Phase diagram of the three-dimensional Anderson model for short-range speckle potentials

We investigate the localization properties of atoms moving in a three-dimensional optical lattice in the presence of a disorder potential having the same probability distribution $P\left(V\right)$ as laser speckles, and a spatial correlation length much shorter than the lattice spacing. We find that the disorder-averaged (single-particle) Green's function, calculated via the coherent-potential approximation, is in very good agreement with exact numerics. Using the transfer-matrix method, we compute the phase diagram in the energy-disorder plane and show that its peculiar shape can be understood from the self-consistent theory of localization. In particular, we recover the large asymmetry in the position of the mobility edge for blue and red speckles, which was recently observed numerically for spatially correlated speckle potentials.

Figure 1

Transfer-matrix results for the 3D Anderson model with a spatially uncorrelated, blue-detuned speckle potential of strength $V_0=8.$ Each curve displays the ratio of the localization length ${\lambda }_M$ of a long bar with cross section $M\times M$ to the bar transverse size $M$, as a function of energy. The various curves from $M=16$ (least steep curve) to $M=31$ (steepest curve) cross at the mobility edge $E_c\approx -9.93$.

Figure 2

Exact localization phase diagram for the 3D Anderson model with a spatially uncorrelated, blue-detuned speckle potential obeying the Rayleigh distribution, Eq. (1), as obtained from the transfer-matrix technique and finite-size scaling. Notice that there are rigorously no states below the solid line corresponding to $E=-6-V_0$. The phase diagram for red-detuned speckles is simply obtained under the change $E\to -E$, as shown in the inset.

Figure 3

Energy dependence of the real (${\mathrm{\Sigma }}^{\prime }$) and imaginary (${\mathrm{\Sigma }}^{\prime \prime }$) parts of the self-energy calculated using the coherent-potential approximation [solid line, Eq. (12)] and the self-consistent Born approximation [dotted line, Eq. (11)] for two values of the disorder strength: $V_0=1$ (left) and $V_0=5$ (right). Also shown is the atomic limit approximation [dashed line, from Eqs. (15) and (16)] for $V_0=5$. In both cases, the CPA is a truly excellent approximation to the exact numerical results (for $k=0$) plotted as circles, reproducing almost all details of both the real and imaginary parts of the self-energy (note that for the numerical computations, system sizes of $M^3$ with $M=28$ were used). In contrast, the SCBA is a rather poor approximation, even at a moderate disorder strength $V_0=1$ where it is supposed to work best.

Figure 4

Density of states as a function of energy for three different values of the disorder strength $V_0=1$ (black, first from the top at $E=0$), $V_0=5$ (blue), and $V_0=10$ (red). Dots correspond to the exact numerical results for system sizes $M^3$ with $M=28$ (for $V_0=1$), and $M=16$ (for $V_0=5,10$). The CPA predictions, shown as solid lines, are in excellent agreement with the numerical data, reproducing very well the existence of an effective band edge at low energy. Only some tiny features above the band edge are not correctly reproduced. The dashed lines correspond to the strong disorder (atomic limit) approximation, Eq. (17), and are reasonably accurate except near the low-energy effective band edge. The dotted lines are the large-energy asymptotic limit, Eq. (18).

Figure 5

Band edge $E_{\text{BE}}$ plotted as a function of disorder strength, calculated within the SCBA (red squares) and the CPA (black circles). Solid lines refer to the corresponding approximate analytical expressions, Eqs. (21) and (26), obtained for weak disorder. The exact transfer-matrix results for the mobility edge are also shown (green diamonds).

Figure 6

Prediction of the self-consistent theory of localization—see Eq. (32)—for the mobility edge of the Anderson model with a Rayleigh potential distribution, using the CPA self-energy (solid line). The open circles correspond to the transfer-matrix results of Fig. 2. The dashed line represents the band edge calculated from CPA, neglecting Lifshitz tails.

Figure 7

Zoom-in of the phase diagram (see Fig. 2) for blue (circles) and red (squares) speckles at the bottom of the band. The shape is qualitatively similar to the one obtained for spatially correlated speckles in Ref. [7], pointing out the crucial role played by the on-site potential distribution $P\left(V\right)$.