  The gapless modes on the edge of four-dimensional (4D) quantum Hall droplets
are known to be anisotropic: they only propagate in one direction, foliating
the 3D boundary into independent 1D conduction channels. This foliation is
extremely sensitive to the confining potential and generically yields chaotic
flows. Here we study the quantum correlations and entanglement of such edge
modes in 4D droplets confined by harmonic traps, whose boundary is a squashed
three-sphere. Commensurable trapping frequencies lead to periodic trajectories
of electronic guiding centers; the corresponding edge modes propagate
independently along $S^1$ fibers, forming a bundle of 1D conformal field
theories over a 2D base space. By contrast, incommensurable frequencies produce
quasi-periodic, ergodic trajectories, each of which covers its invariant torus
densely; the corresponding correlation function of edge modes has fractal
features. This wealth of behaviors highlights the sharp differences between 4D
Hall droplets and their 2D peers; it also exhibits the dependence of 4D edge
modes on the choice of trap, suggesting the existence of observable
bifurcations due to droplet deformations.
