Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials

We investigate the metal-insulator transition occurring in two-dimensional (2D) systems of noninteracting atoms in the presence of artificial spin-orbit interactions and a spatially correlated disorder generated by laser speckles. Based on a high order discretization scheme, we calculate the precise position of the mobility edge and verify that the transition belongs to the symplectic universality class. We show that the mobility edge depends strongly on the mixing angle between Rashba and Dresselhaus spin-orbit couplings. For equal couplings a non-power-law divergence is found, signaling the crossing to the orthogonal class, where such a 2D transition is forbidden.

Figure 1

Numerical calculation of the critical point of the 2D Anderson transition for cold atoms with synthetic Rashba SOC in a blue-detuned speckle. After discretization of the model on a strip-shaped grid of spacing $\mathrm{\Delta }$, height $M$, and length $L\gg M$, we use the transfer-matrix method to calculate the localization length ${\lambda }_M$. The main panel shows the ratio ${\lambda }_M/M$ as a function of energy calculated for increasing values of $M=200$ (top curve on the left), 250, 300, 350, assuming $\mathrm{\Delta }=0.2\pi \sigma$. The crossing point corresponds to the mobility edge, $E=E_c\simeq 0.256E_{\sigma }$, with $E_{\sigma }=1/(m{\sigma }^2)$, $\sigma$ being the correlation length of the speckle; see Eq. (3). The Rashba strength is $m{\lambda }_R\sigma =0.03$ and the disorder amplitude is $V_0=E_{\sigma }$. The inset shows the evaluation of the critical exponent $\nu$ from the scaling behavior of $d{\lambda }_M/dE$ at the mobility edge; see Eq. (5).

Figure 2

Mobility edge $E_c$ of the 2D Anderson transition separating low-energy localized states $(E<E_c)$ from high-energy diffusive states $(E>E_c)$, plotted as a function of the disorder amplitude $V_0$ and for increasing values of the Rashba spin-orbit coupling ${\lambda }_{R}m\sigma =0.03$ (top curve), 0.1, 0.3, and 1.256, assuming ${\lambda }_D=0$. The black line corresponds to the energy bottom $E=-V_0-m{\lambda }_R^2/2$, below which no single particle state exists. Notice the linear behavior of $E_c$ for $V_0/E_{\sigma }\gtrsim 1$.

Figure 3

Mobility edge as a function of the mixing angle $\theta =\arctan ({\lambda }_D/{\lambda }_R)$ between Rashba and Dresselhaus SOC, for a fixed value of the total strength $\sqrt{{\lambda }_R^2+{\lambda }_D^2}m\sigma =0.5$ and disorder amplitude $V_0=E_{\sigma }$. Approaching $\theta =\pi /4$ (vertical dashed line), corresponding to equal strengths of Rashba and Dresselhaus SOC, the mobility edge rises sharply and actually diverges. Indeed, at this special point, spin scattering is absent and the model falls into the orthogonal universality class, for which all states are localized in two dimensions, implying $E_c=+\infty$. Notice that the mobility edge $E_c$ is invariant under the transformation $\theta \to \pi /2-\theta$ exchanging the Rashba and Dresselhaus terms in Eq. (1).