Fractional dimensional Fock space and Haldane's exclusion statistics. q/p case

The discussion of Fractional dimensional Hilbert spaces in the context of Haldane exclusion statistics is extended from the case \cite{IG} of $g=1/p$ for the statistical parameter to the case of rational $g=q/p$ with $q,p$-coprime positive integers. The corresponding statistical mechanics for a gas of such particles is constructed. This procedure is used to define the statistical mechanics for particles with irrational $g$. Applications to strongly correlated systems such as the Hubbard and $t-J$ models are discussed.Comment: 11 pages, latex, no figure

Exclusion statistics: A resolution of the problem of negative weights

We give a formulation of the single particle occupation probabilities for a system of identical particles obeying fractional exclusion statistics of Haldane. We first derive a set of constraints using an exactly solvable model which describes an ideal exclusion statistics system and deduce the general counting rules for occupancy of states obeyed by these particles. We show that the problem of negative probabilities may be avoided with these new counting rules.Comment: REVTEX 3.0, 14 page

Algebra of one-particle operators for the Calogero model

An algebra ${\cal G}$ of symmetric {\em one-particle} operators is constructed for the Calogero model. This is an infinite-dimensional Lie-algebra, which is independent of the interaction parameter $\lambda$ of the model. It is constructed in terms of symmetric polynomials of raising and lowering operators which satisfy the commutation relations of the $S_N$-{\em extended} Heisenberg algebra. We interpret ${\cal G}$ as the algebra of observables for a system of identical particles on a line. The parameter $\lambda$, which characterizes (a class of) irreducible representations of the algebra, is interpreted as a statistics parameter for the identical particles.Comment: 23 pages, LaTe

Thermodynamics for Fractional Exclusion Statistics

We discuss the thermodynamics of a gas of free particles obeying Haldane's exclusion statistics, deriving low temperature and low density expansions. For gases with a constant density of states, we derive an exact equation of state and find that temperature-dependent quantities are independent of the statistics parameter.Comment: 9 pages, Revtex, no figures. References correcte

Symmetries and exact solutions of some integrable Haldane-Shastry like spin chains

By using a class of `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry (HS) spin chain. Lax pairs and conserved quantities for these spin chains are also found and it is established that these models exhibit multi-parameter deformed or nonstandard variants of $Y(gl_M)$ Yangian symmetry. Moreover, by projecting the eigenstates of Dunkl operators in a suitable way, we derive a class of exact eigenfunctions for such HS like spin chain and subsequently conjecture that these exact eigenfunctions would lead to the highest weight states associated with a multi-parameter deformed or nonstandard variant of $Y(gl_M)$ Yangian algebra. By using this conjecture, and acting descendent operator on the highest weight states associated with a nonstandard $Y(gl_2)$ Yangian algebra, we are able to find out the complete set of eigenvalues and eigenfunctions for the related HS like spin-${1\over 2}$ chain. It turns out that some additional energy levels, which are forbidden due to a selection rule in the case of SU(2) HS model, interestingly appear in the spectrum of above mentioned HS like spin chain having nonstandard $Y(gl_2)$ Yangian symmetry.Comment: 35 pages, latex, no figures, minor type errors are corrected, version to appear in Nucl. Phys.

Spin dependent extension of Calogero-Sutherland model through anyon like representations of permutation operators

We consider a $A_{N-1}$ type of spin dependent Calogero-Sutherland model, containing an arbitrary representation of the permutation operators on the combined internal space of all particles, and find that such a model can be solved as easily as its standard $su(M)$ invariant counterpart through the diagonalisation of Dunkl operators. A class of novel representations of the permutation operator $P_{ij}$, which pick up nontrivial phase factors along with interchanging the spins of $i$-th and $j$-th particles, are subsequently constructed. These `anyon like' representations interestingly lead to different variants of spin Calogero-Sutherland model with highly nonlocal interactions. We also explicitly derive some exact eigenfunctions as well as energy eigenvalues of these models and observe that the related degeneracy factors crucially depend on the choice of a few discrete parameters which characterise such anyon like representations.Comment: 25 pages, plain LaTex file, the results of sec.4 are presented in a more explicit way, to appear in Nucl. Phys.

Many-body States and Operator Algebra for Exclusion Statistics

We discuss many-body states and the algebra of creation and annihilation operators for particles obeying exclusion statistics.Comment: 14 pages, plainTex. The first few pages have been modified. Note and references added. (This version will appear in Nucl. Phys. B.

Applications of the Collective Field Theory for the Calogero-Sutherland Model

We use the collective field theory known for the Calogero-Sutherland model to study a variety of low-energy properties. These include the ground state energy in a confining potential upto the two leading orders in the particle number, the dispersion relation of sound modes with a comparison to the two leading terms in the low temperature specific heat, large amplitude waves, and single soliton solutions. The two-point correlation function derived from the dispersion relation of the sound mode only gives its nonoscillatory asymptotic behavior correctly, demonstrating that the theory is applicable only for the low-energy and long wavelength excitations of the system.Comment: LaTeX, 31 page

Bosonic and fermionic single-particle states in the Haldane approach to statistics for identical particles

We give two formulations of exclusion statistics (ES) using a variable number of bosonic or fermionic single-particle states which depend on the number of particles in the system. Associated bosonic and fermionic ES parameters are introduced and are discussed for FQHE quasiparticles, anyons in the lowest Landau level and for the Calogero-Sutherland model. In the latter case, only one family of solutions is emphasized to be sufficient to recover ES; appropriate families are specified for a number of formulations of the Calogero-Sutherland model. We extend the picture of variable number of single-particle states to generalized ideal gases with statistical interaction between particles of different momenta. Integral equations are derived which determine the momentum distribution for single-particle states and distribution of particles over the single-particle states in the thermal equilibrium.Comment: 6 pages, REVTE

Kinetic Approach to Fractional Exclusion Statistics

We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed.Comment: 6 pages, REVTE