Quasiparticle tunneling in the Moore-Read fractional quantum Hall state

In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges $e/4$ (non-Abelian) and $e/2$ (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for $e/2$ and $e/4$ quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge $e/2$ quasihole is exponentially smaller than that for charge $e/4$ quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\nu =5/2$.

Figure 1

(Color online) Theoretical setup for a disk of a fractional quantum Hall liquid, allowing quasiholes to tunnel through the bulk from the center to the edge.

Figure 2

(Color online) tunneling amplitude as a function of number of electrons for (a) $e/4$ and $e/2$ quasiholes in the Moore-Read state at half filling and (b) $e/3$ and $2e/3$ quasiholes in the Laughlin state at $\nu =1/3$.

Figure 3

(Color online) The ratio of tunneling matrix elements for $e/2$ quasiholes to $e/4$ quasiholes in the Moore-Read state at half filling, and for $2e/3$ quasiholes to $e/3$ quasiholes in the Laughlin state at 1/3 filling as a function of number of electrons.

Figure 4

(Color online) (a) The tunneling amplitude ${\Gamma }^{e/4}$ and (b) the tunneling amplitudes ${\Gamma }^{e/2}$ as functions of edge-to-edge distance $d\left(n,N\right)$. Data are shown up to $n=100$ quasiholes and $N=14$ electrons. The data points at $d=0$ or $n\to \infty$ limit are exact results as explained in Appendix .

Figure 5

(Color online) The ratio of tunneling amplitudes ${\Gamma }^{e/2}/{\Gamma }^{e/4}$ as a function of edge-to-edge distance $d\left(n,N\right)$. We also plot Eq. (17) as the solid line to guide the eye. The dashed line is the theoretical estimate based on the charge component only [Eq. (19)].

Figure 6

(Color online) (a) Energies of the $e/4$ and $e/2$ quasihole states $E_{\lambda }^{qh}$, measured from the corresponding ground state in the momentum $M_0$ subspace $E_{\lambda }^0$, as functions of the strength $W$ of the Gaussian trapping potential at the disk center with $s=2.0$ for the $\nu =5/2$ state in a 12-electron system with a mixed Hamiltonian $H_{\lambda }$ $\left(\lambda =0.5\right)$ and $D=0.6l_B$. The $e/4$ quasihole state is energetically favorable for $0.032<W<0.137$. (b) Overlaps of the $e/4$ and $e/2$ quasihole states $|{\Psi }_{\lambda }^{qh}⟩$ for the mixed Hamiltonian with the corresponding variational states $|{\Psi }_{\text{MR}}^{qh}⟩$ [Eqs. (4, 6)].

Figure 7

(Color online) The ratio of the tunneling amplitudes for a mixed Hamiltonian as a function of number of electrons. The mix parameter $\lambda =0.5$ and the width and strength of Gaussian potential are $W=0.1$ and $s=2.0$, respectively. The background potential is located at the distance $D=0.6l_B$. The dotted line is the exponentially decaying trend line [Eq. (13)] for the pure three-body case as shown in Fig. 3.

Figure 8

(Color online) The ratio of tunneling amplitudes ${\Gamma }^{e/2}/{\Gamma }^{e/4}$ as a function of the mixing parameter between the three-body and Coulomb interactions in a 12-electron system at half filling in the case of $D=0.6l_B$. The width and strength of Gaussian potential are $W=0.1$ and $s=2.0$, respectively. The (green) dot at $\lambda =1$ is the value for the short-range three-body interaction case obtained in Sec. 3A.

Figure 9

(Color online) Decoherence length $L_{\varphi }$ as a function of $D$ for both $e/4$ (upper line) and $e/2$ (lower line) quasipaticles in the Moore-Read Pfaffian state. We choose a temperature $T=25\text{mK}$ to allow a direct comparison with experiment (Ref. 50). The broken lines above $D=0.62l_B$ are obtained by extrapolation, as the Moore-Read-like ground state is no longer stable in a system of 12 electrons in 26 orbitals. We note that a stripe phase may emerge below $D=l_B$ (Ref. 18).