Exclusion statistics, operator algebras and Fock space representations

Abstract

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) AbstractWe 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.3

Many-body constraints and non-thermal behavior in 1D open systems with Haldane exclusion statistics

“…This secondquantized description gives an appealingly simple picture of the constraints, in terms of multiple flavors of fermions that are restricted to respect a particular pseudospin ordering in energy space. Though other frameworks for second quantization of particles with novel exclusion statistics exist [56][57][58][59] , ours has the advantage of allowing a straightforward computation of matrix elements, with projections that are easily implemented numerically. As such, this formalism may be useful for investigating the impact of inter-occupancy constraints on other aspects of HES systems.…”

“…Here we develop such a formalism, based on the exact description of the constrained Hilbert space described in Sec. III C. Though other protocols for second quantization of HES particles [56][57][58][59] have been proposed, to the best of our knowledge ours is the first that exactly captures the occupancy constraints.…”

Macroscopic properties ofA-statistics

“…By definition A-statistics is closely related to certain (more precisely, symmetric or Fock) representations of the Lie algebra sl(n + 1), including n = ∞. Apart from that, A-statistics belongs to the class of exclusion statistics as defined in [28,Section 5]. Okubo [29] has also reformulated this in the language of Lie-triple systems.…”

A new method for derivation of statistical weight of the Gentile Statistics

“…al. studied exclusion statistics in the second quantized approach which includes Gentile statistics as a special case [20]. In addition to that, Dai and Xie obtained an operator realization for the angular momentum algebra which naturally leads to Gentile distribution [21].…”