Identification of the fractional quantum Hall edge modes by density oscillations

Na Jiang and Zi-Xiang Hu

Phys. Rev. B 94, 125116 – Published 12 September 2016

Abstract

The neutral fermionic edge mode is essential to the non-Abelian topological property and its experimental detection in $Z_{k}$ fractional quantum Hall (FQH) state for $k > 1$ . Usually, the identification of the edge modes in a finite size system is difficult, especially near the region of the edge reconstruction, due to mixing with the bosonic edge mode and the bulk states as well. We study the edge-mode excitations of the Moore-Read (MR) and Read-Rezayi (RR) states by using Jack polynomials in the truncated subspace. It is found that the electron density, as a detector, has marked different behaviors between the bosonic and fermionic edge modes. As an application, it helps us to identify them near the edge reconstruction and in the RR edge spectrum. On the other hand, we systematically study the edge excitations for the RR state, extrapolating the edge velocities and their related coherence length and temperature in the interferometer experiments.

Received 18 June 2016
Revised 26 August 2016

DOI:https://doi.org/10.1103/PhysRevB.94.125116

Physics Subject Headings (PhySH)

Fractional quantum Hall effect

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Na Jiang and Zi-Xiang Hu^*

Department of Physics, Chongqing University, Chongqing, 401331, People's Republic of China

^*zxhu@cqu.edu.cn

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Issue

Vol. 94, Iss. 12 — 15 September 2016

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Images

Figure 1
(a) The energy spectrum for $N = 12$ electrons with pure Coulomb interaction at 5/2 filling when $d = 0.4 l_{B}$ by exact diagonalization. The global ground state locates in the subspace $Δ M = M - M_{MR} = 0$ where $M_{MR} = N (2 N - 3) / 2$ is the total angular momentum for MR state. (b) The comparison of edge spectrum for 12 electrons both from ED (plotted with bars) and truncated space (points) in the case of $λ = 0.5$ and $d = 0.4 l_{B}$ . The fermionic and bosonic edge states are labeled by circle and square points, respectively. The triangle points label the mixed states.
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Figure 2
The edge spectrum(solid bars) for MR state with 20 electrons (a) and RR state with 24 electrons (b) for pure Coulomb Hamiltonian with a background potential at distance $d = 0.2 l_{B}$ . For large system size, as we can see, the fermionic edge states are gapped from the bosonic ones for small $Δ M$ which indicates a “Bose-Fermi” separation while quasiparticle propagating along the edge.
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Figure 3
The radial density profiles for the MR edge states which include the bosonic states, fermionic states, and their mixture in (a) $Δ M = 2$ and (b) $Δ M = 3$ subspaces. Their corresponding density differentiation $D (r)$ as a function of $r$ are depicted in (c) and (d). The indexes of the state are one-to-one corresponding to the spectrum in Fig. 2.
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Figure 4
The same plots as in Fig. 3 for the RR edge states shown in Fig. 2. The $D (r)$ of the fermionic states is still larger than that of the bosonic edge states.
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Figure 5
(a) The edge spectrum for 16 electrons at $ν = 5 / 2$ after edge reconstruction. The background potential locates at $d = 0.6 l_{B}$ and the first reconstructed state is in $Δ M = 7$ . (b) The dispersion of the reconstructed edge modes for systems ranges from 10 to 20 electrons. The lower one is fermionic and the upper one is bosonic. The radial density profiles for the labeled states in (a) and their corresponding density differentiation $D (r)$ as a function of $r$ are shown in (c) and (d), respectively. From the density profiles and $D (r)$ in (c) and (d), we can identify the first reconstructed state is fermionic rather than bosonic.
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Figure 6
Finite-size scaling (9–24 electrons) of the edge-mode velocities for $ν = 13 / 5$ with Coulomb interaction and background confinement $d = 0.6 l_{B}$ . The charge/neutral mode velocity is fitted by a quadratic/linear function of $1 / N$ . In the thermodynamic limit, the velocity reads $v_{c} ≃ 0.54 e^{2} / ε ℏ$ and $v_{n} ≃ 0.01 e^{2} / ε ℏ$ .
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