Six-dimensional quantum Hall effect and three-dimensional topological pumps

Modern technological advances allow for the study of systems with additional synthetic dimensions. Using such approaches, higher-dimensional physics that was previously deemed to be of purely theoretical interest has now become an active field of research. In this work, we derive from first principles using a semiclassical equation-of-motion approach the bulk response of a six-dimensional Chern insulator. We find that in such a system a quantized bulk response appears with a quantization originating from a six-dimensional topological index: the third Chern number. Alongside this unique six-dimensional response, we rigorously describe the lower even-dimensional Chern-type responses that can occur due to nonvanishing first and second Chern numbers in subspaces of the six-dimensional space. Last, we propose how to realize such a bulk response using three-dimensional topological charge pumps in cold atomic systems.

Figure 1

Illustration of the relationship between a 2D quantum Hall system and a 1D topological charge pump. (a) The 2D quantum Hall effect on a square lattice, also referred to as the Harper-Azbel-Hofstadter (HAH) model [47, 48, 49] [cf. Eq. (30)]. (b) Written in the Landau gauge [as chosen in Eq. (30)], the HAH model can be written with periodic boundary conditions along the $y$ axis, i.e., you can wrap the 2D system onto a cylinder [see Eq. (31)]. Using Faraday's induction law, an electric field perturbation $E_y=\partial {\varphi }_x/\partial t$ is generated by threading a magnetic flux through the cylinder. (c) Using dimensional reduction [16, 37, 39, 40, 50], the 2D HAH model is mapped onto a 1D topological charge pump model, where particles hop on a 1D periodic lattice in the presence of an onsite cosine potential [see Eq. (32)]. As a function of a periodic modulation of the onsite potential, a quantized number of charges is pumped across the 1D system [cf. Eqs. (33) and (34)]. Plotted is the onsite cosine potential with $\overline{\alpha }=13/21$ and ${\varphi }_x=0$. Disks mark the discrete sampling of the potential at lattice sites.

Figure 2

Illustration of the relationship between a 6D quantum Hall system and a 3D topological charge pump. (a) The 6D quantum Hall effect on a 6D hypercube lattice with ${\alpha }_{xw}, {\alpha }_{yu}$, and ${\alpha }_{zv}$ threading the orthogonal $xw,yu$, and $zv$ planes, respectively [cf. Eq. (35)]. Note, we attempt to draw a 6D illustration using a 3D axes system: some imagination is required. An example of a third Chern quantized Hall response involves the generation of $E_v,B_{uz}$, and $B_{wy}$ [see Eq. (29)]. (b) Written in the Landau gauge [as chosen in Eq. (35)], the model can be written with periodic boundary conditions in the $w,u$, and $v$ axes, i.e., you can wrap the 6D system onto three coupled cylinders [see Eq. (37)]. Using Faraday's induction law, an electric field perturbation $E_v=\partial {\varphi }_z/\partial t$ is generated by threading a magnetic flux through the $zv$ cylinder. The perturbing magnetic fields $B_{uz}$ and $B_{wy}$ generate Lorentz forces $F_u$ and $F_w$ in the $u$ and $w$ axes, respectively. (c) Using dimensional reduction, the 6D model (35) is mapped onto a 3D topological charge pump model where particles hop on a 3D periodic lattice in the presence of an “egg-carton” onsite potential composed of a sum of three cosine potentials in the orthogonal physical axes $x,y, z$ [see Eq. (39)]. For clarity, the surface potential is plotted (left) with guiding dashed lines showing the skewness due to the magnetic fields. The bulk “egg-carton” potential (right) is also shown for completeness. As in the 1D pump case, as a function of a periodic modulation of the onsite potential in the $z$ direction, a quantized number of charges is pumped along that axis. Spatial deformations of the potential couple the motion in the $z$ direction onto motion in the $y$ axis, which then induces motion in the $x$ axis, as expected from this Lorentz-type 3D pumping response [cf. Eq. (41)].