Study of off-diagonal disorder using the typical medium dynamical cluster approximation

We generalize the typical medium dynamical cluster approximation and the local Blackman, Esterling, and Berk method for systems with off-diagonal disorder. Using our extended formalism we perform a systematic study of the effects of nonlocal disorder-induced correlations and off-diagonal disorder on the density of states and the mobility edge of the Anderson localized states. We apply our method to the three-dimensional Anderson model with configuration-dependent hopping and find fast convergence with modest cluster sizes. Our results are in good agreement with the data obtained using exact diagonalization and the transfer-matrix and kernel polynomial methods.

Figure 1

The effect of off-diagonal disorder on the average density of states calculated in the DCA scheme with $N_c=4^3$. Our DCA results for $N_c=1$ corresponds to a single-site CPA BEB scheme. We consider two values of local disorder potential below ($V_A=0.4$) and above ($V_A=0.9$) the band-split limit and examine the effect of changing the off-diagonal hopping strength (which amounts to a change in the nonlocal potential). We start with the diagonal-disorder case $t^{AA}=t^{BB}=t^{AB}=1.0$ and then consider two off-diagonal-disorder cases: $t^{AA}=1.5,t^{BB}=0.5$ and $t^{AA}=1.8,t^{BB}=0.2$, respectively. We fix $t^{AB}=t^{BA}=0.5(t^{AA}+t^{BB})$ and $c_A=0.5$. For this parameter range of off-diagonal disorder, we do not observe a significant difference between the CPA ($N_c=1$) and the DCA ($N_c=4^3$) results indicating that nonlocal intersite correlations are weak.

Figure 2

The effect on the average density of states of an increasing diagonal-disorder potential $V_A$ for a fixed off-diagonal disorder calculated with our modified DCA scheme with $t^{AA}=1.5,t^{BB}=0.5,t^{AB}=0.5(t^{AA}+t^{BB})$, and $c_A=0.5$. Results are obtained for $N_c=1$ (corresponding to the CPA) and $N_c=4^3$ cluster sizes. We also compare our DCA average DOS with the DOS obtained using exact diagonalization (ED) for a ${12}^3$ cubic lattice cluster with 48 disorder realizations. For ED results, we used a $\eta =0.01$ broadening in frequency.

Figure 3

The imaginary part of the self-energy vs frequency $\omega$ for $N_c=1$ (red dashed line) and $N_c=4^3$ (solid lines) at various $K$ momenta points: $(0,0,0),(\pi ,0,0),(\pi ,\pi ,0)$, and $(\pi /2,\pi /2,\pi /2)$ for $V_A=0.1$ (top) and $V_A=0.6$ (bottom) diagonal-disorder potential with $t^{AA}=1.5,t^{BB}=0.5,t^{AB}=0.5(t^{AA}+t^{BB})$, and $c_A=0.5$. For small disorder $V_A=0.1$, the self-energy for $N_c=1$ is essentially the same as that of the various $K$ points of the $N_c=4^3$ cluster, indicating that nonlocal effects are negligible for such small disorder. For a larger value of the disorder $V_A=0.6$, the single-site and the finite cluster data differ significantly, which illustrates that at larger disorder, the momentum dependence of the self-energy increases and becomes important.

Figure 4

Diagonal-disorder case: The average density of states (dashed-dotted line) calculated within the DCA for $N_c=1$ (left panel) and $N_c=4^3$ (right panel) and the typical density of states shown as shaded regions for $N_c=1$ (left panel) and $N_c=4^3$ (right panel), and the dashed lines for $N_c=6^3$ (right panel) are calculated within the TMDCA for diagonal disorder $t^{AA}=t^{BB}=t^{AB}=t^{BA}=1,c_A=0.5$, and various values of the local potential $V_A=-V_B$. The TDOS is presented for several cluster sizes $N_c=1,N_c=4^3$, and $N_c=6^3$ in order to show its systematic convergence with $N_c$. The average DOS converges for cluster sizes beyond $N_c=4^3$. The TDOS is finite for the extended states and zero when the states are localized. The mobility edges extracted from the vanishing of the TDOS are marked by the arrows (we show arrows for $N_c=4^3$ only). The extended states region with a finite TDOS is always narrower for $N_c=1$ as compared to the results of the $N_c>1$ clusters, indicating that a single-site TMT tends to overemphasize the localized states.

