Inhomogeneous Metallic Phase in a Disordered Mott Insulator in Two Dimensions

We show that, with increasing randomness, the spectral gap in a 2D Mott-Hubbard insulator is destroyed first at a disorder $V_{c1}$, while antiferromagnetism persists up to a higher $V_{c2}$. Most unexpectedly, between $V_{c1}$ and $V_{c2}$ the system is metallic and is sandwiched between the Mott insulator below $V_{c1}$ and the Anderson insulator above $V_{c2}$. The metal is formed when the spectral gap gets destroyed locally in regions where the disorder potential is high enough to overcome the interelectron repulsion. This generates puddles with enhanced charge fluctuations that percolate with increasing disorder, resulting in a spatially inhomogeneous metallic phase.

Figure 1

(a) Left panels show plots of the local staggered magnetization $m^{†}(i)\equiv \langle S_z(i)\rangle (-1{)}^{i_x+i_y}$ for the disordered Hubbard model at half filling with $U=4t$ on a $28\times 28$ lattice for disorder strengths $V=2t$, $V=3t$, and $V=5t$ for one realization of disorder. The darker regions have AFM order; the defective sites with reduced AFM order are shown in white. There are also sites with negative $m^{†}(i)$ that show up as red sites on a color plot. (b) Probability distribution $P(m^{†})$ for different values of $V$. With increasing $V$, $P(m^{†})$ gets broader and develops weight near 0 indicating the growth of paramagnetic (PM) regions. (c) Probability distribution $P(n)$ of the site occupancy $\langle n_i\rangle$ showing a peak near $\langle n_i\rangle \approx 1$ for $V=1$, which gets broader with increasing $V$ and develops weight for doubly occupied and unoccupied sites.

Figure 2

Correlation between the local disorder potential $V_i$, the local staggered magnetization $m^{†}(i)$, and the local density $n(i)$ for $V=3t$. The distribution $P(m)$ of $m^{†}(i)$ is shown in (b). We define those sites with $m^{†}(i)\ge 0.3$ to be AFM sites (region shown filled); and sites with $m^{†}(i)\le 0.1$ to be PM (shown hatched). The AFM sites correlate with the weakly disordered (WD) (filled) region in the disorder distribution in (a) and the density distribution $P(n)$ centered around unity (filled) in (c), whereas the PM sites correlate with the strongly disordered (SD) (hatched) regions in (a) and a bimodal $P(n)$ with weight near zero and double occupancy (hatched).

Figure 3

(a) Density of states averaged over ten realizations at half filling and $U=4t$. For $V=t$ there is a gap in the spectrum at the chemical potential $E=0$ that closes for $V\ge 2t$. (b) Single particle energy gap as a function of $V$ showing the collapse of the gap at $V_{c1}\sim 2t$ (scale on left). The AFM staggered order parameter $m^{†}$ (scale on right) vanishes beyond $V_{c2}\sim 4t>V_{c1}$. Note that at $V\sim 2t$ the system is still strongly AFM even though the spectral gap has vanished.

Figure 4

Grayscale plots of $|{\psi }_n(\mathbf{r}){|}^2$ corresponding to ${ϵ}_n$ at the Fermi energy. The eigenstate is localized for low disorder $V=2t$ and high $V=5t$ but surprisingly is more extended at intermediate disorder $V=3t$.

Figure 5

(a) Inverse participation ratio $\mathrm{I}\mathrm{P}\mathrm{R}\propto {\xi }_{\mathrm{l}\mathrm{o}\mathrm{c}}^{-2}$, where ${\mathrm{\xi }}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is the localization length for the wave function at the Fermi energy, as a function of disorder $V$. In both regions  I and III IPR increases with disorder or ${\xi }_{\mathrm{l}\mathrm{o}\mathrm{c}}$ decreases with disorder, as expected. In the intermediate region II, the IPR shows a very distinctive nonmonotonic behavior around $VU/2$, which indicates that the wave function, in fact, gets less localized with increasing disorder. The states in region II are extended as seen from (b). (b) Scaling behavior of the inverse participation ratio IPR vs $1/L$ for $L\times L$ systems. In the noninteracting case (open squares) the IPR extrapolates to a finite value for $V=3t$, whereas for the interacting case the IPR extrapolates to zero for $V=3t$ (filled triangles) and $V=3.4t$ (filled circles) indicating a divergence of the localization length in the metallic regime. Further increase to $V=4t$ (open hexagons) once again gives a finite localization length.