  Thanks to recent advances, the 4D quantum Hall (QH) effect is becoming
experimentally accessible in various engineered set-ups. In this paper, we
propose a new type of 4D topological system that, unlike other 2D and 4D QH
models, does not require complicated (artificial) gauge fields and/or
time-reversal symmetry breaking. Instead, we show that there are 4D QH systems
that can be engineered for spinless particles by designing the lattice
connectivity with real-valued hopping amplitudes, and we explain how this
physics can be intuitively understood in analogy with the 2D Haldane model. We
illustrate our discussion with a specific 4D lattice proposal, inspired by the
widely-studied 2D honeycomb and brickwall lattice geometries. This also
provides a minimal model for a topological system in Class AI, which supports
nontrivial topological band invariants only in four spatial dimensions or
higher.
