Quantum Solitons in the Calogero-Sutherland Model

We show that the single quasi-particle Schrödinger 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. 05.30.-d,05.30.Fk Typeset using REVTEX Permanent address: Department of Physics, Science University of Tokyo, Tokyo, Japan 1 Recently, interesting work has been done [1,2] on classical solitons in the one-dimensional many-body system with inverse-square interaction. It has been found that in the h̄ → 0 limit, particle excitations from the quantum ground-state may be regarded as solitons. Two different approaches have been used to investigate this problem. Sutherland and Campbell [1] have examined Newton’s equation of motion for finite systems, and then taken the continuum (thermodynamic) limit for the infinite system. In the second approach, Polychronakos [2] has exploited the quantum collective field formulation of Andric et al. [3] to obtain the one-soliton solution in the h̄ → 0 continuum limit. In this paper, on the other hand, we enquire under what conditions a particle obeying the Schrödinger equation will give rise to stationary solitonic solutions in the Calogero-Sutherland model [4–7](CSM). We find that the one-body Schrödinger equation may be cast in the form of a solvable nonlinear diffusion equation when a certain ansatz for the one-body potential is made. The one-soliton solution of this diffusion equation, first proposed by Satsuma and Mimura [8], is of the same form as found by Polychronakos [2]. We find that soliton solutions may exist for both repulsive or attractive interactions, and these carry positive or negative fractional charge inversely proportional to the strength of the interaction. We begin with the Schrödinger equation for a particle propagating in a one-body potential W (x). The treatment is completely general at this stage, and follows the Feynman Lectures [9] on physics. The ansatz for W (x) will be made later to obtain the soliton solutions. Denoting the one-particle wave-function by ψ(x, t), we have − h̄ 2 2m ∂ψ(x, t) ∂x2 + W (x)ψ(x, t) = ih̄ ∂ψ(x, t) ∂t . (1) Henceforth we shall denote the partial derivatives with respect to x by a prime and with respect to t by a dot. It is useful to multiply Eq. (1) by ψ(x, t), − h̄ 2 2m ψ∗ψ′′ + ψ∗W (x)ψ = ih̄ψ∗ψ̇ , (2) and take the complex conjugate: − h̄ 2 2m ψψ′′ + ψW (x)ψ∗ = −ih̄ψψ̇ . (3) 2 On subtracting Eq. (3) from Eq. (2), we get − h̄ 2 2m (ψ∗ψ′′ − ψ′′ψ) = ih̄(ψ∗ψ̇ + ψψ̇∗) . (4) We now set, quite generally, ψ(x, t) = √ ρ(x, t) e , (5) where ρ and θ are real. Then Eq. (4) reduces to the continuity equation for the “charge” density ρ: ρ̇ + h̄ m ∂ ∂x (ρθ′) = 0 . (6) Thus it is legitimate to rgard the velocity of the “fluid” to be v(x, t) = h̄ m θ(x, t) . (7) The energy density equation is obtained by adding Eqs. (2) and (3), and dividing by 2: τ(x, t) + W (x)ρ = −h̄ρθ̇ , (8) where the kinetic energy density is τ(x, t) = − h̄ 2 4m (ψ∗ψ′′ + ψ′′ψ) = h̄ 2m ( 1 4 (ρ) ρ − 1 2 ρ′′ + ρ(θ) ) . (9) We note, however, that an alternate form of kinetic energy density is given by τ1(x, t) = h̄ 2m (ψ′ψ) = h̄ 2m (