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Particle Propagator of Spin Calogero-Sutherland Model

Ryota Nakai, Yusuke S. Kato

Published Dec 4, 2012• 0 citations
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Abstract

Abstract. We obtained the particle propagator for spin 1/2 Calogero-Sutherland model for the finite system and in thermodynamic limit by Uglov’s method, the details of which are explained in the previous paper about the hole propagator of this model. Corresponding spectral functions in a full energy-momentum space are calculated numerically and the positions of the divergences of intensity are determined. PACS numbers: 02.30.Ik,03.75.Kk,04.20.Jb Submitted to: J. Phys. A: Math. Gen. Particle propagator 2 1.

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Abstract. We obtained the particle propagator for spin 1/2 Calogero-Sutherland model for the finite system and in thermodynamic limit by Uglov’s method, the details of which are explained in the previous paper about the hole propagator of this model. Corresponding spectral functions in a full energy-momentum space are calculated numerically and the positions of the divergences of intensity are determined. PACS numbers: 02.30.Ik,03.75.Kk,04.20.Jb Submitted to: J. Phys. A: Math. Gen. Particle propagator 2 1.
Explicit-exact expressions for the particle propagator of the spin 1/2 Calogero–Sutherland model are derived for the system of a finite number of particles and for that in the thermodynamic limit. Derivation of the expression in the thermodynamic limit is also presented in detail. Combining this result with the hole propagator obtained in earlier studies, we calculate the spectral function of the single particle Green's function in the full range of the energy and momentum space. The resultant spectral function exhibits power-law singularity characteristic to correlated particle systems in one dimension.
The hole propagator of a spin 1/2 Calogero–Sutherland model is derived using Uglov's method, which maps the exact eigenfunctions of the model, called the Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl 2 -Jack polynomials). To apply this mapping method to the calculation of 1-particle Green's function, we confirm that the sum of the field annihilation operator ψ ↑ + ψ ↓ on a Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator ψ on gl 2 -Jack polynomials by the mapping. The resultant expression for the hole propagator for a finite-size system is written in terms of renormalized momenta and spin of quasi-holes, and the expression in the thermodynamic limit coincides with the earlier result derived by another method. We also discuss the singularity of the spectral function for a specific coupling parameter where the hole propagator of the spin Calogero–Sutherland model becomes equivalent to the dynamical colour correlation function of an SU (3) Haldane–Shastry model.
At coupling strengths \ensuremath{\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.