  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 work, we derive from first principles the
electromagnetic response of QH systems in 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 non-commutativity 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.

Strongly Correlated Electrons;Applied Physics