Dual-fermion approach to the Anderson-Hubbard model

We apply the recently developed dual-fermion algorithm for disordered interacting systems to the Anderson-Hubbard model. This algorithm is compared with dynamical cluster approximation calculations for a one-dimensional system to establish the quality of the approximation in comparison with an established cluster method. We continue with a three-dimensional (3D) system and look at the antiferromagnetic, Mott, and Anderson localization transitions. The dual-fermion approach leads to quantitative as well as qualitative improvement of the dynamical mean-field results, and it allows one to calculate the hysteresis in the double occupancy in 3D, taking into account nonlocal correlations.

Figure 1

Two second-order diagrams for the self-energy in the particle-particle channel that contains closed loops (red lines). The diagram on the right contains the “crossed” disorder vertex.

Figure 2

Decomposition of the full vertex $V$ into two purely disordered contributions and all the rest for the particle-hole (top) and particle-particle channel (bottom). Along the green lines inside the boxes spin and energy are conserved. The vertical (crossed) contribution for the particle-hole (particle) channel (second diagram in each case) is unphysical, but it is part of the vertex as defined in Appendix pp3.

Figure 3

Lower-order contributions to the purely disordered vertex functions ${\gamma }^{=}$ and ${\gamma }^p$.

Figure 4

Lower-order contributions to the purely disordered vertex functions ${\gamma }^{\left|\right|}$ and ${\gamma }^{\times }$.

Figure 5

Lower-order contributions to the full vertex function as defined in Appendix pp3. The two diagrams at the bottom would be unphysical without the Coulomb interaction lines. With the Coulomb lines the diagrams are physical, which can be seen as follows: before the disorder average, only Coulomb lines connect the two Green function lines. Additionally, the Green function lines include scattering from an arbitrary number of impurities, in this case one scattering event for each Green function line. Before the disorder average these scattering events are unconnected. After the disorder average, the scattering events become connected and the above diagrams are created.

Figure 6

Summing ${\gamma }^p$ and ${\gamma }^{\times }$ yields a crossing-symmetric vertex function. This gives four different diagrams. The two diagrams in the middle are removed by the replica limit as there are closed loops. The two diagrams with ${\gamma }^p$ and ${\gamma }^{\times }$ are equivalent to the Fock-like diagram. In Eq. (7) the purely disordered part of the dual potential is crossing-asymmetric and carries a factor of $\frac{1}{2}$. If we replace it with the crossing symmetric disorder vertex, a factor of $\frac{1}{4}$ is needed to avoid double counting.

Figure 7

Second-order diagrams for the particle-hole channel. The second and third diagram are topologically equivalent, and therefore a symmetry factor of $\frac{1}{2}$ is associated with these diagrams.

Figure 8

Second-order diagrams for the particle-particle channel. For the first diagram, the two internal Green function lines are equivalent, and therefore the diagram comes with a symmetry factor of $\frac{1}{2}$.

Figure 9

Relative correction from the dual-fermion approach to the local Green function at the lowest Matsubara frequency ($iw=i\pi T$) for various parameters of the 1D lattice. The corrections are minimized for both weak- and large-$U$ limits and maximized for values of $U$ around the bandwidth. The peak position shifts to larger $U$ with increasing disorder strength. This behavior is consistent with DCA results.

Figure 10

Relative correction from the dual-fermion approach to the local Green function at the lowest Matsubara frequency ($iw=i\pi T$) for $U=V$. In this case, the nonlocal corrections are strongly reduced by the disorder. In the first figure the dual-fermion corrections agree quite well with the DCA correction. This is true for DF-2nd and DF-FLEX. For the second figure, the DF-2nd corrections qualitatively reproduce DCA results, and the matching of DF-FLEX to DCA results is nearly perfect.

Figure 11

The $UT$ phase diagram of the 3D Anderson-Hubbard model for various $V$ calculated with the DMFT+CPA, DF-2nd, and DF-FLEX approaches. For small values of $U$, the antiferromagnetic phase is suppressed by disorder. For large values of $U$ the disorder increases $T_N$. The effect of DF is to decrease the transition temperature, especially for the DF-FLEX. We mark $T_N(U=0,V=0)=0$ with a blue quarter-circle. We did not obtain values for $T_N$ in the gray shaded region for $U/W<1/3$.

Figure 12

The double occupancy $D$ of the impurity problem for DMFT and second-order dual fermions. The double occupancy displays a hysteresis which stems from the Mott transition. The effect of disorder is to move the hysteresis to larger values of $U$ and to reduce the area of the hysteresis. The hysteresis from the dual-fermion calculation is moved to smaller values of $U$ and the area is increased compared with DMFT+CPA.

Figure 13

Hysteresis at $V=0$, at $\beta W=120$, and $\beta W=180$. DMFT predicts that $U_{c_1}$ and $U_{c_2}$ increase with decreasing temperature. The DF-2nd result shows that $U_{c_1}$ decreases when nonlocal correlations are taken into account.

Figure 14

DF-2nd results for the hysteresis as a function of temperature at $V=\frac{W}{6}$ and $V=\frac{W}{3}$.

Figure 15

$\text{Im}\delta G_{\mathrm{loc}}$ as a function of $U$ and $V$ at fixed temperature $\beta W=60$ for binary disorder (top panel) and box disorder (bottom panel). We take $\text{Im}\delta G_{\mathrm{loc}}=0$ as an estimate for the phase boundary. For binary disorder we find an insulating phase for large values of $U$ and for large values of $V$. For continuous box disorder we still find the insulating phase for large values of $U$ but not for large values of $V$.

Figure 16

Conductivity on the $UV$ plane. For both binary (top panel) and box disorder (bottom panel) the conductivity gives a phase transition for large $U$ and large $V$.

Figure 17

The real fermion impurity diagrams in (a) contribute to $V_{\alpha \beta }^{p,0}$, while the diagrams in (b) and (c) contribute to $V_{\alpha }^{p,1}$. For these diagrams all replica labels are fixed to $\alpha$ because of the Hubbard interaction, whereas for diagrams in (a) two replica labels $\alpha$ and $\beta$ remain. $\sigma ,{\sigma }^{\prime }$ are independent spin labels, whereas $\overline{\sigma }=-\sigma$.

Figure 18

Two possible connections for the second-order particle-particle channel diagram for the dual Green function.

Figure 19

Four diagrams for the second-order particle-particle channel are of order $m$, i.e., they have only one free replica index $\kappa$. These four diagrams survive the replica limit $m\to 0$. The crossed wiggly lines represent the crossing-symmetric contributions, whereas the cross with the curved lines represents the crossing-asymmetric contributions from disorder scattering only.

Figure 20

Hartree-like diagrams are created from disorder by disorder averaging disconnected diagrams. For quenched disorder all disconnected diagrams are removed before the disorder average, and hence such diagrams do not exist. This property of the real fermion diagrams translates to the dual degrees of freedom. The black dashed line denotes elastic scattering from an impurity.

Figure 21

Physical second-order diagrams for the purely disordered contributions to the particle-hole channel. Both diagrams are equivalent. The diagram on the left shows that complicated connections are necessary to create skeleton diagrams for the particle-hole channel. On the right, the artificially introduced vertical disorder vertex is used. It is more convenient as it allows one to restrict oneself to Hartree-like diagrams. This is particularly helpful for higher-order diagrams.