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College Physics

Algebraic solution of three-dimensional two-particle Calogero-Sutherland module

Tao Yang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The operator system of three-dimensional twoparticle Calogero-Sutherland module(CSM) is obtained by sing the pseudo-angularmomentum operator method(PAMOM);the symmetry of spectrum space is then investigated.The normalized eigenfunction and the analytical expression of Glauber coherent state are obtained.

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Using the collective field technique, we give the description of the spin Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large N limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates. JSPS fellow
The BC-type Calogero–Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica σ-model formalism suitable to treat all chiral RMEs. By exploiting ‘generalized colour–flavour transformation’ we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c = 1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point.
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.
We construct the dynamical symmetry of the quantum Calogero model with particle exchange in a confining Coulomb field. This symmetry is governed by the algebra so(N+1,2)so(N+1,2)so(N+1,2), deformed by exchange (Dunkl) operators, with its invariant sector generated by the Dunkl angular momentum tensor and the modified Laplace-Runge-Lenz vector. The equidistant analogue of the Hamiltonian, with a linear spectrum, is expressed in terms of the conformal subalgebra so(1,2)so(1,2)so(1,2). In addition, the wave functions of the Calogero-Coulomb Hamiltonian are classified into infinite-dimensional lowest-weight so(1,2)so(1,2)so(1,2) multiplets.