Mott and band-insulator transitions in the binary-alloy Hubbard model: Exact diagonalization and determinant quantum Monte Carlo simulations

We use determinant quantum Monte Carlo simulations and exact diagonalization to explore insulating behavior in the Hubbard model with a bimodal distribution of randomly positioned local site energies. From the temperature dependence of the compressibility and conductivity, we show that gapped, incompressible Mott insulating phases exist away from half-filling when the variance of the local site energies is sufficiently large. The compressible regions around this Mott phase are metallic only if the density of sites with the corresponding energy exceeds the percolation threshold, but are Anderson insulators otherwise.

Figure 1

(Color online) Density as a function of chemical potential for $W=0.8$, $U=2$, $\Delta =4$, and $x=0.25$. Solid line: ground-state exact diagonalization of an eight-site cluster. Dashed line: DQMC on $6\times 6$ clusters with temperature $T=1∕16$. In the former case, results are averaged over all configurations with two sites with ${ϵ}_A=-2$ and six sites with ${ϵ}_B=+2$, and over different choices of the boundary phases. In the latter case, results are given for a single realization. The two methods give very similar results, with the DQMC somewhat rounded by finite temperature.

Figure 2

(Color online) Upper panel: participation ratio $P$ as a function of the eigenenergy $E$ for the alloy Hubbard Hamiltonian with $U=0$, $\Delta =4$, and $t=1$. Results are shown for lattice sizes varying from $N=16\times 16$ to $N=64\times 64$. $P\left(E\right)$ is small for the lower alloy band which has only $x=0.25<x_c$ of the sites, but $P\left(E\right)$ in the upper alloy band has a significant fraction of $N$. The inset shows that $P\left(E\right)∕N$ scales to a nonzero value as $N\to \infty$ for the upper band and zero for the lower band. Lower panel: the same for $x=0.45$. Here both alloy subbands have a density below the percolation threshold and both participation ratios scale to zero.

Figure 3

(Color online) The conductivity ${\sigma }_{\mathrm{dc}}$ is shown as a function of chemical potential for several temperatures. Parameter values are as in Fig. 1. The upper alloy band, whose sites have a density larger than the percolation threshold, has a large ${\sigma }_{\mathrm{dc}}$, which also increases as $T$ is reduced (inset). The insulating phases have ${\sigma }_{\mathrm{dc}}\approx 0$. The conductivity near the lower alloy subband is much smaller and much more weakly temperature dependent than the upper.

Figure 4

(Color online) The conductivity ${\sigma }_{dc}$ is shown as a function of temperature for metallic regions of different disorder with interaction energy $U=2$. The conductivity for the nondisordered cases is twice that of the disordered case as the insulating phases as incommensurate filling decrease the size of the metallic regions.

Figure 5

(Color online) The length of the insulating plateaus, $\Delta \mu$, is shown as a function of hopping $t$ for $U=2$, $\Delta =4$, and $x=0.25$ using exact diagonalization of $N=8$ site clusters using boundary condition averaging. The alloy Mott-insulator plateaus at $\rho =x$ and $\rho =1+x$ are more robust than the alloy band-insulator plateau at $\rho =2x$.

Figure 6

Phase diagram of $U∕t$ vs $\Delta ∕t$. I: alloy band insulator at $\rho =0.50$ and no alloy Mott insulator. II: metallic phase. III: alloy band insulator at $\rho =0.50$ and alloy Mott insulators at $\rho =0.25$ and 1.25. IV: alloy Mott insulator at $\rho =0.25$ and 1.25 and no alloy band Insulator. V: $U>\Delta$, alloy band insulator at $\rho =0.25$ and 1.25 and alloy Mott insulator at $\rho =1.00$.