Numerical study of hierarchical quantum Hall edge states in the disk geometry

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 multicomponent Abelian theory and the $W_{1+\infty }$ minimal theory of the incompressible fluids. We introduce two 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 multicomponent Abelian conformal theory. Most notably, we observe that the edge excitations form subfamilies of the low-lying states with a definite pattern, which is explained by the $W_{1+\infty }$ 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 $W_{1+\infty }$ minimal theory, within the finite-size limitations.