The discovery of topological states of matter has greatly improved our understanding of phase transitions in physical systems. Instead of being described by local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example is the two-dimensional (2D) integer quantum Hall effect1: it is characterized by the first Chern number, which manifests in the quantized Hall response that is induced by an external electric field2. Generalizing the quantum Hall effect to four-dimensional (4D) systems leads to the appearance of an additional quantized Hall response, but one that is nonlinear and described by a 4D topological invariant—the second Chern number3,4. Here we report the observation of a bulk response with intrinsic 4D topology and demonstrate its quantization by measuring the associated second Chern number. By implementing a 2D topological charge pump using ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect5,6. Using a small cloud of atoms as a local probe, we fully characterize the nonlinear response of the system via in situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probing higher-dimensional quantum Hall systems, in which additional strongly correlated topological phases, exotic collective excitations and boundary phenomena such as isolated Weyl fermions are predicted4.
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We acknowledge discussions with M. Aidelsburger and I. Carusotto. This work was funded by the European Commission (UQUAM, SIQS), the Deutsche Forschungsgemeinschaft (DIP, FOR2414) and the Nanosystems Initiative Munich. M.L. was additionally supported by the Elitenetzwerk Bayern (ExQM), H.M.P. by the European Commission (FET Proactive, grant no. 640800 ‘AQuS’, and Marie Skłodowska–Curie Action, grant no. 656093 ‘SynOptic’) and the Autonomous Province of Trento (SiQuro), and O.Z. by the Swiss National Science Foundation.
The authors declare no competing financial interests.
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Extended data figures and tables
Slope of the nonlinear response at and θ = 0.54(3) mrad as a function of Vs,y, with all other lattice parameters as in Figs 3 and 4. with is the maximum intra-double-well tunnelling rate along y, which is calculated from the corresponding lattice depth. The solid line indicates the theoretically expected slope and the error bars show the fit error for the slope. The dashed line at Vs,y = 6.25Er,s marks the point at which a topological transition occurs in the first excited subband along y, indicating the transition between the Landau regime for Vs,y < 6.25Er,s and the Hofstadter regime for Vs,y > 6.25Er,s.
The 4D tight-binding parameter space (δJx, Δx, δJy, Δy) is visualized using the transformation in equation (4). a, Changing the pump parameter φx leads to a periodic modulation of δJx and Δx along a closed trajectory, as shown in the inset for a full pump cycle φx = 0 → 2π. This pump path (green) encircles the degeneracy point at the origin (grey), at which the gap between the two lowest subbands of the Rice–Mele model closes. The surface in the main plot shows the same trace transformed according to equation (4) and with φy ∈ [0.46π, 0.54π]. The spacing of the mesh grid illustrating φx is π/10. b, For a given φx, a large system simultaneously samples all values of φy. This corresponds to a closed path in δJy–Δy parameter space, in which a singularity also occurs at the origin (inset). The main plot shows the transformed path for φx ∈ [0.46π, 0.54π]. c, In a full pump cycle, such a system therefore covers a closed surface in the 4D parameter space by translating the path shown in b along the trajectory from a. d, In the transformed parameter space, the singularities at (δJx = 0, Δx = 0) and (δJy = 0, Δy = 0) correspond to two planes that touch at the origin. e, Cut around r3 = 0 showing both the pump path from c (red/blue) and the singularities from d (grey). Whereas they intersect in the 3D space (r1, r2, r3), the value of r4 is different on both surfaces and the 4D pump path thus fully encloses the degeneracy planes.
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Lohse, M., Schweizer, C., Price, H. et al. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018). https://doi.org/10.1038/nature25000
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