Finite-cluster typical medium theory for disordered electronic systems

We use the recently developed typical medium dynamical cluster (TMDCA) approach [Ekuma et al., Phys. Rev. B 89, 081107 (2014)] to perform a detailed study of the Anderson localization transition in three dimensions for the box, Gaussian, Lorentzian, and binary disorder distributions, and benchmark them with exact numerical results. Utilizing the nonlocal hybridization function and the momentum resolved typical spectra to characterize the localization transition in three dimensions, we demonstrate the importance of both spatial correlations and a typical environment for the proper characterization of the localization transition in all the disorder distributions studied. As a function of increasing cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge for disorder distributions with finite variance, obtaining the correct critical disorder strengths, and shows that the order parameter critical exponent for the Anderson localization transition is universal. The TMDCA is computationally efficient, requiring only a small cluster to obtain qualitative and quantitative data in good agreement with numerical exact results at a fraction of the computational cost. Our results demonstrate that the TMDCA provides a consistent and systematic description of the Anderson localization transition.

Figure 1

The self-consistent loop of the TMDCA.

Figure 2

(Left) The average density of states calculated using the DCA for small and large disorder strengths [26]. (Right) The probability of an electron remaining on a site after time $t,P\left(t\right)=〈{\left|G(l,l,t)\right|}^2〉$. The insets in the plots on the right are the probability of an electron remaining on a site for all time $P(t\to \infty ,\eta )={lim}_{t\to \infty }〈{\left|G(l,l,t)\right|}^2〉={lim}_{\eta \to 0}\frac{\eta }{\pi }{\int }_{-\infty }^{\infty }d\epsilon 〈{\left|G(l,l,\epsilon +i\eta )\right|}^2〉$ for the $N_c=1$ (black) and 64 (red), respectively. The ADOS is not critical at the Anderson transition. The noncritical behavior in the DCA can also be inferred from $P\left(t\right)$, which decays rather fast instead of remaining constant, and in that $P(t,\eta )$ extrapolates to zero instead of a finite value when many of the states are localized. The DCA only shows a precursor to the Anderson localization as manifested in $P\left(t\right)$ for large disorder strength for the finite cluster sizes.

Figure 3

A comparison of the phase diagrams of the Anderson localization transition in 3D obtained from different cluster approximations with $N_c=38$ using the CTMT and the TMDCA (linear-log and log-log) schemes. Observe that in the CTMT, as a consequence of self-averaging, the higher disorder behaviors that are captured in our TMDCA are totally missed and the critical disorder strength is also severely overestimated.

Figure 4

The evolution of the ADOS and TDOS at various disorder strengths $W$ for the single-site TMT and the TMDCA with cluster size $N_c=64$ and 125. At low disorder, where all the states are metallic, the shape of 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 re-entrance behavior. 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

Comparison of the average (typical) density of states calculated with the DCA (TMDCA) and the kernel polynomial method (KPM) for the box disorder distribution at various strengths for the cluster size $N_c=64$. The kernel polynomial method used 4096 moments on a ${48}^3$ cubic lattice, and 1000 independent realizations generated with 32 sites randomly sampled from each realization.

Figure 6

The TDOS $(\omega =0)$ vs disorder strength $W$ at different cluster sizes $N_c=1,64,$ and 125 for the uniform (box) disorder distribution. The TDOS $(R=0,\omega =0)$ vanishes at the critical disorder strength $W_c$. At $N_c=1,W_c^{N_c=1}\approx 1.65$. 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. The inset shows the integrated hybridization function [$-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$] as a function of disorder strength $W$. Observe that $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ vanishes at the same disorder strength as the TDOS. The dashed lines are the $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ from the DCA. Observe that it is a constant regardless of the disorder strength and cluster size. This shows that the DCA, even though it incorporates spatial correlations, does not describe the Anderson localization transition. Moreover, near the critical region the TDOS $(R=0,\omega =0)$ data can be fitted to a power law, with TDOS $(R=0,\omega =0)=a_0{|W-W_c^{\mathrm{fit}}|}^{\beta }$. The obtained critical exponent for large enough clusters $\beta \approx 1.62\pm 0.10$ is in good agreement with exact results. Note that the $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ data for $N_c=64$ and 125 has been normalized with that of $N_c=1$.

