Quantum anomalous Hall states in the $p$-orbital honeycomb optical lattices

We study the quantum anomalous Hall states in the $p$-orbital bands of the honeycomb optical lattices loaded with single-component fermions. Such an effect has not yet been realized in both condensed-matter and cold-atom systems. By applying the available experimental techniques to rotate each lattice site around its own center, the band structures become topologically nontrivial. At a certain rotation angular velocity $\Omega$, a flat band structure appears with localized eigenstates carrying chiral current moments. By imposing the soft confining potential, the density profile exhibits a wedding-cake-shaped distribution with insulating plateaus at commensurate fillings. Moreover, the inhomogeneous confining potential induces dissipationless circulation currents, the magnitudes and chiralities of which vary with the distance from the trap center. In the insulating regions, the Hall conductances are quantized, and in the metallic regions, the directions and magnitudes of chiral currents can not be described by the usual local-density approximation. The quantum anomalous Hall effects are robust at temperature scales that are small compared to band gaps, which increase the feasibility of experimental realizations.

Figure 1

(a) The sketch of the honeycomb optical lattice. The three laser beams cross each other at 120${}^{°}$ in the $\mathit{xy}$ plane. Phase modulators are placed in the paths of the two beams. (b) The illustration of the on-site rotation by Gemelke et al. [56, 57, 58]. The entire lattice takes the motions of a fast oscillation with the frequency ${\omega }_{\text{RF}}$ and a slow precession with the frequency $\Omega$, as schematically plotted with the red solid lines in one lattice site. After taking the time average of the fast oscillation, atoms feel that each site is rotating around its own center with the precession frequency $\Omega$, which is plotted with the red dot-dashed line around each site.

Figure 2

The pattern of the induced complex-valued NNN hopping at $\Omega \gg t_{\parallel }$, which is generated by the virtual hopping between orbitals with opposite chiralities (from Wu [53]).

Figure 3

The band structures of the Hamiltonian equation (11). With increasing rotation velocity from (a) $\Omega /t_{\parallel }=0$ to (f) $\Omega /t_{\parallel }=1.7$. The flat bands appear at $\Omega /t_{\parallel }=\frac{3}{4}$. Energies are in units of hopping strength $t_{\parallel }$ and wave vectors are in the units of $1/a$, which is the reciprocal of the lattice constant. The energies of these two flat bands are $E/t_{\parallel }=\pm \frac{3}{4}$.

Figure 4

Configurations of the localized eigenstates of the two flat bands at $\Omega /t_{\parallel }=\frac{3}{4}$ for (a) $E/t_{\parallel }=-\frac{3}{4}$ and (b) $E/t_{\parallel }=\frac{3}{4}$, respectively. $\omega =e^{i\frac{\pi }{3}}$ is the relative phase factor between neighboring sites. The plaquette currents directions are clockwise for both (a) and (b) in the opposite direction of the $\Omega$.

Figure 5

The Berry curvature $F_{\mathit{xy}}$ (in units of $a^2$) distribution at $\Omega /t_{\parallel }=\frac{3}{4}$ for the (a) first and (b) second bands. The Chern numbers for the first and second bands are 1 and 0, respectively. The unit of the wave vector is $1/a$.

Figure 6

The band structures with both $\pi$ bonding $(t_{\perp }/t_{\parallel }=0.05)$ and on-site rotation $\Omega$. Only the lower two bands are presented, and the spectra of the other two bands are symmetric with respect to zero energy. (a) At $\Omega /t_{\parallel }=0$, the two bands remain touching at the center of the BZ. The lowest band develops dispersion at the order of $t_{\parallel }$. (b) At $\Omega /t_{\parallel }=0.2$, the gap opens between the lower two bands. The lowest band is topologically nontrivial and nearly flat.

Figure 7

The DOS of the $p_{x,y}$-orbital model [Eqs. (5) and (4)] in the homogeneous system. The red solid lines are the DOS at $\Omega /t_{\parallel }=\frac{1}{2}$, whereas the blue dashed lines are the DOS at $\Omega /t_{\parallel }=\frac{3}{4}$.

Figure 8

The particle-density distributions $\langle n(\mathrm{r⃗})\rangle$ vs the radius $r$ at (a) $\Omega /t_{\parallel }=\frac{1}{2}$ and (b) $\frac{3}{4}$, which exhibit a four-layer wedding-cake shape (the blue dots). The radius is in the units of the lattice constant $a$. The DOS at the local chemical potential ${\mu }_{\text{loc}}(r)$ in the LDA approximation is plotted with the black dashed lines. ${\mu }_{\text{loc}}(r)/t_{\parallel }=2.3$ at the center of the trap and ${\omega }_T/t_{\parallel }=0.1$.

Figure 9

The effective local Hall conductance ${\sigma }_{\rho \theta }(r)$ vs $r$ defined in Eq. (31) at (a) $\Omega /t_{\parallel }=\frac{1}{2}$ and (b) $\frac{3}{4}$. The radius is in the unit of the lattice constant $a$. The results from diagonalizing the free Hamiltonian with the trapping potential is marked with asterisks, and those from the modified LDA are plotted with dashed lines. ${\sigma }_{\rho \theta }$ is quantized in the insulating plateaus with commensurate fillings of $\langle n(r)\rangle$.

Figure 10

The pattern of the anomalous Hall current in the honeycomb lattice with the confining trap at the rotation of $\Omega /t_{\parallel }=0.75$. The blue dotted (red solid) lines represent the counterclockwise (clockwise) anomalous Hall currents, respectively. The color depth indicates the magnitude of the current. The reversed direction of the anomalous Hall currents (red solid lines) in the metallic regions between two neighboring plateaus can be explained as the anomalous contribution from the gradient of $\langle n(r)\rangle$.

Figure 11

The particle-density distributions $\langle n(\mathrm{r⃗})\rangle$ vs the radius $r$ at (a) $\Omega /t_{\parallel }=\frac{3}{2}$ and (b) $2$ (the blue dots). The radius is in the units of the lattice constant $a$. The DOS at the local chemical potential ${\mu }_{\text{loc}}(r)$ in the LDA approximation is plotted with the black dashed lines. ${\mu }_{\text{loc}}(r)/t_{\parallel }=3.5$ at the center of the trap and ${\omega }_T/t_{\parallel }=0.1$.

Figure 12

The effective local Hall conductance ${\sigma }_{\rho \theta }(r)$ vs $r$ defined in Eq. (31) at (a) $\Omega /t_{\parallel }=\frac{3}{2}$ and (b) $2.$ The radius is in the units of the lattice constant $a$. The results from diagonalizing the free Hamiltonian with the trapping potential is marked with asterisks, and those from the modified LDA are plotted with dashed lines.

Figure 13

The radial distributions of (a) the anomalous Hall conductance ${\sigma }_{\rho \theta }(\mathrm{r⃗},T)$ and (b) the entropy $S(\mathrm{r⃗},T)$ with $\Omega /t_{\parallel }=\frac{3}{4}$. The black dot, blue plus sign, magenta diamonds, and red asterisks of the data represent temperatures at $T=0$, $T/t_{\parallel }=0.1$, $T/t_{\parallel }=0.2$, and $T/t_{\parallel }=0.5$, respectively.