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A matrix model for a quantum hall droplet with manifest particle-hole symmetry

David Berenstein

TL;DR

The paper presents a fermionic gauged matrix quantum mechanics framework that realizes a quantum Hall droplet on a sphere with explicit particle-hole symmetry. It develops two complementary constructions: a pair of rectangular fermionic matrices with $U(N)\times U(M)$ symmetry, and an orbifolded harmonic oscillator model whose edge dynamics map to a $c=1/2$ fermionic edge; both yield finite, symmetric particle/hole sectors described by Young tableaux and Schur polynomials. By mapping states to $SU(2)$ representations and edge modes to traces like $\mathrm{tr}((a^\dagger b^\dagger)^n)$, the authors reveal how particle-hole symmetry can be embedded in the matrix degrees of freedom and how a sphere geometry naturally emerges. They outline a route to fractional quantum Hall physics via SU(2)-invariant double-trace perturbations and discuss the potential for a string theory dual, highlighting the significance of manifest symmetries in matrix-model realizations of topological quantum states.

Abstract

We find that a gauged matrix model of rectangular fermionic matrices (a matrix version of the fermion harmonic oscillator) realizes a quantum hall droplet with manifest particle-hole symmetry. The droplet consists of free fermions on the topology of a sphere. It is also possible to deform the Hamiltonian by double trace operators, and we argue that this device can produce two body potentials which might lead the system to realize a fractional quantum hall state on the sphere. We also argue that a single gauged fermionic quantum mechanics of hermitian matrices realizes a droplet with an edge that has $c=1/2$ CFT on it.

A matrix model for a quantum hall droplet with manifest particle-hole symmetry

TL;DR

The paper presents a fermionic gauged matrix quantum mechanics framework that realizes a quantum Hall droplet on a sphere with explicit particle-hole symmetry. It develops two complementary constructions: a pair of rectangular fermionic matrices with symmetry, and an orbifolded harmonic oscillator model whose edge dynamics map to a fermionic edge; both yield finite, symmetric particle/hole sectors described by Young tableaux and Schur polynomials. By mapping states to representations and edge modes to traces like , the authors reveal how particle-hole symmetry can be embedded in the matrix degrees of freedom and how a sphere geometry naturally emerges. They outline a route to fractional quantum Hall physics via SU(2)-invariant double-trace perturbations and discuss the potential for a string theory dual, highlighting the significance of manifest symmetries in matrix-model realizations of topological quantum states.

Abstract

We find that a gauged matrix model of rectangular fermionic matrices (a matrix version of the fermion harmonic oscillator) realizes a quantum hall droplet with manifest particle-hole symmetry. The droplet consists of free fermions on the topology of a sphere. It is also possible to deform the Hamiltonian by double trace operators, and we argue that this device can produce two body potentials which might lead the system to realize a fractional quantum hall state on the sphere. We also argue that a single gauged fermionic quantum mechanics of hermitian matrices realizes a droplet with an edge that has CFT on it.

Paper Structure

This paper contains 10 sections, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The gauged $U(N)$ harmonic oscillator and the QHE
  3. Description of the excitations of the system
  4. The type 0A harmonic oscillator matrix model.
  5. A fermionic matrix quantum mechanics with $U(N)\times U(M)$ symmetry
  6. Description of the spectrum in terms of Young tableaux.
  7. Fermionization of the "eigenvalues" and $SU(2)$.
  8. Towards the FQHE: adding interactions.
  9. A system with a $c=1/2$ edge
  10. Conclusion

Figures (4)

  • Figure 1: Young tableaux describing the state $|6,4,2,1\rangle$, $\tilde{n}_1 =6, \tilde{n}_2=4, \tilde{n}_3=2, \tilde{n}_4 =1, \tilde{n}_k=0 \ \hbox{for all}\ k>4$
  • Figure 2: Young tableaux describing a single particle and a single hole state
  • Figure 3: Flipping the tableaux: upper $U(N)$ indices and lower $U(M)$ indices of tensors made of products of $a^\dagger$ are related by Fermi statistics by flipping the tableaux through the diagonal.
  • Figure 4: An allowed Young tableaux for the fermionic matrix model