Excitonic Insulator and the Extended Falicov--Kimball Model Away from Half-Filling

TL;DR

This work investigates the extended Falicov–Kimball model away from half-filling using Hartree–Fock mean-field theory to understand uniform excitonic states and phase separation. It identifies a robust excitonic insulator at n=1 and demonstrates that a uniform excitonic metal with n≠1 can be the lowest-energy uniform state yet is typically unstable to phase separation, suggesting long-range Coulomb interactions could stabilize it. The phase diagrams reveal broad PS1 and PS2 regions, with EM often appearing as a competing uniform phase near n≈1 and expanding as the narrow-band hopping |t'| decreases. The findings indicate that excitonic correlations affect the doped regime significantly and may influence transport via percolation in phase-separated states, with potential relevance to materials such as 1T-TiSe₂ and Ta₂NiSe₅.

Abstract

We consider an extended spinless Falicov--Kimball model at an arbitrary doping level, focusing on the range of parameter values where a uniform excitonic insulator is stabilised at half-filling. We compare the properties of possible uniform phases and construct the Hartree--Fock phase diagrams, which include sizeable phase separation regions. It is seen that the excitonic insulator can appear as a component phase in a mixed-phase state in a broad interval of doping levels. In addition, in a certain range of parameter values the excitonic metal (doped excitonic insulator) is identified as the lowest-energy uniform phase. We suggest that this phase, which is unstable with respect to phase separation, may be stabilised when the phase separation is suppressed by the long-range Coulomb interaction. Overall, we find that excitonic correlations can affect the behaviour of the system relatively far away from half-filling.

Figures (7)

• Figure 1: Excitonic metal solution for $U=2$, $E_d=0.4$, $t^\prime=-0.15$(a) and for $U=0.5$, $E_d=0.4$, $t^\prime=-0.015$(b). Dashed and dotted lines show the dependence of $n_d$ and $\Delta$, respectively, on the carrier density $n$. Solid line (right scale) corresponds to the chemical potential $\mu$.
• Figure 2: Chemical potential near half filling in the excitonic metal phase for $U=0.5$, $E_d=0.4$, and $t^\prime=-0.015$. Solid, dashed, and dashed-dotted lines corresponds to $T=10^{-4}$, $T=10^{-3}$, and $T=3 \cdot 10^{-3}$ in the Fermi distributions in Eqs. (\ref{['eq:delta']}--\ref{['eq:ntot']}). The diamonds correspond to respective chemical potential values crossing out of the hybridisation gap, whose width is about $0.02$. The outer pair of diamonds refer to $T=3 \cdot 10^{-3}$, and the middle one -- to $T=10^{-3}$.
• Figure 3: Single-band and semimetal solutions for $U=2$, $E_d=0.4$, $t^\prime=-0.15$(a) and for $U=0.5$, $E_d=0.4$, $t^\prime=-0.015$(b). At a given value of carrier density $n$, dashed and dotted lines show the value of $n_d$ (left scale) for the lowest-energy single-band and semimetal solutions, respectively. Solid and dashed dotted lines represent the corresponding values of chemical potential $\mu$ (right scale).
• Figure 4: Energies of single-band (solid line) and semimetal (dashed) solutions for $U=2$, $E_d=0.4$, $t^\prime=-0.15$(a) and for $U=0.5$, $E_d=0.4$, $t^\prime=-0.015$(b). Insets show the energy differences between these solutions and the excitonic metal, the latter always corresponding to the lowest energy.
• Figure 5: Typical energy dependence of the quasiparticle density of states (DOS) for an excitonic metal solution (solid lines). Dashed lines show the contributions of broad-band electrons to the net DOS, whereas the dotted lines correspond to the DOS in the absence of hybridisation (obtained by formally setting $\Delta=0$; bold dotted line represents the localised band). The data correspond to $n=1.2$, $U=2$, $E_d=0.4$. We present results obtained for $t^\prime \rightarrow 0$, as corrections due to a finite value of $t^\prime$ are not significant for our purposes here.