Calculation of two-particle quantities in the typical medium dynamical cluster approximation

The mean-field theory for disordered electron systems without interaction is widely and successfully used to describe equilibrium properties of materials over the whole range of disorder strengths. However, it fails to take into account the effects of quantum coherence and information of localization. Vertex corrections due to multiple backscatterings may drive the electrical conductivity to zero and make expansions around the mean field in strong disorder problematic. Here, we present a method for the calculation of two-particle quantities which enables us to characterize the metal-insulator transitions in disordered electron systems by using the typical medium dynamical cluster approximation. We show how to include vertex corrections and information about the mobility edge in the typical mean-field theory. We successfully demonstrate the application of the developed method by showing that the conductivity formulated in this way properly characterizes the metal-insulator transition in disordered systems.

Figure 1

Cartoon showing the general idea of the TMDCA. Instead of solving the original lattice disorder system directly (left), the TMDCA maps the lattice problem to a periodic finite-sized cluster embedded in a self-consistently determined typical mean field (right). We can calculate the susceptibility of the system by applying the field $h$ to either the lattice problem on the left or the embedded cluster problem on the right.

Figure 2

Representation of the two-particle quantities in the particle-hole (a) and the particle-particle channel (b). Note that for disorder systems, only elastic scatterings exist, and therefore only two frequencies are needed (in the figure, following the dashed lines, the energy is conserved). For the retarded-advanced component, we assign the above line $\omega$ as the retarded fermionic line and the lower line ${\omega }^{\prime }$ as the advanced fermionic line. In addition, we focus mainly on the dc conductivity at zero temperature, so we only need zero frequencies, namely $\omega =0$ and ${\omega }^{\prime }=0$, for all the two-particle quantities involved. This would greatly simplify our two-particle calculation.

Figure 3

Bethe-Salpeter equation relating the two-particle Green's function $\chi$ and the irreducible vertex $\mathrm{\Gamma }$. While $k,k^{\prime }$, and $q$ represent momentum indices, $\omega$ and $\nu$ represent frequency indices (for fermionic and bosonic frequencies, respectively). Note that for disorder systems considered here, the scatterings are elastic and thus the energy is conserved following any fermionic Green's function line. Therefore, we only need two frequency indices to represent the frequency degree of freedom of the system.

Figure 4

The evolution of the ADOS, TDOS, and dc conductivity at various disorder strengths $W$ for the single-site TMT and the TMDCA with cluster size $N_c=64$. Here, for dc conductivity, $\omega$ corresponds to the chemical potential used in the calculation. At low disorder, where all the states are metallic, the shape of the TDOS is the same as that of the ADOS. As $W$ increases, in the case of single-site TMT, the TDOS gets suppressed and the mobility edge moves towards $\omega =0$ monotonically. In the TMDCA, the TDOS is also suppressed, but the mobility edge first moves to higher energy, and only with a further increase of $W>1.8$, it starts moving towards the band center, indicating that the TMDCA can successfully capture the reentrance behavior [16]. Arrows indicate the position of the mobility edge, which separates the extended electronic states from the localized ones and the colored region indicates the TDOS.

Figure 5

The dc conductivity of the 3D Anderson model at $T=0$ and $\mu =0$(band center) vs disorder $W$ for different cluster size $N_c=1,10,12,64,92$. The dc conductivity vanishes at $W_c$ where all states become localized. For $Nc=1$ (TMT), the critical disorder strength $W_c^{N_c=1}\approx 1.65$. As the cluster size increases, $W_c$ systematically increases with $W_c^{N_c\gg 12}\approx 2.10\pm 0.10$, showing a quick convergence with the cluster size. At large enough $N_c$, our results converge to those obtained by KPM.