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arXiv:hep-th/0309085

Anisotropic Quantum Hall Matrix Model

Ahmed Jellal, Rudolf A. Römer, Michael Schreiber

Published Sep 8, 2003• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.hep-th/0309085

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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.
We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at ν = q i K −1 ij q j , where K ij is a 2 × 2 matrix and q i is a vector. Our model allows us to reproduce several well-known wavefunctions. We show that the wavefunction Ψ ( m , m , n ) constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor and in particular Ψ (3,3,1) at filling can be obtained from our vacuum configuration. The unpolarized Halperin wavefunction and especially that for the fractional quantum Hall state at filling factor can also be recovered from our approach. Generalization to more than two layers is straightforward.
We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at \\nu=q_iK_{ij}^{-1}q_j, where K_{ij} is a 2\\times 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wave functions. We show that the wave function \\Psi_{(m,m,n)} constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor {2\\over m+n} and in particular \\Psi_{(3,3,1)} at filling {1\\over 2} can be obtained from our vacuum configuration. The unpolarized Halperin wave function and especially that for the fractional quantum Hall state at filling factor {2\\over 5} can also be recovered from our approach. Generalization to more than 2 layers is straightforward.
Recent experiments indicate the presence of new anisotropic fractional quantum Hall states at regimes not anticipated before. These experiments raise many fundamental questions regarding the inner nature of the electronic system that leads to such anisotropic states. Interplay between electron mass anisotropy and electron-electron correlation effects in a magnetic field can create a rich variety of possibilities. Several anisotropic electronic states ranging from anisotropic quantum Hall liquids to anisotropic Wigner solids may stabilize due to such effects. The electron mass anisotropy in a two-dimensional electron gas effectively leads to an anisotropic Coulomb interaction potential between electrons. An anisotropic interaction potential may strongly influence the stability of various quantum phases that are close in energy since the overall stability of an electronic system is very sensitive to local order. As a result there is a possibility that various anisotropic electronic phases may emerge even in the lowest Landau level in regimes where one would not expect them. In this work we study the state with filling factor 1/6 in the lowest Landau level, a state which is very close to the critical filling factor where the liquid-solid transition takes place. We investigate whether an anisotropic Coulomb interaction potential is able to stabilize an anisotropic electronic liquid state at this filling factor. We describe such an anisotropic state by means of a liquid crystalline wave function with broken rotational symmetry which can be adiabatically connected to the actual wave function for the corresponding isotropic phase. We perform quantum Monte Carlo simulations in a disk geometry to study the properties of the anisotropic electronic liquid state under consideration. The findings indicate stability of liquid crystalline order in presence of an anisotropic Coulomb interaction potential. The results are consistent with the existence of an anisotropic electronic liquid state in the lowest Landau level.
We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].
The present understanding of nonperturbative ground states in the fractional quantum Hall effect is based on effective theories of the Jain ‘composite fermion’ excitations. We review the approach based on matrix variables, i.e. D0 branes, originally introduced by Susskind and Polychronakos. We show that the Maxwell–Chern–Simons matrix gauge theory provides a matrix generalization of the quantum Hall effect, where the composite-fermion construction naturally follows from gauge invariance. The matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the Laughlin and Jain hierarchical states. The matrix theory possesses a physical limit for commuting matrices that could be reachable while staying in the same phase.
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.