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arXiv:cond-mat/9506115 · Physics Letters A

Fractional dimensional Fock space and Haldane's exclusion statistics: q/p case and t−J model

Kirill Ilinski, J.M.F. Gunn

Published Jan 1, 1996• 3 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(96)80005-3

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We study the impact of the interlevel energy constraints imposed by Haldane exclusion statistics on energy relaxation processes in one-dimensional systems coupled to a bosonic bath. By formulating a second-quantized description of the relevant Fock space, we identify certain universal features of this relaxation dynamics, and show that it is generically slower than that of spinless fermions. Our study focuses on the Calogero-Sutherland model, which realizes Haldane Exclusion statistics exactly in one dimension; however, our results apply to any system that has the associated pattern of interlevel occupancy constraints in Fock space.
We propose that the notion of Haldane's dimension and the corresponding statistics be treated in a probabilistic spirit. Motivated by the experience of dimensional regularization, the dimension of a space is defined as a trace of a diagonal ``unit operator'' where the diagonal matrix elements are not in unity in general but are probabilities to find a system in a given state. These probabilities are then uniquely defined by the rules of Haldane's statistics. The partition function for an ideal gas of the particles, a state-counting procedure, the entropy and a distribution function for this probabilistic definition are investigated and compared with previous works in the context of g particles. The corresponding creation and annihilation operators are introduced and Hamiltonians for interacting g particles are constructed. The stability of the thermodynamical properties of a gas of particles with exclusion statistics under the influence of hopping (or scattering) between states is investigated. \textcopyright{} 1996 The American Physical Society.
We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio.We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Green's realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.