Numerical Study of Hierarchical Hall Edge States on the Disk Geometry

TL;DR

This paper numerically analyzes the edge states of hierarchical quantum Hall systems on a disk, confirming predictions of conformal field theories and introducing new criteria for identifying edge excitations.

Contribution

It provides a detailed numerical study of edge excitations in hierarchical quantum Hall states, validating conformal field theory descriptions and introducing novel identification methods.

Findings

Jain states accurately reproduce bulk and edge excitations.

Edge excitations form patterns explained by W-infinity minimal theory.

The two conformal theories are connected by a projection mechanism.

Abstract

We present a detailed analysis of the exact numerical spectrum of up to ten interacting electrons in the first Landau level on the disk geometry. We study the edge excitations of the hierarchical plateaus and check the predictions of two relevant conformal field theories: the multi-component Abelian theory and the W-infinity minimal theory of the incompressible fluids. We introduce two new criteria for identifying the edge excitations within the low-lying states: the plot of their density profiles and the study of their overlaps with the Jain wave functions in a meaningful basis. We find that the exact bulk and edge excitations are very well reproduced by the Jain states; these, in turn, can be described by the multi-component Abelian conformal theory. Most notably, we observe that the edge excitations form sub-families of the low-lying states with a definite pattern, which is explained by the W-infinity minimal conformal theory. Actually, the two conformal theories are related by a projection mechanism whose effects are observed in the spectrum. Therefore, the edge excitations of the hierarchical Hall states are consistently described by the W-infinity minimal theory, within the finite-size limitations.