Four-Dimensional Quantum Hall Effect with Ultracold Atoms

We propose a realistic scheme to detect the 4D quantum Hall effect using ultracold atoms. Based on contemporary technology, motion along a synthetic fourth dimension can be accomplished through controlled transitions between internal states of atoms arranged in a 3D optical lattice. From a semiclassical analysis, we identify the linear and nonlinear quantized current responses of our 4D model, relating these to the topology of the Bloch bands. We then propose experimental protocols, based on current or center-of-mass-drift measurements, to extract the topological second Chern number. Our proposal sets the stage for the exploration of novel topological phases in higher dimensions.

Figure 1

(a) The 4D lattice in the absence of perturbing fields ($E_{\mu }$, $B_{\mu \nu }=0$). Atoms move in a 3D optical lattice, with hopping amplitude $J$. A Hofstadter model [58] is realized in the $x\text{−}z$ plane, with flux $2\pi {\mathrm{\Phi }}_1$ per unit cell, from $x$-dependent Peierls phase factors in the hopping along the $z$ direction [5, 6, 7]. A fourth (synthetic) dimension [11, 12], with coordinate $w$, is accessible at each lattice site, as Raman coupling induces internal-state transitions [35]. A second Hofstadter model, with flux $2\pi {\mathrm{\Phi }}_2$, is realized in the $y\text{−}w$ plane by adding a $y$-dependent phase factor to the internal-state transitions, i.e., by aligning the Raman wave vector along the $y$ direction [12, 15, 16]. (b) Energy spectrum $\mathcal{E}(k_x,k_y)$ for ${\mathrm{\Phi }}_{1,2}=1/4$, and for many values $k_{z,w}$ over the Brillouin zone; the 2CN of the lowest (nondegenerate) band is indicated.

Figure 2

(a) Current density $j^{\mu }(t)$ after ramping up the electric field to $E_y=-0.2J/a$. The fluxes were taken ${\mathrm{\Phi }}_{1,2}=1/4$, and the Fermi energy $E_{\mathrm{F}}=-5J$ was set in the first bulk energy gap [see Fig. 1]. The perturbing magnetic field was chosen as $B_{zw}=-2\pi \tilde{\mathrm{\Phi }}/a^2=-\pi /5a^2$. Simulations were performed on a $8\times 10\times 10\times 4$ lattice, with periodic boundary conditions along the $x$, $z$, $w$ directions. The component $j^w$ was also calculated for ${\mathrm{\Phi }}_1=1/5$ (red dotted line). Black lines show predictions [Eqs. (8) and (9)] for ${\nu }_2={\nu }_1^{yw}=-1$. (b) Center-of-mass trajectory $x_{\mathrm{c}.\mathrm{m}.}(t)$ after ramping up the electric field to $E_y=0.2J/a$; the nonlinear response is in the opposite direction to (a) as $E_y$ is reversed. Simulations are for a $12\times 12\times 10\times 4$ lattice, with periodic boundary conditions along the $z$, $w$ directions. The initial radius of the cloud was $r_0=4a$ in the $x\text{−}y$ plane. The fluxes are ${\mathrm{\Phi }}_{1,2}=1/4$, with all other parameters the same as in (a). The red dotted curve shows the predicted drift, $v_{\mathrm{c}.\mathrm{m}.}^x=j^x\times 16a^4=-8a^2{\nu }_2{E}_y\tilde{\mathrm{\Phi }}/\pi$, for ${\nu }_2=-1$.