delta }}_Q$ (dotted line), and $m_Q$ (dashed-dotted line) for $V=0$ (the FKM) and for $U=1.0$ as a function of temperature $k_{B}T$ (left panel) and $k_{B}T=0.06$ as a function of onsite repulsion $U$ (right panel). Vertical dashed-dotted lines indicate the transitions. The locations of the ${\mathrm{AFI}}_2$-AFM and AFM-${\mathrm{AFI}}_1$ boundaries are determined by vanishing of $\rho \left({\varepsilon }_F\right)$, which dependence is not shown in the figure.

Figure 12

Top panels: temperature dependencies of parameter $d$ (solid line), parameter $|d_1|$ (dashed line), DOS at the Fermi level $\rho \left({\varepsilon }_F\right)$ (dashed-dotted line), and energy gap at the Fermi level $\mathrm{\Delta }\left({\varepsilon }_F\right)$ (dashed-dotted-dotted line). Lower panels: temperature dependencies of free energy $F$ per site and entropy $S$ per site (as indicated). Left panels are obtained for $U=1.0$ (${\mathrm{AFI}}_2$-AFM-NOM sequence of continuous transitions) whereas right panels are obtained for $U=2.5$ (continuous ${\mathrm{AFI}}_1-{\mathrm{NOI}}_{\mathrm{B}}$ transition); $V=0.1$ (cf. Fig. 5). Vertical dashed-dotted lines indicate the transitions. Note that the ${\mathrm{AFI}}_2$-AFM transition is not a second-order transition in the usual sense.

Figure 13

Top panels: $U$ dependencies of parameter $d$ (solid line), parameter $|d_1|$ (dashed line), DOS at the Fermi level $\rho \left({\varepsilon }_F\right)$ (dashed-dotted line), energy gap at the Fermi level $\mathrm{\Delta }\left({\varepsilon }_F\right)$ (dashed-dotted-dotted line). Bottom panels: $U$ dependencies of total free energy $F$ per site (solid line). The dotted lines indicate the dependencies of mentioned quantities in the metastable phases. Left panels are obtained for $V=0.6$ and $k_{B}T=0.085$ (discontinuous ${\mathrm{COI}}_3$-AFM transition, cf. Fig. 7), whereas right panels are obtained for $V=1.0$ and $k_{B}T=0.4$ (discontinuous ${\mathrm{COI}}_3$-NOM transition, cf. Fig. 8). Vertical dashed-dotted lines indicate the transitions. The gray shadow indicates the coexistence regions in the neighborhood of the discontinuous phase transitions.