Duality and quasiparticles in the Calogero-Sutherland model Some exact results.pdf

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Duality and quasiparticles in the Calogero-Sutherland model Some exact results.pdf

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Duality and quasiparticles in the Calogero-Sutherland model Some exact results

a r X i v : h e p - t h / 0 0 1 0 0 3 3 v 1 5 O c t 2 0 0 0 Duality and quasiparticles in the Calogero-Sutherland model: Some exact results Ivan Andric? and Larisa Jonke ? Theoretical Physics Division, Rudjer Bos?kovic? Institute, P.O. Box 180, HR-10002 Zagreb, CROATIA Abstract The quantum-mechanical many-body system with the potential proportional to the pairwise inverse-square distance possesses a strong-weak coupling du- ality. Based on this duality, particle and/or quasiparticle states are described as SU(1,1) coherent states. The constructed quasiparticle states are of hier- archical nature. PACS numbers: 03.65.Nk 05.30.Pr 05.45.Yv I. INTRODUCTION The one-dimensional pairwise inverse-square distance interaction ofN particles, and their generalizations under the name of Calogero-Sutherland-Moser models (CSMM) [1–3] have so far appeared in a variety of different physical contexts. They are related to the random matrix model [4] and the two-dimensional Yang-Mills theory [5]. They also represent an example of generalized exclusion statistics [6], and quantum spin chains with long-range ?e-mail address: andric@thphys.irb.hr larisa@thphys.irb.hr 1 interactions [7]. CSMM’s describe edge states in the quantum Hall system [8] and the Chern-Simons theory [9]. We still lack a local canonical field-theoretical formulation of CSMM’s [10], but a collective-field theory [11] can be established [12,13] in the large-N limit, and it connects CSMM’s with 2d gravity [14]. A deeper understanding of the models can be gained by exploring various solutions. CSMM’s are exactly solvable and integrable, both classically and quantum-mechanically. From the quantum Lax formulation [15] we can find infinitely many commuting conserved operators, and the underlying algebraic structure should reveal the large degeneracy structure of CSMM’s. Its eigenfunctions are known to be symmetric polynomials [16,17] but, to these days, the only explicit form have been the original wave functions found