Non-Abelian Optical Lattices: Anomalous Quantum Hall Effect and Dirac Fermions

We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments. In particular, we show that the underlying Minkowski space-time can also be modified, reaching anisotropic regimes where a remarkable anomalous quantum Hall effect and a squeezed Landau vacuum could be observed.

Figure 1

(a) Square lattice subjected to a non-Abelian gauge potential. This external field induces state-dependent hoppings described by the U(2) operators $U_x$ and $U_y$. (b) Energy bands close to the $\pi$-flux regime (${\Phi }_{\alpha }=\pi /2+0.1$, ${\Phi }_{\beta }=\pi /2-0.1$), with vanishing Abelian flux $\Phi =0$. The bands touch at four Dirac points inside the first Brillouin zone (BZ), where the energy scales linearly with momenta $E\sim k$.

Figure 2

(a) Density of states (DOS) in the $\pi$-flux regime ${\Phi }_{\alpha }={\Phi }_{\beta }=\pi /2$ when $\Phi =0$. (b) Hall conductivity in units of $h^{-1}$ as a function of the Fermi energy in the same regime for $\Phi =1/41$. Black arrows designate the VHS. (c) DOS close to the $\pi$-flux regime ${\Phi }_{\alpha }=\pi /2+0.1$ and ${\Phi }_{\beta }=\pi /2-0.1$ when $\Phi =0$. (d) Hall conductivity $h{\sigma }_{xy}=h{\sigma }_{xy}(E_F)$ in the same regime as (c) for $\Phi =1/41$. Dark red and light green arrows, respectively, designate the VHS $E_{\mathrm{red}}^{\mathrm{VHS}}$ and $E_{\mathrm{green}}^{\mathrm{VHS}}$ [cf. Eq. (6)].

Figure 3

Vortex-like single-particle wave functions of the LLL ${\varphi }_{\mathrm{LLL}}^m(x,y)$ for $m=4$. (a), (b) Isotropic limit $c_x=c_y$. (c), (d) Anisotropic regime $c_y=2c_x$. Note that distances are measured in units of the magnetic length $l_B$.