Variational Monte Carlo study of Anderson localization in the Hubbard model

We have studied the effects of interactions on persistent currents in half-filled and quarter-filled Hubbard models with weak and intermediate strength disorder. Calculations are performed using a variational Gutzwiller ansatz that describes short-range correlations near the Mott transition. We apply an Aharonov-Bohm magnetic flux, which generates a persistent current that can be related to the Thouless conductance. The magnitude of the current depends on both the strength of the screened disorder potential and the strength of electron-electron correlations, and the Anderson localization length can be extracted from the scaling of the current with system size. At half-filling, the persistent current is reduced by strong correlations when the interaction strength is large. Surprisingly, we find that the disorder potential is strongly screened in the large interaction limit, so that the localization length grows with increasing interaction strength even as the magnitude of the current is suppressed. This supports earlier dynamical mean-field-theory predictions that the elastic scattering rate is suppressed near the Mott transition.

Figure 1

Schematic phase diagram for the Anderson-Hubbard model in three dimensions at half-filling. Thin lines indicate continuous transitions between metallic, Anderson insulating, and Mott insulating phases; the thick solid line indicates a region of the phase boundary where DMFT predicts a discontinuity in the density of states.

Figure 2

Variational energies of the PMGW and DFSGW states for a one-dimensional chain with $L=50$ and $W=4t$. Results are at half-filling and are for a single configuration of disorder. The energy per electron is shown. Inset: Energy difference $\Delta E=E_{\mathrm{DFSGW}}-E_{\mathrm{PMGW}}$ between the two variational states.

Figure 3

Distribution of the persistent currents for an ensemble of one-dimensional chains with $L=50$, $W=4t$, and $U=8t$. Here and throughout this work, the Aharanov-Bohm flux is $\alpha =0.25$. Results are shown for (a) paramagnetic Gutzwiller and (b) paramagnetic Hartree Fock approximations. Smooth curves are Gaussian fits to the data for comparison.

Figure 4

Persistent current as a function of $U$ for PMGW and DFSGW ground states for both a clean ($W=0$) and disordered ($W=8t$) two-dimensional plane. Results are shown at half-filling ($n=1$) for a single disorder configuration with linear dimension $L=8$. Results with $g$ set to 1 by hand include screening but not strong correlations; these are labeled PMHF (paramagnetic Hartree-Fock, obtained from PMGW) and DFS (disordered Fermi sea, obtained from DFSGW). The DFS curves have the same value of $\epsilon$ as the DFSGW curves. The inset shows the dependence of the variational parameters on $U$.

Figure 5

As in Fig. 4, but for a one-dimensional chain with $L=50$ and $W=4t$. The inset shows PMGW and DFSGW results for a short ($L=8$) chain.

Figure 6

As above, but at quarter-filling ($n=0.5$). Results are for clean ($W=0$) and disordered ($W=D$) systems for (a) a two-dimensional lattice with $L=8$ and $D=8t$ and (b) a one-dimensional lattice with $L=50$ and $D=4t$.

Figure 7

Scaling of $\ln J_{\mathrm{typ}}$ with system size in one dimension. Typical currents are shown for half-filling with (a) $U\lesssim W$ and (b) $U>W$. Data are for the PMGW approximation, solid lines are fits to the data from Eq. (10). Data are calculated for $10000/L$ disorder configurations, and error bars give the statistical uncertainty in the typical current. Where error bars are not shown, they are smaller than the symbol size. (c) The scaling function $\beta$ for PMGW at half-filling (left) and DFSGW at quarter-filling (right).

Figure 8

Localization length as a function of $U$ for (a) PMGW and DFSGW at half-filling and (b) DFSGW at quarter-filling. PMHF results are shown for comparison.