Figure 7

(Right) The probability that an electron remains localized at all times $P(\infty ,\eta )$ for the Anderson model for $N_c=1$ and 64 at varying disorder strengths. We used the fact that $P(t\to \infty ,\eta )={lim}_{\eta \to 0}\frac{-2i\eta }{N_c}{\sum }_l{\int }_{-\infty }^{\infty }d\omega d{\omega }^{\prime }〈\frac{\overline{A}(l,\omega )\overline{A}(l,\omega )}{\omega -{\omega }^{\prime }-2i\eta }〉$, where $\overline{A}(l,\omega ) = -1/\pi \Im \overline{G}(l,l,\omega )$ is the local coarse-grained (but not disorder averaged) spectral function. As $\eta \to 0,P(\infty ,\eta )$ extrapolates to zero for small disorder strength indicating metallic behavior but does not extrapolate to zero anymore as the disorder strength is systematically increased towards the critical disorder strength leading to the transition (see the inset where this is manifestly illustrated). (Left) The probability of an electron on a site thereby remaining trapped at finite time $t$ for $N_c=1$ and 64 for the same parameter as $P(\infty ,\eta )$ on a semilogarithmic plot.

Figure 8

The evolution with disorder strength of the probability distribution of the density of states at different cluster cells for $N_c=38$. Utilizing the irreducible wedge property and particle-hole symmetry, the original 38 cells are reduced to four cells. For small disorder strength, the cells show Gaussian distributions whereas close to the critical disorder strength $\simeq 2.0$, all the cells show log-normal distributions.

Figure 9

The phase diagram of the Anderson localization transition in 3D for the box disorder distribution obtained from TMDCA simulations. A systematic improvement of the trajectories of the mobility edge is achieved as the cluster size increases. At large enough $N_c$ and within computation error, our results converge to those determined by the transfer matrix method (TMM). The TMM data is for system sizes of length $L=1\times {10}^6$ (number of multiplications), the range of system widths used is $M=[2,16]$ and reorthogonalization is done every 5 transfer matrix multiplications (see Appendix pp1 for details). The error was determined in a point in the region of most discrepancy between the methods (upper bound of re-entrance) by a finite size scaling analysis (see Appendix pp1). The black line with filled circles denotes the Lifshitz boundaries (extracted from the ADOS calculated within the DCA).

Figure 10

The scaling of the imaginary typical hybridization function [$-\Im \mathrm{\Gamma }(\mathbf{K},\omega )$] for the 64 site cluster at disorder strengths of $W=1.8$ and 2.0 on a semi-logarithmic scale. The labels A–F and their associated momenta $\mathbf{K}$ correspond to each of the six distinct cells obtained using the point-group and particle-hole symmetry ($\rho (\mathbf{K},\omega )=\rho (\mathbf{Q}-\mathbf{K},-\omega )$, with $\mathbf{Q}=(\pi ,\pi ,\pi ))$ of the cluster). Observe that the mobility edges may be collapsed on top of each other by multiplying each of the hybridization functions with a constant such that $-\Im \mathrm{\Gamma }(\mathbf{K},\omega )=-\alpha \left(\mathbf{K}\right)\times \Im \mathrm{\Gamma }(\mathbf{K},\omega )$, where $\alpha \left(\mathbf{K}\right)$ is a scaling constant, in agreement with Mott's idea of energy selective Anderson localization transition.

Figure 11

Comparison of the average (typical) density of states calculated with the DCA (TMDCA) and the kernel polynomial method (KPM) for the binary alloy system $\left(W_b=-W_a\right)$ at various values of the local potential $W_a$ and concentrations $c_a$ for the cluster size $N_c=125$. The kernel polynomial method uses 2048 moments on a ${48}^3$ cubic lattice, and 200 independent realizations generated with 32 sites randomly sampled for each realization.

Figure 12

Disorder-energy $(W_a-\omega )$ phase diagram of the Anderson localization transition in 3D for the binary alloy system ${\mathrm{A}}_{c_a}{B}_{1-c_a}$ for $c_a=0.5$. Observe the systematic improvement of the trajectories of the phase diagram for the clusters ($N_c=64$ and 216) in basic agreement with the numerical results from TMM. The system widths used for TMM are $M=[6,12]$ and the length is scaled with the width as $L=M\times {10}^4$. See Appendix pp1 for details.

Figure 13

Comparison of the average (typical) density of states calculated with the DCA (TMDCA) and the kernel polynomial method (KPM) for the Gaussian disorder distribution at various values of the disorder strength for cluster size $N_c=64$. The kernel polynomial method uses 4096 moments on a ${48}^3$ cubic lattice, and 1000 independent realizations generated with 32 sites randomly sampled from each realization.

