Electromagnetic response of quantum Hall systems in dimensions five and six and beyond

Quantum Hall (QH) states are arguably the most ubiquitous examples of nontrivial topological order, requiring no special symmetry and elegantly characterized by the first Chern number. Their higher dimension generalizations are particularly interesting from both mathematical and phenomenological perspectives, and have attracted recent attention due to a few high profile experimental realizations. In this paper, we derive from first principles the electromagnetic response of QH systems in an arbitrary number of dimensions, and elaborate on the crucial roles played by their modified phase-space density of states under the simultaneous presence of magnetic field and Berry curvature. We provide new mathematical results relating this phase-space modification to the noncommutativity of phase space, and show how they are manifested as a Hall conductivity quantized by a higher Chern number. When a Fermi surface is present, additional response currents unrelated to these Chern numbers also appear. This unconventional response can be directly investigated through a few minimal models with specially chosen fluxes. These models, together with more generic 6D QH systems, can be realized in realistic 3D experimental setups like cold atom systems through possibly entangled synthetic dimensions.

Figure 1

Cartoon picture of electrons moving under the simultaneous influence of magnetic field and Berry curvature. They exhibit “cylotron” orbits in phase space, thereby behaving like “smudged” particles with uncertainties in their coordinates proportional to the commutator between their phase-space coordinates.

Figure 2

The energy spectrum of the 6D AAH model [Eq. (42)] with ${\lambda }_1={\lambda }_2={\lambda }_3=3t=1.5$ and $b=1/4$ numerically solved with a $4\times 4\times 4$ supercell for (a) periodic and (b) open boundary conditions, where gapless states (small black boxes) that traverse the bulk gap appear, indicative of a nontrivial Chern number. The density plot of the energy eigenvalues of every lattice point in our $4\times 4\times 4$ supercell is plotted in (c) at $k_w=k_t=k_x=k_y=k_z=0,k_s=-0.783$ for open boundaries in $x$ direction. The boundary states are clearly localized in the boundaries of $x$ direction, i.e., the Y-Z plane.

Figure 3

The energy spectrum of the entangled 6D AAH model [Eq. (45)], also with ${\lambda }_{=}{\lambda }_2={\lambda }_3=3t=1.5$ and $b=1/4$, for (a) periodic and (b) open boundary conditions. Due to the nontrivial coupling between the different copies of AAH models, numerous additional subbands appear. Gapless degenerate topological boundary states (small black boxes) that traverse the bulk gap appear, and are similarly plotted in (c) as in the previous figure except $k_s=-1.553$. The boundary states are also clearly localized in the Y-Z plane.

Figure 4

The cold-atom setup for the 6D quantum Hall (QH) system of Eq. (46), with external magnetic fields $B_{xw}$ and $B_{yt}$, illustrated for the unentangled case $R={\mathbb{I}}_{3\times 3}$. (a) 3D cartoon of the cold-atom system. The lattice sites are formed by the potential in Fig. 4, with each sphere denoting a cold atom. (b) The system is modulated periodically by the sum of two subpotentials (orange), as given by Eq. (46). The blue potential denotes the on-site modulation we set to the system and the red potential denotes the potential that pins the cold atoms to the lattice sites.