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arXiv:cond-mat/0511674

Incidence of the boundary shape in the tunnelling exponent of electrons into fractional quantum Hall edges

D. C. Cabra, Nicolás Grandi

Published Nov 28, 2005• 2 citations
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Abstract

Experiments on tunneling into fractional quantum Hall droplets systematically found tunneling exponents smaller than those predicted by the ordinary chiral Luttinger liquid theory. In this note, by considering the effects of a smooth boundary, we propose a modification of the theory that predicts a reduced exponent, in qualitative agreement with experiments. Such modification consists on a self-interaction cubic term, containing higher derivatives of the chiral boson field, that provides a non-universal one loop correction to the propagator.

DOI
10.48550/arxiv.cond-mat/0511674

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Experiments on tunneling into fractional quantum Hall droplets systematically found tunneling exponents smaller than those predicted by the ordinary chiral Luttinger liquid theory. In this note, by considering the effects of a smooth boundary, we propose a modification of the theory that predicts a reduced exponent, in qualitative agreement with experiments. Such modification consists on a self-interaction cubic term, containing higher derivatives of the chiral boson field, that provides a non-universal one loop correction to the propagator.
The effective field theory of the fractional quantum Hall edge is reformulated from microscopic dynamics. Noncommutative Chern-Simons theory is a microscopic description for the quantum Hall fluid. We use it for reference. Considering relabeling symmetry of the electrons and incompressibility of the fluid, we obtain a constraint and derive a chiral Tomonaga-Luttinger liquid theory containing interaction terms. We calculate one-loop corrections to the phonon and electron propagators and get a new tunneling exponent. It agrees with experiments.
We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.
This study builds upon the work of Palacios and MacDonald [Phys. Rev. Lett. 76, 118 (1996)], wherein they identify the bosonic excitations of Wen's approach for the edge of the 1∕31∕31∕3 fractional quantum Hall state with certain operators introduced by Stone. Using a quantum Monte Carlo method, we extend this approach to larger systems containing up to 40 electrons and obtain more accurate thermodynamic limits for various matrix elements for a short-range interaction. The results are in agreement with those of Palacios and MacDonald for small systems, but offer further insight into a detailed approach to the thermodynamic limit. For the short range interaction, the results are consistent with chiral Luttinger liquid predictions. We also study excitations using the Coulomb ground state for up to nine electrons to ascertain the effect of interactions on the results; in this case, our tests of the chiral Luttinger liquid approach are inconclusive.
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 describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
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.