Ergodic Edge Modes in the 4D Quantum Hall Effect

TL;DR

This paper explores the complex behavior of edge modes in 4D quantum Hall systems under harmonic confinement, revealing how trapping frequencies influence their propagation, correlations, and entanglement, with implications for observable bifurcations.

Contribution

It introduces a detailed analysis of 4D quantum Hall edge modes under harmonic traps, showing how their dynamics depend on trap parameters and exhibit fractal and ergodic features.

Findings

Periodic trajectories form bundles of 1D conformal field theories.

Incommensurable frequencies lead to ergodic, fractal correlation functions.

Edge mode behavior varies sharply with trap deformation.

Abstract

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.