Finite matrix model of quantum hall fluids on S2

DOI10.1017/S0004972700039502zbMATH Open1130.81057arXivhep-th/0308040OpenAlexW2074599095MaRDI QIDQ5756326

Yi-Xin Chen, Yao-Zhong Zhang, Mark D. Gould

Publication date: 4 September 2007

Published in: Bulletin of the Australian Mathematical Society (Search for Journal in Brave)

Abstract: Based on Haldane's spherical geometrical formalism of two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of $S^2$ and the two-dimensional quantum Hall fluids is exhibited. If the number of particles $N$ is infinitely large, two-dimensional quantum Hall physics can be precisely described in terms of the noncommutative U(1) Chern-Simons theory proposed by Susskind, like in the case of plane. However, for the finite number of particles on two-sphere, the matrix-regularized version of noncommutative U(1) Chern-Simons theory involves in spinor oscillators. We establish explicitly such a finite matrix model on two-sphere as an effective description of fractional quantum Hall fluids of finite extent. The complete sets of physical quantum states of this matrix model are determined, and the properties of quantum Hall fluids related to them are discussed. We also describe how the low-lying excitations are constructed in terms of quasiparticle and quasihole excitations in the matrix model. It is shown that there consistently exists a Haldane's hierarchical structure of two-dimensional quantum Hall fluid states in the matrix model. These hierarchical fluid states are generated by the parent fluid state for particles by condensing the quasiparticle and quasihole excitations level by level, without any requirement of modifications of the matrix model.

Full work available at URL: https://arxiv.org/abs/hep-th/0308040

zbMATH Keywords

matrix modelnoncommutative Chern-Simons theoryHopf fibrationfuzzy spherequantum Hall fluid

Mathematics Subject Classification ID

Yang-Mills and other gauge theories in quantum field theory (81T13) Many-body theory; quantum Hall effect (81V70)

Cites Work

• The fuzzy sphere

Related Items (3)

Symmetry aspects and finite-size scaling of quantum Hall fluids ⋮ A MATRIX MODEL FOR $\nu_{k_1k_2} = {k_1+k_2 \over k_1 k_2}$ FRACTIONAL QUANTUM HALL STATES ⋮ Exact modular \(S\) matrix for \(\mathbb{Z}_k\) parafermion quantum Hall Islands and measurement of non-abelian anyons

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