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arXiv:quant-ph/0109108 · International Journal of Modern Physics A

COHERENT STATES AND GEOMETRIC PHASES IN THE CALOGERO–SUTHERLAND MODEL

Dae-Yup Song, JeongHyeong Park

Published Jan 20, 2002• 4 citations
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Abstract

Exact coherent states in the Calogero–Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the AN-1 Calogero–Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.

DOI
10.1142/s0217751x0200561x

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Exact coherent states in the Calogero–Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the AN-1 Calogero–Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.
For the models of N-body identical harmonic oscillators interacting through potentials of homogeneous degree -2, the unitary operator that transforms a system of time-dependent parameters into that of unit spring constant and unit mass of different time scale is found. If the interactions can be written in terms of the differences between positions of two particles, it is also shown that the Schr\"odinger equation is invariant under a unitary transformation. These unitary relations can be used not only in finding coherent states from the given stationary states in a system, but also in finding exact wave functions of the Hamiltonian systems of time-dependent parameters from those of time-independent Hamiltonian systems. Both operators are invariant under the exchange of any pair of particles. The transformations are explicitly applied for some of the Calogero-Sutherland models to find exact coherent states.
A similarity transformation is constructed, through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension can be mapped identically to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion, and a linear ${W}_{1+\ensuremath{\infty}}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.
We examine a collection of particles interacting with inverse-square two-body potentials in the thermodynamic limit. We find explicit large-amplitude density waves and soliton solutions for the motion of the system. Waves can be constructed as coherent states of either solitons or phonons. Therefore, either solitons or phonons can be considered as the fundamental excitations. The generic wave is shown to correspond to a two-band state in the quantum description of the system, while the limiting cases of solitons and phonons correspond to particle and hole excitations.