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arXiv:hep-th/9411054 · Physical Review Letters

Waves and Solitons in the Continuum Limit of the Calogero-Sutherland Model

Alexios P. Polychronakos

Published Jun 26, 1995• 91 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.74.5153

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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.
The Calogero Sutherland model is system of particle moving on a line and interacting with long-range forces. In this thesis we consider the classical case where the particles may or may not possess a spin degree of freedom. We demonstrate the intimate connection between the Calogero-Sutherland system and the Benjamin Ono equation. We then directly obtain a classical hydrodynamical limit of both the spineless and spinful Calogero system. The continuum limit of the spinless system is known to exhibit solition solutions. We show numerically that the spinful system also exhibits localized solutions with the soliton property. This is a strong evidence that the continuum spin-Calogero model is exactly integrable.
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.
We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the
A one-dimensional quantum many-body system consisting of particles confined in a harmonic potential and subject to finite-range two-body and three-body inverse-square interactions is introduced. The range of the interactions is set by truncation beyond a number of neighbors and can be tuned to interpolate between the Calogero-Sutherland model and a system with nearest and next-nearest neighbors interactions discussed by Jain and Khare. The model also includes the Tonks-Girardeau gas describing impenetrable bosons as well as an extension with truncated interactions. While the ground state wave function takes a truncated Bijl-Jastrow form, collective modes of the system are found in terms of multivariable symmetric polynomials. We numerically compute the density profile, one-body reduced density matrix, and momentum distribution of the ground state as a function of the range rrr and the interaction strength.
We show that the kinetic theory of quantum and classical Calogero particles reduces to the free-particle Boltzmann equation. We reconcile this simple emergent behaviour with the strongly interacting character of the model by developing a Bethe-Lax correspondence in the classical case. This demonstrates explicitly that the freely propagating degrees of freedom are not bare particles, but rather quasiparticles corresponding to eigenvectors of the Lax matrix. We apply the resulting kinetic theory to classical Calogero particles in external trapping potentials and find excellent agreement with numerical simulations in all cases, both for harmonic traps that preserve integrability and exhibit perfect revivals, and for anharmonic traps that break microscopic integrability. Our framework also yields a simple description of multi-soliton solutions in a harmonic trap, with solitons corresponding to sharp peaks in the quasiparticle density. Extensions to quantum systems of Calogero particles are discussed.