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arXiv:hep-th/0210059

Quasiparticle excitations and hierarchies of 4-dimensional quantum Hall fluid states in the matrix models

Yi-Xin Chen

Published Oct 7, 2002• 4 citations
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Abstract

We investigate the condensate mechanism of the low-lying excitations in the matrix models of 4-dimensional quantum Hall fluids recently proposed by us. It is shown that there exist some hierarchies of 4-dimensional quantum Hall fluid states in the matrix models, and they are similar to the Haldane's hierarchy in the 2-dimensional quantum Hall fluids. However, these hierarchical fluid states appear consistently in our matrix models without any requirement of modifications of the matrix models.

DOI
10.48550/arxiv.hep-th/0210059

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We investigate the condensate mechanism of the low-lying excitations in the matrix models of 4-dimensional quantum Hall fluids recently proposed by us. It is shown that there exist some hierarchies of 4-dimensional quantum Hall fluid states in the matrix models, and they are similar to the Haldane's hierarchy in the 2-dimensional quantum Hall fluids. However, these hierarchical fluid states appear consistently in our matrix models without any requirement of modifications of the matrix models.
Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.
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.
The coupled-wire construction provides a useful way to obtain microscopic Hamiltonians for various two-dimensional topological phases, among which fractional quantum Hall states are paradigmatic examples. Using the recently introduced flux attachment and vortex duality transformations for coupled wires, we show that this construction is remarkably versatile to encapsulate phenomenologies of hierarchical quantum Hall states: the Jain-type hierarchy states of composite fermions filling Landau levels and the Haldane-Halperin hierarchy states of quasiparticle condensation. The particle-hole conjugate transformation for coupled-wire models is also given as a special case of the hierarchy construction. We also propose coupled-wire models for the composite Fermi liquid, which turn out to be compatible with a sort of the particle-hole symmetry implemented in a nonlocal way at $\ensuremath{\nu}=\frac{1}{2}$. Furthermore, our approach shows explicitly the connection between the Moore-Read Pfaffian state and a chiral ppp-wave pairing of the composite fermions. This composite-fermion picture is also generalized to a family of the Pfaffian state, including the anti-Pfaffian state and Bonderson-Slingerland hierarchy states.
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.