Topological insulators and metals in atomic optical lattices

We propose the realization of topological quantum states in a cold-atom system, using a two-dimensional hexagonal optical lattice and a light-induced periodic vector potential. A necessary condition for observing the topological states is the realization of a confining potential with a flat bottom and sharp boundaries. To probe the topological states, we propose to load bosons into the characteristic edge states and image them directly. The possibility of mapping out the edge states and controlling the optical lattice and vector potentials offers opportunities for exploring topological phases with no equivalent in condensed-matter systems.

Figure 1

(Color online) Optical lattice potential formed by the superposition of three standing waves (see main text) and generating a two-dimensional hexagonal lattice with lattice constant $a$. The effective confining potential along the segment $\left(0,0\right)-\left(0,2a\right)$ [vertical red (dark gray) line in the upper right panel] is shown in the absence of a vector potential (blue circles) and for $\mathbf{A}\ne \mathbf{0}$ (red triangles). Inset: typical cluster used in the calculations consisting of a disk-shaped piece of hexagonal lattice with radius $R\approx 39a$.

Figure 2

(Color online) Spectrum for a cluster with a hard-wall boundary in the presence of a periodic vector potential with $\alpha =2$ (panel 1). The coordinates of each dot represent the orbital momentum in some arbitrary units (horizontal axis) and the energy $ϵ$ (vertical axis) of a particular state. The “edge” vs “bulk” character of each state is shown by the color code, which represents the relative boundary contribution to the norm ${\gamma }_n$ (see main text). ${\gamma }_n$ ranges from 1 for edge states (blue: the edge mode containing state B) to 0 for purely bulk states (red: the ”butterfly wings” corresponding to the upper and lower bands). Notice the chiral nature of the edge mode that populates the gap, which is responsible for the well-defined sign of the edge state orbital momentum. The DOS corresponding to this spectrum is shown in panel 2. For comparison, in panel 3 we show the DOS in the absence of a gauge potential.

Figure 3

(Color online) Left panels: contour plots of ${\rho }_n\left(\mathbf{r}\right)={\left|{\psi }_n\left(\mathbf{r}\right)\right|}^2$ for the states marked in Fig. 2. The quantity ${\rho }_n\left(\mathbf{r}\right)$ is the product between a common factor associated with the underlying hexagonal lattice structure and a state-dependent envelope function. The corresponding envelope functions are shown in the right panels. (a) represents the ground state, (b) is a typical edge state, and (c) is the lowest-energy bulk state from the upper band. The weak edge contributions in (c) are due to finite-size effects and vanish in the large cluster limit.

Figure 4

(Color online) Soft boundary confining potentials (top panels): (a) quartic wall, (b) linear step, and (c) harmonic trap plus infinite wall. The turquoise (gray, $V_c<0.025$) region corresponds to energies smaller than the gap ${\Delta }_{\alpha }$, while in the yellow (light gray, $V_c<0.125$) region the energy is smaller than the bandwidth $W$. A confining potential with characteristic length $d_c$ on the order of lattice spacing [(a) and (b)] preserves the bulk gap (topological insulator). The in-gap features appearing in the density of states (middle panels) are all due to edge states (see also Fig. 5). In a smoothly varying confining potential [case (c)], the bulk gap collapses for $V_c^c\left(r\right)>{\Delta }_{\alpha }$ (metal, see lower panel). In cases (a) and (b), the lower band is unaffected by the details of the confining potential, while the in-gap structures (middle panels) are similar. The rapid oscillations at low energies in case (c) indicate the formation of harmonic-oscillator levels.

Figure 5

(Color online) In-gap states for a system with soft boundary (quartic wall). Color code is the same as in Fig. 2. Note that all the in-gap states have edgelike character. In addition to the typical chiral edge states (a), notice the presence of Tamm-type edge states (b). These states can mix. For softer boundaries, proliferation of and mixing with the Tamm states will eventually destroy the topological insulating phase.