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Quantum Hall (QH) states are arguably the most ubiquitous examples ofnontrivial topological order, requiring no special symmetry and elegantlycharacterized by the first Chern number. Their higher dimension generalizationsare particularly interesting from both mathematical and phenomenologicalperspectives, and have attracted recent attention due to a few high profileexperimental realizations. In this work, we derive from first principles theelectromagnetic response of QH systems in arbitrary number of dimensions, andelaborate on the crucial roles played by their modified phase space density ofstates under the simultaneous presence of magnetic field and Berry curvature.We provide new mathematical results relating this phase space modification tothe non-commutativity of phase space, and show how they are manifested as aHall conductivity quantized by a higher Chern number. When a Fermi surface ispresent, additional response currents unrelated to these Chern numbers alsoappear. This unconventional response can be directly investigated through a fewminimal models with specially chosen fluxes. These models, together with moregeneric 6D QH systems, can be realized in realistic 3D experimental setups likecold atom systems through possibly entangled synthetic dimensions.

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