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arXiv:hep-th/0409115 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Matrix model for a quantum Hall droplet with manifest particle-hole symmetry

David E Berenstein

Published Apr 1, 2005• 41 citations
arXivHTMLPDF

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 $c=1/2\text{ }\text{ }\mathrm{CFT}$ on it.

DOI
10.1103/physrevd.71.085001

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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  CFTc=1/2\text{ }\text{ }\mathrm{CFT}c=1/2  CFT on it.
The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.
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The particle-hole (PH) symmetry of electrons is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. We investigate an emergent symmetry in the fractional quantum Hall effect, namely, the PH symmetry of composite fermions, which relates states at composite fermion filling factors ${\ensuremath{\nu}}^{*}=n+\overline{\ensuremath{\nu}}$ and ${\ensuremath{\nu}}^{*}=n+1\ensuremath{-}\overline{\ensuremath{\nu}}$, where the integer nnn is the $\mathrm{\ensuremath{\Lambda}}$-level index and $0\ensuremath{\le}\overline{\ensuremath{\nu}}\ensuremath{\le}1$. Detailed calculations using the microscopic theory of composite fermions demonstrate the following for low-lying $\mathrm{\ensuremath{\Lambda}}$ levels (small nnn): (i) The two-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\mathrm{\ensuremath{\Lambda}}$ level; and (ii) the three-body interaction for composite fermions is an order of magnitude smaller than the two-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\mathrm{\ensuremath{\Lambda}}$ levels, which is also supported by exact-diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\ensuremath{\nu}=\frac{n+\overline{\ensuremath{\nu}}}{2(n+\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$ and $\ensuremath{\nu}=\frac{n+1\ensuremath{-}\overline{\ensuremath{\nu}}}{2(n+1\ensuremath{-}\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\mathrm{\ensuremath{\Lambda}}$-level index, the two-body and three-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\ensuremath{\nu}=1/2$.
We have carried out a numerical study of elementary excitations in a finite droplet of an incompressible quantum Hall liquid by using the few-particle exact diagonalization technique on disks and spheres in the presence of an external confining potential. We study both weak and strong confinements in a spherical droplet, finding in the latter case a possible destruction of the incompressibility gap for high-enough confinement strength. In the disk geometry, we find evidence for a linear gap mode and for a second edge mode at a fractional filling of 2/5.
In this paper we propose a model of the fractional quantum Hall effect within conventional one-dimensional bosonization. It is shown that in this formalism the resulting bosonized fermion operator corresponding to momenta of Landau gauge wave function is effectively two-dimensional. At special filling factors the bulk gets gapped, and the theory is described by a sine-Gordon model. The edges are shown to be gapless, chiral, and carrying a fractional charge. The hierarchy of obtained fractional charges is consistent with existing experiments and theories. It is also possible to draw a connection to composite fermion description and to the Laughlin many-body wave function.
We consider the anisotropic effect in the quantum Hall systems by applying a confining potential that is not of parabolic type. This can be done by extending Susskind--Polychronakos's approach to involve the matrices of two coupled harmonic oscillators. Starting from its action, we employ a unitary transformation to diagonalize the model. The operators for building up the anisotropic ground state and creating the collective excitations can be constructed explicitly. Evaluating the area of the quantum Hall droplet, we obtain the corresponding filling factor which is found to depend on the anisotropy parameter and to vary with the magnetic field strength. This can be used to obtain the observed anisotropic filling factors, i.e. {9\over 2}, {11\over 2} and others.