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arXiv:cond-mat/9806238 · Physical review. B, Condensed matter

Numerical study of hierarchical quantum Hall edge states in the disk geometry

Andrea Cappelli A, Carlos Méndez, J. Simonin, Guillermo R. Zemba

Published Dec 15, 1998• 19 citations
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Abstract

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.

DOI
10.1103/physrevb.58.16291

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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 W1+\ensuremath∞{W}_{1+\ensuremath{\infty}}W1+\ensuremath∞​ 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 W1+\ensuremath∞{W}_{1+\ensuremath{\infty}}W1+\ensuremath∞​ 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 W1+\ensuremath∞{W}_{1+\ensuremath{\infty}}W1+\ensuremath∞​ minimal theory, within the finite-size limitations.
Abstract We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let’s us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s i for each level, parameterize a Casimir effect or chemical potential shift that could be experimentally observed. These results are generalized to fractional and hierarchical fillings by exploiting the W-infinity symmetry of incompressible Hall fluids.
We show how two-dimensional incompressible quantum fluids and their excitations can be viewed as ${\mathit{W}}_{1+\mathrm{\ensuremath{\infty}}}$ edge conformal field theories, thereby providing an algebraic characterization of incompressibility. The Kac-Radul representation theory of the ${\mathit{W}}_{1+\mathrm{\ensuremath{\infty}}}$ algebra leads then to a purely algebraic complete classification of hierarchical quantum Hall states, which encompasses all measured fractions. Spin-polarized electrons in single-layer devices can only have Abelian anyon excitations.
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.
The abelian hierarchy of quantum Hall states accounts for most of the states in the lowest Landau level, and there is evidence of a similar hierarchy of non-abelian states emanating from the {\nu} = 5/2 Moore-Read state in the second Landau level. Extending a recently developed formalism for hierarchical quasihole condensation, we present a theory that allows for the explicit construction of the ground state wave function, as well as its quasiparticle excitations, for any state based on the abelian hierarchy. We relate our construction to structures in rational conformal field theory and stress the importance of using coherent state wave functions which allows us to formulate an extension of the bulk - edge correspondence that was conjectured by Moore and Read. Finally, we study the proposed ground state wave functions in the limiting geometry of a thin torus and argue that they coincide with the known exact results.
The Abelian hierarchy of quantum Hall states accounts for most of the states in the lowest Landau level, and there is evidence of a similar hierarchy of non-Abelian states emanating from the ν=5/2 Moore–Read state in the second Landau level. Extending a recently developed formalism for hierarchical quasihole condensation, we present a theory that allows for the explicit construction of the ground state wave function, as well as its quasiparticle excitations, for any state based on the Abelian hierarchy. We relate our construction to structures in rational conformal field theory and stress the importance of using coherent state wave functions, which allows us to formulate an extension of the bulk–edge correspondence that was conjectured by Moore and Read. Finally, we study the proposed ground state wave functions in the limiting geometry of a thin torus and argue that they coincide with the known exact results.
Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter τττ, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor ν=2/5ν=2/5ν=2/5 Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex τττ-plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.