Metal-insulator transition induced by random dipoles

We study the localization properties of a test dipole feeling the disordered potential induced by dipolar impurities trapped at random positions in an optical lattice. This random potential is marked by correlations which are a convolution of short-range and long-range ones. We show that when short-range correlations are dominant, extended states can appear in the spectrum. Introducing long-range correlations, the extended states, if any, are wiped out and localization is restored over the whole spectrum. Moreover, long-range correlations can either increase or decrease the localization length at the center of the band, which indicates a richer behavior than previously predicted.

Figure 1

Schematic representation of the physical model. Dipolar impurities (green spheres) are trapped at the minima of the optical lattice, occupying random positions. The test dipole (red upper sphere), excited to another internal level, feels a shallower optical potential and a disordered potential due to the dipolar interaction with the impurities.

Figure 2

Site energies ${\lambda }_0$, ${\lambda }_1$, and ${\lambda }_2$, and hopping energy $\theta$ as function of ${\sigma }_{\perp }$ in units of $d_L$, for the case of dysprosium atoms with $D^2/d_L^3=0.016E_R$, $\alpha =\pi /2$, ${\lambda }_L=2d_L=570$ nm, $s_{\left(T\right)}=6$, and $s_{\left(I\right)}=30$. The colored vertical lines labeled by different letters correspond, respectively, to (a) ${\sigma }_{\perp }/d_L=0.158$, (b) ${\sigma }_{\perp }/d_L=0.184$, (c) ${\sigma }_{\perp }/d_L=0.226$, and (d) ${\sigma }_{\perp }/d_L=0.302$ and identify the set of parameters that we used for the calculation of the localization properties of the system.

Figure 3

Top panel: localization length ${\mathcal{L}}_{\mathrm{loc}}$ in units of the lattice spacing $d_L$ as a function of the energy in units of $J$ for $\mathcal{C}=1/4$, system sizes up to ${10}^7$ lattice sites, and averaged over 100 configurations. The black dashed lines correspond to the localization length calculated in the Born approximation ${\mathcal{L}}_{\mathrm{loc}}^{\left(2\right)}$. Bottom panel: reflection coefficient $\mathcal{R}$ of the single impurity as a function of the energy in units of $J$. The vertical dashed lines indicate the energies for which the reflection coefficient vanishes [$\mathcal{R}\left(E\right)=0$]. From left to right, different plots refer to increasing values of ${\sigma }_{\perp }$, corresponding to the vertical lines in Fig. 2.

Figure 4

Phase diagram induced by short-range correlations extracted from the reflection coefficient in Eq. (23) for the single impurity case. The different regions correspond to different localization regimes obtained from the solution of Eq. (24). No real solutions of Eq. (24) correspond to the (a) red region. If Eq. (24) has real solutions, we can distinguish three cases depending on the number of solutions lying inside the spectrum: two solutions [(b) blue region], one solution [(c) green region], and no solutions [(d) yellow region]. The four markers in the diagram correspond to the simulations presented in Fig. 3 and to the values of ${\sigma }_{\perp }$ indicated by vertical lines in Fig. 2.

Figure 5

Localization length ${\mathcal{L}}_{\mathrm{loc}}$ in units of the lattice spacing $d_L$ as a function of the energy in units of $J$ for $\beta =1,3,\infty$, namely, different types of long-range correlations identified by the asymptotic decay of the tails of the two-point correlation function $C_{\varepsilon }\left(\ell \right)$. The left and right panels correspond to two different localization regimes induced by short-range correlations as shown in Figs. 3a and 3b for $\beta =3$ (see text for more details).