The Anderson Transition in a Two-Dimensional Ultracold Gas

The wave-nature of electrons is responsible for the long mean free path in regular lattices because of destructive interference from back-scattered waves. This leads to the high conductivity of metals. In disordered systems, interference can suppress transport in all directions leading to insulating behaviour, (Anderson Localisation).

In one-dimension (1D) in the thermodynamic limit, however weak the disorder, all single particle states are localised, and the system is an insulator. In three-dimensions (3D), states with low energy can be localised, whilst those with higher momenta and energy can be extended. There then exists a quantum phase transition depending upon the location of the Fermi Surface relative to the Mobility Edge, the boundary between localised and extended states. Two-dimensions (2D) represents the marginal dimension in which the system asymptotically approaches the metallic phase as the strength of disorder approaches zero such that, again, for any finite disorder, the states are always localised and the system an insulator, but the localisation length can be very large.

Direct observation of the localisation of matter waves in 2D, particularly in ultracold atomic gases, had remained elusive. Using a geometry based upon a circular source region of atoms and a similar drain region, connected by a 2D channel containing a series of optical “spike” potentials representing the disorder, we were able to observe, the characteristic exponential decay of atom density, and hence conclude that we had observed Anderson Localisation in a 2D ultracold atom system.

For continuous phase transitions, such as the Anderson transition, there exists a single parameter that diverges exponentially at the transition point, determined by a critical exponent unique to the specific universality class. For time-reversal and spin-rotational symmetries (orthogonal symmetry class) an Anderson transition occurs for a finite level of disorder in three (and higher) dimensions, below which the system is a conductor and above which it is an insulator. In 1D, as stated above, any disorder, however weak, always localises states, and (2D), for weakening disorder the localisation length gets longer and longer, but again for large enough systems, the states are always localised, and the system is an insulator.

Spin-orbit coupling breaks the spin rotational symmetry, e.g., leads to the splitting of the sodium D-line. Such reduction of symmetry yields a system in the symplectic symmetry class and permits the development of a well-defined Anderson transition even in 2D. In the symplectic class the critical exponent was first accurately evaluated by Slevin’s group. More recently, in a Letter published in Physical Review Research, we have shown how the critical exponent can be extracted from experimentally available momentum-space signatures. Both our group, and that of Slevin, have also characterised the temporal and finite-size properties of the symplectic 2D system around the phase transition.

This investigation has raised several questions regarding the nature of this novel metal-insulator phase transition in a 2D system and has implications for other fields including electron transport in Si MOSFETs.

We present results from our recent work both experimentally observing Anderson Localisation in a two-dimensional ultracold atomic gas within the orthogonal symmetry class and theoretically investigating the Anderson Transition in a 2D symplectic system with spin-orbit coupling.