Realizing and Detecting the Quantum Hall Effect without Landau Levels by Using Ultracold Atoms

We design an ingenious scheme to realize Haldane’s quantum Hall model without Landau levels by using ultracold atoms trapped in an optical lattice. Three standing-wave laser beams are used to construct a wanted honeycomb lattice, where different on site energies in two sublattices required in the model can be implemented through tuning the phase of one laser beam. The staggered magnetic field is generated from the light-induced Berry phase. Moreover, we establish a relation between the Hall conductivity and the atomic density, enabling us to detect the Chern number with the typical density-profile-measurement technique.

Figure 1

(a) Illustration of the Honeycomb lattice structure of graphene, where open and solid circles represent sites in sublattices $\overline{A}$ and $\overline{B}$. ${\mathbf{a}}_1=(\frac{1}{2}a,-\frac{\sqrt{3}}{2}a)$, ${\mathbf{a}}_2=(\frac{1}{2}a,\frac{\sqrt{3}}{2}a)$ are the unit vectors of the underlying triangular sublattice. $s_1$, $s_2$, and $s_3$ are three vectors pointing from a $\overline{B}$ site to its three nearest-neighbor sites. (b) and (c) show the contours of the potential V for $\chi =2\pi /3$ and $\chi =39\pi /60$, respectively. The vertical (horizontal) axis represents $y{k}_0^L/\pi$ ($x{k}_0^L/\pi$). (d) Contours of the magnetic field defined by Eq. (2).

Figure 2

(a) Phase diagram of the system, where the phase boundary (solid line) corresponds to $h_3=0$. (b)–(e) Charge density in units $\mathcal{B}/{\varphi }_0$ for $|m_{-}|=0.5|m_{+}|$ and $e\mathcal{B}\hslash v_F^2=4|m_{+}|$ as a function of normalized chemical potential $\mu /|m_{+}|$ for four different cases: (b) $m_{+}<0<m_{-}$, (c) $m_{+}>0>m_{-}$, (d) $m_{+}<m_{-}<0$, and (e) $m_{+}>m_{-}>0$.