Probe of Three-Dimensional Chiral Topological Insulators in an Optical Lattice

We propose a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice, which is characterized by an integer topological invariant distinct from the conventional ${\mathbb{Z}}_2$ topological insulators and has a remarkable macroscopic zero-energy flat band. To probe its property, we show that its characteristic surface states—the Dirac cones—can be probed through time-of-flight imaging or Bragg spectroscopy and the flat band can be detected via measurement of the atomic density profile in a weak global trap. The realization of this novel topological phase with a flat band in an optical lattice will provide a unique experimental platform to study the interplay between interaction and topology and open new avenues for application of topological states.

Figure 1

Schematics of the laser configuration to realize the Hamiltonian in Eq. (3). Panel (a) shows the propagation direction (big arrows) and the polarization (small arrows) of each laser beam. (b) A linear tilt ${\mathrm{\Delta }}_{x,y,z}$ per site in the lattice along each direction. The detuning in each direction matches the frequency offset of the corresponding Raman beams, which are shown in panels (c), (d), and (e). Polarizations of each beam are shown in brackets. Rabi frequencies for each beam are: ${\mathrm{\Omega }}_1^{\pi }={\mathrm{\Omega }}_0{e}^{ikx}$, ${\mathrm{\Omega }}_2^{\pi }={\mathrm{\Omega }}_0{e}^{iky}$, ${\mathrm{\Omega }}_1^x=i\sqrt{2}{\mathrm{\Omega }}_0{e}^{ikz}$, ${\mathrm{\Omega }}_2^x=-i\sqrt{2}{\mathrm{\Omega }}_0{e}^{ikz}$, ${\mathrm{\Omega }}_1^y=-\sqrt{2}{\mathrm{\Omega }}_0{e}^{ikz}$, ${\mathrm{\Omega }}_2^y=\sqrt{2}{\mathrm{\Omega }}_0{e}^{ikz}$, ${\mathrm{\Omega }}^z=2i{\mathrm{\Omega }}_0{e}^{ikz}$(see the Supplemental Material [22]). (a) laser configuration. (b) tilted optical lattice. (c) $x$ direction. (d) $y$ direction. (e) $z$ direction.

Figure 2

(a) The topological index $\mathrm{\Gamma }$ as a function of the parameter $h$. (b) Energy dispersion for three bulk bands (surface plot) and surface states (mesh plot) at the boundary along the $z$ direction for $h=2$. (c) Quasimomentum distribution ${\rho }_{\text{cry}}(\mathbf{k})$ for various $h=0,0.5,1,1.5,2$ at a fixed chemical potential $\mu /2t=-2$ (see the Supplemental Material [22]). One hundred layers are taken along the $z$ direction with open boundaries in (b) and (c).

Figure 3

The atomic density profile $n$ as a function of the radial distance $r$ under the LDA. (a) $h=0$, ${\mu }_0/2t=3$. (b) $h=1$, ${\mu }_0/2t=4$. ${}^{40}\mathrm{K}$ is used and $t/\hslash$ is taken to be $2\pi \times 40\mathrm{Hz}$.