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arXiv:cond-mat/9408089 · Physical review. B, Condensed matter

Laughlin state on stretched and squeezed cylinders and edge excitations in the quantum Hall effect

Edward H. Rezayi, F. D. M. Haldane

Published Dec 15, 1994• 138 citations
arXivHTMLPDF

Abstract

We study the Laughlin wave function on the cylinder. We find it only describes an incompressible fluid when the two lengths of the cylinder are comparable. As the radius is made smaller at fixed area, we observe a continuous transition to the charge-density-wave Tao-Thouless state. We also present some exact properties of the wave function in its polynomial form. We then study the edge excitations of the quantum Hall incompressible fluid modeled by the Laughlin wave function. The exponent describing the fluctuation of the edge predicted by recent theories is shown to be identical with numerical calculations. In particular, we obtain the occupation amplitudes of edge state n(k) for four to ten electron-size systems. When plotted as a function of the scaled wave vector they become essentially free of finite-size effects. The resulting curve obtains a very good agreement with the appropriate infinite-size Calogero-Sutherland model occupation numbers. Finally, we numerically obtain n(k) of the edge excitations for some pairing states which may be relevant to the \ensuremath{\nu}=5/2 incompressible Hall state.

DOI
10.1103/physrevb.50.17199

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We study the Laughlin wave function on the cylinder. We find it only describes an incompressible fluid when the two lengths of the cylinder are comparable. As the radius is made smaller at fixed area, we observe a continuous transition to the charge-density-wave Tao-Thouless state. We also present some exact properties of the wave function in its polynomial form. We then study the edge excitations of the quantum Hall incompressible fluid modeled by the Laughlin wave function. The exponent describing the fluctuation of the edge predicted by recent theories is shown to be identical with numerical calculations. In particular, we obtain the occupation amplitudes of edge state n(k) for four to ten electron-size systems. When plotted as a function of the scaled wave vector they become essentially free of finite-size effects. The resulting curve obtains a very good agreement with the appropriate infinite-size Calogero-Sutherland model occupation numbers. Finally, we numerically obtain n(k) of the edge excitations for some pairing states which may be relevant to the \ensuremath{\nu}=5/2 incompressible Hall state.
We present a detailed analysis of the exact numerical spectrum of up to ten interacting electrons in the first Landau level on the disk geometry. We study the edge excitations of the hierarchical plateaus and check the predictions of two relevant conformal field theories: the multicomponent Abelian theory and the ${W}_{1+\ensuremath{\infty}}$ minimal theory of the incompressible fluids. We introduce two criteria for identifying the edge excitations within the low-lying states: the plot of their density profiles and the study of their overlaps with the Jain wave functions in a meaningful basis. We find that the exact bulk and edge excitations are very well reproduced by the Jain states; these, in turn, can be described by the multicomponent Abelian conformal theory. Most notably, we observe that the edge excitations form subfamilies of the low-lying states with a definite pattern, which is explained by the ${W}_{1+\ensuremath{\infty}}$ minimal conformal theory. Actually, the two conformal theories are related by a projection mechanism whose effects are observed in the spectrum. Therefore, the edge excitations of the hierarchical Hall states are consistently described by the ${W}_{1+\ensuremath{\infty}}$ minimal theory, within the finite-size limitations.
Abstract The description of chiral quantum incompressible fluids by the W∞ symmetry can be extended from the edge, where it encompasses the conformal field theory approach, to the non-conformal bulk. The two regimes are characterized by excitations with different sizes, energies and momenta within the disk geometry. In particular, the bulk quantities have a finite limit for large droplets. We obtain analytic results for the radial shape of excitations, the edge reconstruction phenomenon and the energy spectrum of density fluctuations in Laughlin states.
We study fractional quantum Hall states in the cylinder geometry with open boundaries. We focus on principal fermionic $\ensuremath{\nu}=1/3$ and bosonic $\ensuremath{\nu}=1/2$ fractions in the case of hard-core interactions. The gap behavior as a function of the cylinder radius is analyzed. By adding enough orbitals to allow for edge modes, we show that it is possible to measure the Luttinger parameter of the nonchiral liquid formed by the combination of the two counterpropagating edges when we add a small confining potential. Although we measure a Luttinger exponent consistent with the chiral Luttinger theory prediction for the full hard-core interaction, the exponent remains nontrivial in the Tao-Thouless limit as well as for simple truncated states that can be constructed on the cylinder. If the radius of the cylinder is taken to infinity, the problem becomes a Tonks-Girardeau one-dimensional interacting gas in Fermi and Bose cases. Finally, we show that the Tao-Thouless and truncated states have an edge electron propagator, which decays spatially with a Fermi-liquid exponent, even if the energy spectrum can still be described by a nontrivial Luttinger parameter.
We study the universal long distance behavior of the Laughlin state for the fractional quantum Hall effect and the ground state of the Calogero-Sutherland model (one-dimensional 1/r2 interaction model). In particular, it is shown that these two wave functions coincide exactly when Laughlin state is confined in a narrow cylinder geometry. The seeming difference of dimensionality is merely a difference of representation of wave functions. We also give a recipe to interpret operators acting on states in the lowest Landau level in terms of the usual one-dimensional fermion operators, which is important for extracting the Tomonaga-Luttinger liquid behavior of the edge states.
Once the fractional quantum Hall (FQH) state for a finite-sized system is put on the surface of a cylinder, the distance between the two ends with open boundary conditions can be tuned by varying the aspect ratio γ . It scales linearly with increasing the system size and therefore has a larger adjustable range than that on a disk. The previous study of the quasi-hole tunneling amplitude on a disk by Hu et al (2011 New J. Phys. 13 035020 ) indicates that the tunneling amplitudes have a scaling behavior as a function of the tunneling distance and the scaling exponents are related to the scaling dimension and the charge of the transported quasiparticles. However, the scaling behaves poorly due to the narrow range of the tunneling distance on the disk. Here we systematically study the quasiparticle tunneling amplitudes of the Laughlin state in the cylinder geometry which shows a much better scaling behavior. In particular, there are some crossover behaviors at the two length scales when the two open edges are close to each other. These lengths are also reflected in the bipartite entanglement and the electron Green's function as either a singularity or a crossover. These two critical length scales of the edge–edge distance, and are found to be related to the dimension reduction and back scattering point respectively.
We study the universal long-distance behaviour of the Laughlin state for the fractional quantum Hall effect and the ground state of the Calogero-Sutherland model (one dimensional 1/r^2 interaction model). In particular, it is shown that these two wave functions coincide exactly when Laughlin state is confined in a narrow cylinder geometry. The seeming difference of dimensionality is merely a difference of representation of wave functions. We also give a recipe to interpret operators acting on states in the lowest Landau level in terms of the usual one dimensional Fermion operators, which is important for extracting the Tomonaga-Luttinger liquid behaviour of the edge states.