Construction of edge states in fractional quantum Hall systems by Jack polynomials

We study the edge-mode excitations of a fractional quantum Hall droplet by expressing the edge-state wave functions as linear combinations of Jack polynomials with a negative parameter. We show that the exact diagonalization within a subspace of Jack polynomials can be used to generate the chiral edge-mode excitation spectrum in the Laughlin phase and the Moore-Read phase with realistic Coulomb interaction. The truncation technique for the edge excitations simplifies the procedure to reliably extract the edge-mode velocities, which avoids the otherwise complicated analysis of the full spectrum that contains both edge and bulk excitations. Generalization to the Read-Rezayi state is also discussed.

Figure 1

The comparison of the low-energy excitation spectra from the exact diagonalization in the full Hilbert space (red bars) and from the diagonalization within the edge Jack polynomial subspace (blue bars with an artificial shift in momentum for clarity). (a) The Laughlin case with nine electrons. (b) and (c) are for the Moore-Read case with 12 electrons. Here, $\lambda$ is the parameter for the mixed Hamiltonian with three-body interaction $H=(1-\lambda )H_{\text{Coulomb}}+\lambda H_{3B}$, as in Ref. [19]. $\lambda =0$ is the case of pure Coulomb interaction.

Figure 2

Finite-size scaling of the edge-mode velocities for $\nu =5/2$ with Coulomb interaction and confinement from neutralizing background charge. The charge-mode velocity is fitted by a quadratic function in $1/N$ and, in the thermodynamic limit, the velocity reads $0.44e^2/\epsilon \hslash$ (based on data from 8 to 20 particles). The neutral-mode velocity is fitted by a linear dependence on $1/N$. In the thermodynamic limit, the velocity reads $0.033e^2/\epsilon \hslash$.

Figure 3

Ferres diagrams for the constraints on the edge partitions for $k=2$ (left) and $k=3$ in the Read-Rezayi series with $r=2$ (right). The left Ferres diagram illustrates how an edge partition is determined for $k=2$. In the example, the edge partition can be bipartitioned into ${\eta }^{\left(1\right)}=\{7,6,5,5,2\}$ and an ${\eta }^{\left(2\right)}$ to be determined. The subpartition represented by the white cells (including the gray cells) is ${\eta }^{\left(1\right)}$ and the partition represented by the gray is a partition obtained by shifting ${\eta }^{\left(1\right)}$ upward by 1 and leftward by $r=2$. As discussed in the text, an allowed ${\eta }^{\left(2\right)}$ should be dominated by ${\eta }^{\left(1\right)}$ but dominates the gray-cell partition. The right Ferres diagram illustrates the case for $k=3$, assuming that ${\eta }^{\left(2\right)}$ is already selected based on the left Ferres diagram, e.g., ${\eta }^{\left(2\right)}=\{4,3,3\}$ (gray and dark-gray cells) for the same ${\eta }^{\left(1\right)}=\{7,6,5,5,2\}$ (white, gray, and dark-gray cells). ${\eta }^{\left(3\right)}$ should be dominated by ${\eta }^{\left(2\right)}$, but dominate the partition represented by dark grey cells, obtained by shifting ${\eta }^{\left(2\right)}$ upward by 1 and leftward by $r=2$.