Quantum Solitons in the Calogero-Sutherland Model

TL;DR

This paper demonstrates that a specific one-body Schr"odinger equation with a potential derived from the Calogero-Sutherland model admits stationary soliton solutions, linking quantum solitons to Coulomb gas models.

Contribution

It introduces a novel connection between quantum solitons in the Calogero-Sutherland model and classical Coulomb gas dynamics, providing new insights into their mathematical relationship.

Findings

Stationary soliton solutions exist for the single quasi-particle Schr"odinger equation.

The equation aligns with Dyson's Coulomb gas steady-state equation under certain conditions.

Charge magnitude is unity only for the semion case.

Abstract

We show that the single quasi-particle Schr\"odinger equation for a certain form of one-body potential yields a stationary one soliton solution. The one-body potential is assumed to arise from the self- interacting charge distribution with the singular kernel of the Calogero-Sutherland model. The quasi-particle has negative or positive charge for negative or positive coupling constant of the interaction. The magnitude of the charge is unity only for the semion. It is also pointed out that for repulsive coupling, our equation is mathematically the same as the steady-state Smoluchowski equation of Dyson's Coulomb gas model.