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arXiv:1201.2105 · International Journal of Modern Physics Conference Series

BULK AND EDGE QUASIHOLE TUNNELING AMPLITUDES IN THE LAUGHLIN STATE

ZI-XIANG HU, KI HOON LEE, XIN WAN

Published Jan 1, 2012• 3 citations
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Abstract

The tunneling between the Laughlin state and its quasihole excitations are studied by using the Jack polynomial. We find a universal analytical formula for the tunneling amplitude, which can describe both bulk and edge quasihole excitations. The asymptotic behavior of the tunneling amplitude reveals the difference and the crossover between bulk and edge states. The effects of the realistic coulomb interaction with a background-charge confinement potential and disorder are also discussed. The stability of the tunneling amplitude manifests the topological nature of fractional quantum Hall liquids.

DOI
10.1142/s2010194512006174

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The tunneling between the Laughlin state and its quasihole excitations are studied by using the Jack polynomial. We find a universal analytical formula for the tunneling amplitude, which can describe both bulk and edge quasihole excitations. The asymptotic behavior of the tunneling amplitude reveals the difference and the crossover between bulk and edge states. The effects of the realistic coulomb interaction with a background-charge confinement potential and disorder are also discussed. The stability of the tunneling amplitude manifests the topological nature of fractional quantum Hall liquids.
In a Laughlin fractional quantum Hall state, one- and two-quasihole states can be obtained by diagonalizing the many-body Hamiltonian with a trapping potential or, for larger systems, from the linear combination of the edge Jack polynomials. The quasihole states live entirely in the subspace of the lowest-energy branch in the energy spectrum with a fixed number of orbits, or a hard-wall confinement. The reduction in the Hilbert space dimension facilitates the study of time evolution of the quasihole states after, say, the removal of the trapping potential. We explore the quench dynamics under a harmonic external potential, which rotates the quasiholes in the droplet, and discuss the effect of long-range interaction and more realistic confinement. Accurate evaluation of the mutual statistics phase of anyons for a wide range of anyon separation can be achieved from the Berry-phase calculation.
Fractional quantum Hall systems are often described by model wave functions, which are the ground states of pure systems with short-range interaction. A primary example is the Laughlin wave function, which supports Abelian quasiparticles with fractionalized charge. In the presence of disorder, the wave function of the ground state is expected to deviate from the Laughlin form. We study the disorder-driven collapse of the quantum Hall state by analyzing the evolution of the ground state and the single-quasihole state. In particular, we demonstrate that the quasihole tunneling amplitude can signal the fractional quantum Hall phase to insulator transition.
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/4e/4e/4 (non-Abelian) and e/2e/2e/2 (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for e/2e/2e/2 and e/4e/4e/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/2e/2e/2 quasihole is exponentially smaller than that for charge e/4e/4e/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 $\ensuremath{\nu}=5/2$.
In a microscopic model of fractional quantum Hall liquids with electron-electron interactions and confinement, we calculate the edge Green's function via exact diagonalization. Our results for nu=1/3 and 2/3 suggest that, in the presence of Coulomb interaction, "external" parameters such as the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. In particular, we do not find any direct dependence of this exponent on the range of the interaction potential as suggested by recent calculations in contradiction to the topological nature of the edge.
A general description of weak and strong tunneling fixed points is developed in the chiral-Luttinger-liquid model of quantum Hall edge states. Tunneling fixed points are a subset of ``termination'' fixed points, which describe boundary conditions on a multicomponent edge. The requirement of unitary time evolution for bounded edges gives a nontrivial consistency condition for possible low-energy boundary conditions. The effects of interactions and random hopping on fixed points are studied through a perturbative renormalization-group (RG) approach which generalizes the Giamarchi-Schulz RG for disordered Luttinger liquids to broken left-right symmetry and multiple modes. We apply our approach to a number of examples, such as tunneling between a quantum Hall edge and a superconductor and tunneling between two quantum Hall edges in the presence of interactions. Interactions are shown to induce a continuous renormalization of effective tunneling charge for the integrable case of tunneling between two Laughlin states.
We consider a series of problems regarding quantum Hall edges, focusing on both dynamics and the mathematical structure of edge states. We begin in Chapter 3 with a limiting case of the Laughlin state placed in a very steep confining potential, but which is weak compared to the interactions. We find that the eigenstates have a Jack polynomial structure and an energy spectrum which is extremely different from the well-known Luttinger liquid edge. In Chapter 5 we analyse the inner products of edge state wavefunctions, using an effective description given by a large-N expansion ansatz proposed by J. Dubail, N. Read and E. Rezayi, PRB 86, 245310 (2012). As noted by these authors, the terms in this ansatz can be constrained using symmetry, a procedure we perform to high orders. We then check the conjecture by calculating overlaps exactly for small system sizes and comparing the numerics with our high-order expansion to find excellent agreement. Finally, Chapter 6 considers the behaviour of quantum Hall edges close to the Luttinger liquid fixed point that occurs in the low energy, large system limit. We construct effective Hamiltonians using a local field theory description and then consider the effect of bulk symmetries on this edge. The symmetry analysis produces remarkable simplifications which allow for very accurate descriptions of the low-energy edge physics even relatively far away from the Luttinger liquid fixed point.