Two-dimensional disordered Mott metal-insulator transition

We studied several aspects of the Mott metal-insulator transition in the disordered case. The model on which we based our analysis is the disordered Hubbard model, which is the simplest model capable of capturing the Mott metal-insulator transition. We investigated this model through statistical dynamical mean-field theory (statDMFT). This theory is a natural extension of dynamical mean-field theory (DMFT), which has been used with relative success in the past several years with the purpose of describing the Mott transition in the clean case. As is the case for the latter theory, statDMFT incorporates the electronic correlation effects only in their local manifestations. Disorder, on the other hand, is treated in such a way as to incorporate Anderson localization effects. With this technique, we analyzed the disordered two-dimensional Mott transition, using the quantum Monte Carlo algorithm to solve the associated single-impurity problems. We found spinodal lines at which the metal and insulator ceased to be metastable. We also studied spatial fluctuations of local quantities, such as self-energy and local Green's function, and showed the appearance of metallic regions within the insulator and vice versa. We carried out an analysis of finite-size effects and showed that, in agreement with the theorems of Imry and Ma [Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975).], the first-order transition is smeared in the thermodynamic limit. We analyzed transport properties by means of a mapping to a random classical resistor network and calculated both the average current and its distribution across the metal-insulator transition.

Figure 1

The clean noninteracting density of states of our model. Note that the van Hove singularities occur away from the band center.

Figure 2

$T-n$ phase diagram for the Mott transition and the occupation number as a function of chemical potential at $T=0$ in the Hubbard model.

Figure 3

Square resistor network in which each resistor couples two neighboring sites.

Figure 4

Resistors associated with a $3\times 3$ square network.

Figure 5

Hysteresis curves for different temperature values, below the critical point of the clean Mott transition. As the temperature increases, the hysteresis loops become smaller.

Figure 6

Hysteresis curves for different values of temperature. Notice how the disorder shifts the hysteresis curve to higher values of interaction energy, while the coexistence region shrinks.

Figure 7

Imaginary part of the Green's function at the first Matsubara frequency for each site of the square lattice with $T=0.024D$ in the neighborhood of the Mott transition when going from the metal to the insulator. See video mi.avi in the Supplemental Material [77].

Figure 8

Imaginary part of the Green's function at the first Matsubara frequency for each site of the lattice for $T=0.024D$ and a scan of values of $U$ in the neighborhood of the Mott transition going from the insulator to the metal. See video im.avi in the Supplemental Material [77].

Figure 9

For four different realizations of disorder, we show $\mathrm{Im}{G}_i\left({\omega }_1\right)$ via a color scale. The red color represents the lattice sites that have metallic behavior. Insulating behavior corresponds to blue regions. We used $W=0.52D, U=2.27D$, and $T=0.024D$.

Figure 10

Hysteresis loops for different lattice sizes, below the critical point of the Mott transition. As the temperature increases, the hysteresis loops become smaller.

Figure 11

Finite-size effects: Spatial pattern of $\mathrm{Im}{G}_i\left(i{\omega }_1\right)$ for different sizes of the square lattice. Metallic and insulating bubbles proliferate as the lattice increases at $U=2.27D$ in the upper branch of the hysteresis loop.

Figure 12

Imaginary part of self-energy for the first Matsubara frequency for $T=0.024D$.

Figure 13

Spatial mapping of the current in the vicinity of the Mott transition for $T=0.024D$.

Figure 14

Average current as a function of the Coulomb potential in the transition region of Mott for different temperature values.