Four-dimensional topological lattices through connectivity

Thanks to recent advances, the 4D quantum Hall (QH) effect is becoming experimentally accessible in various engineered setups. In this paper, we propose a 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.