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A Calogero model for the non-Abelian quantum Hall effect

Jean-Emile Bourgine, Yutaka Matsuo

Abstract

A model of the non-Abelian fractional quantum Hall effect is obtained from the diagonalization of the matrix model proposed by Dorey, Tong, and Turner (DTT). The Hamiltonian is reminiscent of a spin Calogero-Moser model but involves higher-order symmetric representations of the non-Abelian symmetry. We derive the energy spectrum and show that the Hamiltonian has a triangular action on a certain class of wave functions with a free fermion expression. We deduce the expression of the ground states eigenfunctions and show that they solve a Knizhnik-Zamolodchikov equation. Finally, we discuss the emergence of Kac-Moody symmetries in the large $N$ limit using the level-rank duality and confirm the results obtained previously by DTT.

A Calogero model for the non-Abelian quantum Hall effect

Abstract

A model of the non-Abelian fractional quantum Hall effect is obtained from the diagonalization of the matrix model proposed by Dorey, Tong, and Turner (DTT). The Hamiltonian is reminiscent of a spin Calogero-Moser model but involves higher-order symmetric representations of the non-Abelian symmetry. We derive the energy spectrum and show that the Hamiltonian has a triangular action on a certain class of wave functions with a free fermion expression. We deduce the expression of the ground states eigenfunctions and show that they solve a Knizhnik-Zamolodchikov equation. Finally, we discuss the emergence of Kac-Moody symmetries in the large limit using the level-rank duality and confirm the results obtained previously by DTT.
Paper Structure (45 sections, 166 equations)

This paper contains 45 sections, 166 equations.

Table of Contents

  1. Introduction
  2. Derivation of the Hamiltonian
  3. Definition of the model
  4. Abelian model
  5. Spin Calogero-Moser
  6. Remark
  7. Diagonalization of the DTT matrix model
  8. Diagonalization
  9. Quantization
  10. Spectrum and eigenfunctions
  11. Abelian model
  12. Non-Abelian model
  13. Wedge states
  14. Eigenfunctions as a product of wedge states
  15. Ground state wave function and degeneracies
  16. ...and 30 more sections

Theorems & Definitions (1)

  • proof