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  • 1
    ISSN: 1572-9613
    Keywords: Quantum Hall effect ; Kac-Moody algebras ; Abelian Chern-Simons theory ; integral lattices ; quadratic forms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The purpose of this paper is to present a rather comprehensive classification of incompressible quantum Hall states in the limit of large distance scales and low frequencies. In this limit, the description of low-energy excitations above the groundstate of an incompressible quantum Hall fluid is intimately connected to the theory of integral quadratic forms on certain lattices which we call quantum Hall lattices. This connection is understood with the help of the representation theory of algebras of gapless, chiral edge currents or, alternatively, from the point of view of the bulk effective Chern-Simons theory. Our main results concern the classification of quantum Hall lattices in terms of certain invariants and their enumeration in low dimensions and for a limited range of values of those invariants. Among physical consequences of our analysis we find explicit, discrete sets of plateau values of the Hall conductivity, as well as the quantum numbers of quasiparticles in fluids corresponding to any one among those quantum Hall lattices. Furthermore, we are able to predict transitions between structurally different quantum Hall fluids corresponding to the same filling factor. Our general results are illustrated by explicitly considering the following plateau values: σ H =N/(2N±1),N=1, 2, 3,..., σ N =1/2.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1572-9613
    Keywords: Quantum Hall effect ; classification of universality classes of quantum Hall fluids ; integral lattices ; quadratic forms ; lattice embeddings ; phase transitions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this paper, the key ideas of characterizing universality classes of dissipationfree (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification of quantum Hall lattices are stated and their physical implications are discussed. Physically relevant subclasses of quantum Hall lattices are defined and completely classified. The results are carefully compared with experimental data and also with other theoretical schemes (the hierarchy schemes). Several proposals for new experiments are made which could help to settle interesting issues in the theory of the (fractional) quantum Hall effect and thus would lead to a deeper understanding of this remarkable effect.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1572-9613
    Keywords: Lattice gas ; percolation ; ferrofluid
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A lattice gas on ℤ3 consisting of hard spheres with exclusions extending through third neighbors is proved to undergo a percolation transition. If spins with ferromagnetic couplings are attached to the spheres, spontaneous magnetization is proved to occur. This may provide a model for a “ferrofluid,” a system which exhibits spontaneous magnetization without crystalline order. Similar results are also obtained for an analogous model on ℤ2.
    Type of Medium: Electronic Resource
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