Figure 14

The TDOS $(\omega =0)$ vs disorder strength $W$ at different cluster sizes $N_c=1,64,$ and 125 for the Gaussian disorder distribution. The TDOS $(\omega =0)$ vanishes at the critical disorder strength $W_c$. At $N_c=1$, the critical disorder strength $W_c^{N_c=1}\approx 4.0$. As the cluster size increases, the critical strength systematically increases with $W_c^{N_c\gg 12}\approx 5.30\pm 0.10$, showing a quick convergence with the cluster size. The inset shows the integrated hybridization function $\left(-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega \right)$ as a function of disorder strength $W$. Observe that $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ vanishes at the same disorder strength as the TDOS. The dashed lines are the $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ from the DCA. Observe that it is a constant regardless of the disorder strength. This shows that the DCA, even though it incorporates spatial correlations does not describe the Anderson localization transition. Moreover, near the critical region, the TDOS $(\omega =0)$ data can be fitted to the power-law TDOS $(\omega =0)=a_0{|W-W_c^{\mathrm{fit}}|}^{\beta }$. The obtained critical exponent for large enough clusters $\beta \approx 1.57\pm 0.10$ is in good agreement with the numerically exact results [46]. Note that the $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ data for $N_c=64$ and 125 has been normalized with that of $N_c=1$.

Figure 15

The phase diagram of the Anderson transition in 3D for the Gaussian disorder distribution obtained from TMDCA simulations. A systematic improvement of the trajectories of the mobility edge is achieved as the cluster size increases. At large enough $N_c$, our results converge to those determined by the TMM which are calculated for widths $M=[2,16]$ and for a system length (number of transfer matrix multiplications) $L=1\times {10}^6$, where the matrix products are reorthogonalized every five transfer matrix multiplications (see Appendix pp1). The deviation between the TMDCA and TMM results is consistent with the behavior observed for the box and Lorentzian disorders and can be attributed to the fact that a finite grid in energy is used for the TMDCA which tends to cause the typical density of states to be larger, hence slightly overemphasizing the metallic behavior and as such, the mobility edge is slightly larger when compared to TMM in certain frequency ranges near the band edge. The effect is most pronounced here due to the small density of states near the band edge. In addition, near the re-entrance regime, the TMM also has difficulties due to an increase in finite size effects (see Appendix pp1).

Figure 16

The TDOS $(\omega =0)$ vs disorder strength $W$ at cluster sizes $N_c=1$, 64, and 125 for the Lorentzian disorder distribution. The TDOS $(\omega =0)$ vanishes at the critical disorder strength $W_c$. At $N_c=1$, the critical disorder strength $W_c^{N_c=1}\approx 0.6$. As the cluster size increases, the critical strength systematically increases until $W_c^{N_c\gg 12}\approx 0.94\pm 0.10$, showing a quick convergence with the cluster size. The inset shows the integrated hybridization function [$-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$] as a function of disorder strength $W$. Observe that $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ vanishes at the same disorder strength as the TDOS. The dashed lines are the $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ from the DCA. Observe that it is a constant regardless of the disorder strength in agreement with the observations we have for other disorder distributions. We can fit the TDOS $(\omega =0)$ data near the critical region to the power-law, with TDOS $(\omega =0)=a_0{|W-W_c^{\mathrm{fit}}|}^{\beta }$. The obtained critical exponent for large enough clusters $\beta \approx 1.60\pm 0.10$ is in good agreement with numerically exact results [46]. Note that the $-\Im \int \mathrm{\Gamma }(\mathbf{K},\omega )d\mathbf{K}d\omega$ data for $N_c=64$ and 125 have been normalized with that of $N_c=1$.

Figure 17

The phase diagram of the Anderson localization transition in 3D for the Lorentzian disorder distribution obtained from TMDCA simulations. The TMM data were computed for a system length of $L=8\times {10}^5$ for system widths of $M=[2,16]$ and re-orthogonalization is done every four multiplications (see Appendix pp1 for details). The discrepancy between the TMDCA and TMM is due to the nature of the Lorentzian distribution as explained in Sec. 3f.

Figure 18

The band center of the typical density of states ($\mathrm{TDOS}\left(R=0;\omega =0\right)$) at various disorder strengths $\left(W\right)$ for the 125 site cluster for the Box, Gaussian, and Lorentzian disorder distributions. The linear region of the data can be fit with a scaling ansatz: TDOS $(\omega =0)=a_0\times {|W-W_c^{\mathrm{fit}}|}^{\beta }$, with $\beta \sim 1.62$ for the box disorder distribution, $\beta \sim 1.57$ for the Gaussian disorder distribution, and $\beta \sim 1.60$ for the Lorentzian disorder distribution in good agreement with the recently reported value of $\beta \approx 1.67$ [46, 50]. The circles in the plot depict the data point where the fit starts to deviate from the data.

Figure 19

The plot of the typical density of states (TDOS) at $\omega =0.0$ and $1.3$ for various disorder strengths for the (a) box, (b) Gaussian, and (c) Lorentzian disorder distributions.