Figure 5

Diagonal-disorder case. Comparison of the average and typical DOS calculated with the DCA/TMDCA and kernel polynomial methods (KPMs) [47] for diagonal disorder with $t^{AA}=t^{BB}=t^{AB}=t^{BA}=1$ at various values of local potential $V_A$ and concentrations $c_A$ for cluster size $N_c=6^3$. The kernel polynomial method used 2048 moments on a ${48}^3$ cubic lattice and 200 independent realizations generated with 32 sites randomly sampled from each realization.

Figure 6

Off-diagonal-disorder case. The left panel displays results for $N_c=1$, and the right panel displays results for $N_c>1$. The average density of states (dashed-dotted line) and the typical density of states (shaded regions) for $N_c=1$ (left panel), $N_c=4^3$ (right panel), and blue dashed lines for $N_c=5^3$ (left panel) for various values of the local potential $V_A$ with off-diagonal-disorder parameters: $t^{AA}=1.5,t^{BB}=0.5,t^{AB}=0.5(t^{AA}+t^{BB})$, and $c_A=0.5$. As in Fig. 4, we show the TDOS for several cluster sizes $N_c=1,4^3$, and $6^3$ in order to show its systematic convergence with increasing cluster size $N_c$. The average DOS converges for cluster sizes beyond $N_c=4^3$. The TDOS is finite for the extended states and zero for the localized states. The mobility edges are extracted as described in Fig. 4.

Figure 7

Disorder-energy phase diagram for the diagonal-disorder case. Parameters used are as follows: $t^{AA}=t^{BB}=t^{AB}=1.0$, and $c_A=0.5$. We compare the mobility edges obtained from the TMT $N_c=1$ (black dashed line), TMDCA with $N_c=4^3$ (green dot-dashed line) and $N_c=6^3$ (red solid line), and the transfer-matrix method (TMM) (blue dotted line). The single site $N_c=1$ results strongly underestimate the extended states region when compared with TMDCA results for $N_c>1$. The mobility edges obtained from the finite cluster TMDCA ($N_c>1$) show good agreement with those obtained from the TMM, in contrast to single-site TMT. See the text for parameters and details of the TMM implementation.

Figure 8

Disorder-energy phase diagram for the off-diagonal-disorder case. Parameters used are $t^{AA}=1.5,t^{BB}=0.5,t^{AB}=1.0$, and $c_A=0.5$. The mobility edges obtained from the TMT $N_c=1$ (black dashed line), TMDCA $N_c=3^3$ (green dot-dashed line), $N_c=4^3$ (purple double-dot-dashed line) and $N_c=5^3$ (red solid line), and the TMM (blue dotted line). The single site $N_c=1$ strongly underestimates the extended states region especially for higher values of $V_A$. The mobility edges obtained from the finite cluster TMDCA ($N_c>1$) converge gradually with increasing $N_c$ and show good agreement with those obtained from the TMM, in contrast to single-site TMT. See the text for parameters and details of the TMM implementation.

Figure 9

The average DOS (dot-dashed lines) and the typical DOS (shaded regions) for various values of the concentration $c_A$ with off-diagonal-disorder parameters $t^{AA}=1.1,t^{BB}=0.9$, and $t^{AB}=1.0$ at fixed local potential $V_A=1.0$ for $N_c=1$ (left panel) and $N_c=5^3$ (right panel).

Figure 10

The evolution of the typical density of states for $N_c=1$ (left panel) and $N_c=5^3$ (right panel) with the change in the concentration $0<c_A<1$ at fixed diagonal- and off-diagonal-disorder parameters: $t^{AA}=1.1,t^{BB}=0.9,t^{AB}=1.0$, and $V_A=1.0$.