Ergodic edge modes in the 4D quantum Hall effect

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^1S1 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.

Aspects of higher dimensional quantum Hall effect: Bosonization, entanglement entropy

“…In the presence of edge degrees of freedom the entanglement entropy for two-dimensional systems develops subleading logarithmic contributions 𝑆 edge ∼ 𝑐 6 log 𝑙, where 𝑐 is the central charge of the gapless edge modes and 𝑙 is the segment of the entangling boundary intersecting the chiral droplet. This was indeed confirmed in the case of the two-dimensional 𝜈 = 1 QHE droplets whose edge dynamics is characterized by a chiral scalar of central charge 𝑐 = 1 [23] and was recently extended to the case of a four-dimensional QHE analog with abelian magnetic field [24]. We have previously analyzed in detail higher dimensional QHE droplets, their corresponding edge effective actions and their spectrum [4,5].…”

Equipartition of Entanglement in Quantum Hall States

“…In our case, dealing with overlaps rather than the correlation operator (13) turns out to be simpler, as this trades the diagonalization of a continuous kernel for that of a discrete matrix. Translation invariance along y actually trivializes the problem since the overlap matrix (15) of LLL states (10) is automatically diagonal, with eigenvalues λ m labelled by momentum: where erfc is the complementary error function. Note the spectral flow (or charge pumping) that can be read off from (16): the spectrum returns to itself when the flux Φ increases by one unit, mapping λ m → λ m+1 .…”

“…For instance, the entanglement entropy (EE) of gapped phases of matter obeys an area law reminiscent of black hole entropy [7,8], while that of one-dimensional (1D) critical systems exhibits an anomalous logarithmic growth sensitive to the central charge [9][10][11]. In the quantum Hall effect (QHE) to be studied here, entanglement detects intrinsic topological order [12,13] and identifies gapless edge modes at the boundary [14,15] or at the interface between different fractional quantum Hall states [16][17][18]. It is thus a crucial theoretical probe of the properties of topological phases of matter.…”

Fractional Quantum Hall States on CP2 Space

“…Interactions give raise to exotic many-body states including the Laughlin phase 5 , which exhibits fractional Hall conductivity, and fractionalized particles or anyons 6 . Theoretically Landau levels have been predicted to exist in even dimensions 1,2,[7][8][9][10][11][12] , and examples have been realized in optical and cold atom experiments [13][14][15][16][17] .…”