Gapless regime in the charge density wave phase of the finite dimensional Falicov-Kimball model

The ground-state density of states of the half-filled Falicov-Kimball model contains a charge-density-wave gap. At finite temperature, this gap is not immediately closed, but is rather filled in by subgap states. For a specific combination of parameters, this leads to a stable phase where the system is in an ordered charge-density-wave phase, but there is high density of states at the Fermi level. We show that this property can be, in finite dimensions, traced to a crossing of sharp states resulting from the single particle excitations of the localized subsystem. The analysis of the inverse participation ratio points to a strong localization in the discussed regime. However, the pronounced subgap density of states can still lead to a notable increase of charge transport through a finite size system. We show this by focusing on the transmission in heterostructures where a Falicov-Kimball system is sandwiched between two metallic leads.

Figure 1

Simplified phase diagram of the spinless FKM on a square 2D lattice with ordered CDW phase (OP) and disordered phases in weak (DPw) and strong (DPs) interaction regimes. The insets illustrate typical $d$-electron DOS in the respective phases.

Figure 2

Schematic picture of the heterostructure. The black part represents the two-dimensional FKM system with nearest-neighbor hopping $t$. The red parts are noninteracting leads with hopping $t_{L,R}$ and the hybridization interaction with hopping ${\gamma }_{L,R}$ is indicated in blue. The leads are characterized by elliptic surface DOSs. The voltage drop $V$ is introduced by mutual shift of ${\epsilon }_{L,R}$ where the condition ${\mu }_{L,R}={\epsilon }_{L,R}$ is used to keep the bands half filled at any applied voltage.

Figure 3

Details of the subgap density of states $\left(\left|\varepsilon \right|<U/2\right)$ for three values of of $U$ and various values of temperature. All data have been calculated on a two-dimensional square lattice with size $L=24\times 24$ and periodic boundary conditions. The sharp states, calculated for particular configurations $w$ in the sampling process, have been broadened by a Gaussian broadening (see text) with $b=0.002t$. The arrows signal the positions of the bound states belonging to single $f$-electron excitations discussed in the text. Their color coding is the same as in Fig. 4. The inset in panel (b) is the detail of the DOS close to the Fermi level calculated for $L=24\times 24,30\times 30$ and $b=0.002t$ at $T=0.095t$.

Figure 4

(a)–(c) Position of the subgap states for disrupted checkerboard ordering of $f$ electrons in different dimensions. Three single $f$-electron excitations are considered. Namely, adding one additional $f$ electron to an unoccupied lattice point (add), removing one $f$ electron (rem) and displacing one electron from its position to a nearest-neighboring unoccupied point (dis) as illustrated for $D=2$ case in the top three panels. The results have been obtained using the analytical and semianalytical formulas from Appendix pp2 and represent thermodynamic limit solutions $\left(L\to \infty \right)$. (d)–(e) Difference between the minimal energy of the disrupted and perfect checkerboard ordering of $f$ electrons. The energy differences for add and rem cases are identical. The dotted line represents the energy of the system where the half-filling $N_f+N_d=L$ is enforced. Solid red line breaks this condition below $U_c$ by one particle $N_f+N_d=L\pm 1$.

Figure 5

(a) Density of states at the Fermi level as a function of $U$ for a square lattice $L=20\times 20$. The inset shows details of the peak at $U_c=2.5t$ where the bound states from Fig. 4 cross each other. (b) Finite size scaling of the DOS at the Fermi level for various temperatures calculated with broadening $b=0.002t$. (c) Dependence of the DOS at the Fermi level on the used artificial broadening.

Figure 6

(a) Density of states at the Fermi level as a function of temperature for $D=2$ at $U=2.5t$ calculated with broadening parameter $b=0.002t$ (solid lines) for various lattice sizes and with $b=0.004t$ for $L=24\times 24$ (dashed line). The vertical black dotted line signals the critical temperature of the order-disorder phase transition. (b) Dependence of the statistical weight of the configurations with perfect CDW ordering (green dashed line calculated for $L=20\times 20)$ and configurations with a single $f$ particle added or removed from CDW.

Figure 7

(a) System size scaling of the averaged inverse participation ratio for $U=2.5t$ at $T=0.095t$ (red circles) and $T=1t$ (blue circles) and system size scaling of the generalized averaged inverse participation ratio for the same parameters and $\gamma =2t$ (black and green squares). (b) Dependence of the inverse participation on the artificial broadening of the delta functions. The IPR for $T=0.5t$ is multiplied by factor of 10.

Figure 8

Averaged DOS of the coupled system calculated for $U=2.5t,L=20\times 20$ and three values of $\gamma$. The position of the maximum $T\sim 0.095t$ coincides with the position of the maxima in Fig. 6. The inset represents the finite size scaling of DOS at $T=0.095t$ (maximum) and $1t$ (high temperature).

Figure 9

gIPR as a function of $\gamma$ for $T=0.095t$ and $T=1t$ and central system size $L=20\times 20$. The gIPR for $T=1t$ is multiplied by factor of 10 for the sake of clarity.

Figure 10

(a) Dependence of the equilibrium transmission function on temperature calculated at Fermi level for a heterostructure with $U=2.5t$ and the system size $L=20\times 20$ coupled to two semi-infinite noninteracting leads with semielliptical surface density of states and band half-width $B=20t$. The position of the humps signalled by an arrow coincides with the maxima in Fig. 6. The inset is an example of the full transmission function for $\gamma =1t$ and $T=0.095t$. The subgap region is multiplied by 100 for the sake of visibility. (b) Nonequilibrium transmission function for $\varepsilon =0$ as a function of voltage drop rescaled by $U=2.5t$. (c) Dependence of the equilibrium transmission function at the Fermi level multiplied by the linear size of the system on the total lattice size. The values for $T=0.095t$ were scaled by factor 10.