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Many-body States and Operator Algebra for Exclusion Statistics.pdf

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Many-body States and Operator Algebra for Exclusion Statistics

a r X i v : c o n d - m a t / 9 4 1 0 1 1 7 v 3 6 M a r 1 9 9 5 IASSNS-HEP-94/88 CCNY-HEP 94/9 October 1994 Many-body States and Operator Algebra for Exclusion Statistics DIMITRA KARABALI Institute for Advanced Study Princeton, NJ 08540 V.P. NAIR Physics Department City College of the City University of New York New York, New York 10031 Abstract We discuss many-body states and the algebra of creation and annihilation operators for particles obeying exclusion statistics. E-mail addresses: karabali@sns.ias.edu, vpn@ajanta.sci.ccny.cuny.edu Recently Haldane introduced a variant of fractional statistics for which the notion of exclusion, a generalization of the Pauli principle, rather than exchange or braiding properties, is the prime characteristic[1]. This exclusion statistics appears naturally in the fractional quantum Hall effect, spin-1 2 antiferromagnetic chains and the Calogero- Sutherland model[1?5]. Exclusion principle implies that the number of available one-particle states should change with increasing occupation of a state and hence exclusion statistics can be char- acterized by the change ?di in available states as the occupation number is changed by ?Ni, i.e., by ?di = ? ∑ j gij ?Nj (1) gij are the parameters characterizing the statistics. di is the number of one-particle states available to the Ni-th particle with quantum numbers i, holding the labels of the (Ni ? 1) particles fixed. Particles obeying exclusion statistics were called g-ons in ref.[6], a name we shall also use. The thermodynamic distributions for particles obeying exclusion statistics can be set up generalizing the long familiar combinatorial calculation of entropy for bosons and fermions. A number of thermodynamic properties have been studied in this framework[4?6]. A question which naturally arises in this context is whether one can go beyond the ther- modynamic formulation, attempting a microscopic description by explicitly constructing the many-body Hilbert space and further intro