Dynamical correlation functions in the Calogero-Sutherland model

We compute the dynamical Green function and density-density correlation in the Calogero-Sutherland model for all integer values of the coupling constant. An interpretation of the intermediate states in terms of quasi-particles is found.Comment: 20pgs, (1 reference added

Single particle Green's function in the Calogero-Sutherland model for rational couplings β=p/q\beta=p/q

We derive an exact expression for the single particle Green function in the Calogero-Sutherland model for all rational values of the coupling $\beta$. The calculation is based on Jack polynomial techniques and the results are given in the thermodynamical limit. Two type of intermediate states contribute. The firts one consists of a particle propagating out of the Fermi sea and the second one consists of a particle propagating in one direction, q particles in the opposite direction and p holes.Comment: 9 pages, RevTeX, epsf.tex, 4 figures, files uuencode

Dynamical Correlation Functions and Finite-size Scaling in Ruijsenaars-Schneider Model

The trigonometric Ruijsenaars-Schneider model is diagonalized by means of the Macdonald symmetric functions. We evaluate the dynamical density-density correlation function and the one-particle retarded Green function as well as their thermodynamic limit. Based on these results and finite-size scaling analysis, we show that the low-energy behavior of the model is described by the $C=1$ Gaussian conformal field theory under a new fractional selection rule for the quantum numbers labeling the critical exponents.Comment: 27 pages, PS file, to be published in Nucl.Phys.

Excited States of Calogero-Sutherland Model and Singular Vectors of the WNW_N Algebra

Using the collective field method, we find a relation between the Jack symmetric polynomials, which describe the excited states of the Calogero-Sutherland model, and the singular vectors of the $W_N$ algebra. Based on this relation, we obtain their integral representations. We also give a direct algebraic method which leads to the same result, and integral representations of the skew-Jack polynomials.Comment: LaTeX, 29 pages, 2 figures, New sections for skew-Jack polynomial and example of singular vectors adde

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.

Green Function of the Sutherland Model with SU(2) internal symmetry

We obtain the hole propagator of the Sutherland model with SU(2) internal symmetry for coupling parameter $\beta=1$, which is the simplest nontrivial case. One created hole with spin down breaks into two quasiholes with spin down and one quasihole with spin up. While these elementary excitations are energetically free, the form factor reflects their anyonic character. The expression for arbitrary integer $\beta$ is conjectured.Comment: 13pages, Revtex, one ps figur

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.

Nonrelativistic Factorizable Scattering Theory of Multicomponent Calogero-Sutherland Model

We relate two integrable models in (1+1) dimensions, namely, multicomponent Calogero-Sutherland model with particles and antiparticles interacting via the hyperbolic potential and the nonrelativistic factorizable $S$-matrix theory with $SU(N)$-invariance. We find complete solutions of the Yang-Baxter equations without implementing the crossing symmetry, and one of them is identified with the scattering amplitudes derived from the Schr\"{o}dinger equation of the Calogero-Sutherland model. This particular solution is of interest in that it cannot be obtained as a nonrelativistic limit of any known relativistic solutions of the $SU(N)$-invariant Yang-Baxter equations.Comment: 4 pages, latex(uses Revtex), one figur

Single-particle Green's functions of the Calogero-Sutherland model at couplings \lambda = 1/2, 1, and 2

At coupling strengths lambda = 1/2, 1, or 2, the Calogero-Sutherland model (CSM) is related to Brownian motion in a Wigner-Dyson random matrix ensemble with orthogonal, unitary, or symplectic symmetry. Using this relation in conjunction with superanalytic techniques developed in mesoscopic conductor physics, we derive an exact integral representation for the CSM two-particle Green's function in the thermodynamic limit. Simple closed expressions for the single-particle Green's functions are extracted by separation of points. For the advanced part, where a particle is added to the ground state and later removed, a sum of two contributions is found: the expected one with just one particle excitation present, plus an extra term arising from fractionalization of the single particle into a number of elementary particle and hole excitations.Comment: 19 REVTeX page

Exact Dynamical Correlation Functions of Calogero-Sutherland Model and One-Dimensional Fractional Statistics

One-dimensional model of non-relativistic particles with inverse-square interaction potential known as Calogero-Sutherland Model (CSM) is shown to possess fractional statistics. Using the theory of Jack symmetric polynomial the exact dynamical density-density correlation function and the one-particle Green's function (hole propagator) at any rational interaction coupling constant $\lambda = p/q$ are obtained and used to show clear evidences of the fractional statistics. Motifs representing the eigenstates of the model are also constructed and used to reveal the fractional {\it exclusion} statistics (in the sense of Haldane's ``Generalized Pauli Exclusion Principle''). This model is also endowed with a natural {\it exchange } statistics (1D analog of 2D braiding statistics) compatible with the {\it exclusion} statistics. (Submitted to PRL on April 18, 1994)Comment: Revtex 11 pages, IASSNS-HEP-94/27 (April 18, 1994