Photonic topological boundary pumping as a probe of 4D quantum Hall physics

A 2D topological charge pump in a photonic waveguide array is used to observe boundary physics associated with the 4D quantum Hall effect experimentally. Quantum Hall annexes fourth dimension The quantum Hall effect, discovered in the 1980s, is an important fundamental effect in condensed matter physics that links topological states with electronic properties in two-dimensional systems. The quantized conductance is prescribed by an integer global topological invariant and is therefore protected against perturbations. Such invariants are characterized by a so-called Chern number. Two papers in this issue experimentally confirm the prediction that the quantum Hall effect can be generalized to a four-dimensional (4D) system. Immanuel Bloch and colleagues implement the 4D quantum Hall system in a superlattice of ultracold bosonic atoms, and Mikael Rechtsman and colleagues achieve the same in a photonic waveguide array. Both groups find that their system harbours a second Chern number, as expected. The studies show an intriguing advance towards new physics provided by topological protection in higher dimensions. When a two-dimensional (2D) electron gas is placed in a perpendicular magnetic field, its in-plane transverse conductance becomes quantized; this is known as the quantum Hall effect 1 . It arises from the non-trivial topology of the electronic band structure of the system, where an integer topological invariant (the first Chern number) leads to quantized Hall conductance. It has been shown theoretically that the quantum Hall effect can be generalized to four spatial dimensions 2 , 3 , 4 , but so far this has not been realized experimentally because experimental systems are limited to three spatial dimensions. Here we use tunable 2D arrays of photonic waveguides to realize a dynamically generated four-dimensional (4D) quantum Hall system experimentally. The inter-waveguide separation in the array is constructed in such a way that the propagation of light through the device samples over momenta in two additional synthetic dimensions, thus realizing a 2D topological pump 5 , 6 , 7 , 8 . As a result, the band structure has 4D topological invariants (known as second Chern numbers) that support a quantized bulk Hall response with 4D symmetry 7 . In a finite-sized system, the 4D topological bulk response is carried by localized edge modes that cross the sample when the synthetic momenta are modulated. We observe this crossing directly through photon pumping of