# Topological Insulators

Most quantum phenomena are notoriously difficult to observe, and therefore to manipulate, measure and, ultimately, understand. That is especially true for a newly discovered class of condensed-matter states called topological insulators (TIs). But now PFC scientists* are devising a method that could allow direct observation of these exotic entities.

TIs form in certain materials that, in bulk, have the distinctive physical signature of insulators: That is, the permitted energy levels (or “bands”) in their component atomic structures are characterized by a full valence band and an empty conduction band, with a substantial gap of forbidden energies in between. However, at the boundaries with other materials or the vacuum, they develop gapless regions that permit conduction along one-dimensional paths or the edges of two-dimensional surfaces – but in a single direction only. These states, discovered only a few years ago, are highly promising candidates for use in quantum computation because they are robust in the face of disorder or inhomogeneities.

The chief reason is their topological nature, which in this context refers to the quantum characteristics of the material as a whole, as distinct from its local properties at any point. TIs belong to one topological set, and conventional insulators to another. PFC theorist Tudor Stanescu puts it this way: “Topologically, a sphere, a cube and a pyramid are identical. That is, they can all be smoothly transformed into one another without ‘tearing’ the surface. Similarly, a torus (donut shape) and a coffee cup are identical. There are many types of standard insulators, but they are all topologically identical, like the sphere, cube and pyramid. Topological insulators belong to a different class, like the torus and coffee cup, and you can’t deform them into the other class.”

At the boundary between the two classes of material, peculiar phenomena arise which are of immediate interest to the PFC because one of its major research activities involves the study of correlated and topological matter in cold atoms, typically arrayed in optical lattices, to model condensed-matter systems. That approach is ideally suited to the investigation of TIs.

Perhaps the most familiar manifestation of a TI occurs in the quantum Hall effect: Gapless regions form along the edges of two-dimensional structures, with the direction of charge movement dependent on the spin of the particle. The Hall effect, of course, requires an externally applied magnetic field. But recently, physicists posited that in some materials, analogous states could form in the absence of any external field, resulting only from the “effective” magnetic field produced by spin-orbit coupling. Graphene, the two-dimensional form of graphite, has been identified as a particularly illustrative case. Theory indicates that it will act as an insulator in bulk, but with conducting states at the edges.

The scheme involves the superposition of three standing waves to create a lattice in which the energy minima form a hexagonal unit cell. [Figure 1]

That phenomenon would be extremely difficult to isolate in a conventional condensed-matter system, and direct observation of the edge states would likely be nearly impossible. The PFC theory team, however, is in the process of completing a model experimental protocol using ultracold atoms suspended in a two-dimensional optical lattice of intersecting laser beams to simulate a graphene-like arrangement which would make topological insulators both detectable and adjustable.

The beams are arranged so that the entire structure has a periodic “vector potential” (that is, an overall directional tendency) that acts, in effect, as a magnetic field. As the atoms interact with both the lattice structure and the vector potential, they take on well-defined boundary shapes that are the equivalent of edge states in bulk-matter TIs. These formations are chiral (left- or right-handed, like a mirror image) in the sense that they permit momentum in only one direction.

Figure 2 (right) shows a plot of atomic states in a TI lattice array. The red and yellow dots represent atoms within ordinary energy bands. The blue dots represent atoms in the band gap, constituting the lattice equivalent of one TI edge state. Note that the in-gap atoms only cross the gap in one direction, owing to the chiral nature of the arrangement.

*JQI/PFC investigators on the TI simulation project are Tudor Stanescu, Victor Galitski, Jay Vaishnav, Charles Clark and Sankar Das Sarma. Galitski, Clark and Das Sarma are JQI Fellows.

*Phys. Rev. A*,

**79**, 053639 (2009)

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