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Modern Physics Letters B

Ground State for the Impurity Calogero Sutherland Model

Yue-lin Shen, Xiuting Wang

Published Sep 10, 1998• 0 citations
Publisher

Abstract

We propose a Calogero–Sutherland model with an impurity. We prove that when the interaction constants have certain relations, the ground state of the model can be abtained with the aid of a set of generalized annihilation and creation operators. Further we present the differential equation for the excited state, with one special solution given.

DOI
10.1142/s0217984998000998

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We propose a Calogero–Sutherland model with an impurity. We prove that when the interaction constants have certain relations, the ground state of the model can be abtained with the aid of a set of generalized annihilation and creation operators. Further we present the differential equation for the excited state, with one special solution given.
We propose a series of models with inverse-square interactions, for which the ground states have an exact closed form, that describe one localized spin-1/21/21/2 Kondo impurity coupled to Luttinger liquids of itinerant bosons or fermions in the continuum or on a lattice. The model Hamiltonians contain two parts: a two-component Calogero-Sutherland model with an open boundary in the continuum or a long-range $t\text{\ensuremath{-}}J$-type model on the lattice, and long-range Kondo couplings between the impurity and the particles. The wave functions of the ground states in a certain basis have Jastrow product forms with powers depending on the particle statistics. It is shown that the wave functions can be expressed as chiral correlators of certain conformal field theories, which reveals that the particles are fractionalized to fully screen the Kondo impurity. This insight would be very useful in future studies on the very difficult problem of Kondo physics with interacting baths.
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.
INTRODUCTION The Calogero-Sutherland model has recently attracted some attention mainly because, in spite of its simplicity, it yields nontrivial results which contribute to shape our understanding of fractional statistics [1]. Its most remarkable property is that its ground state is given by a Jastrow wave-function which is the one dimensional analogue of the Laughlin wave-function. The excitations are described by quasiparticles which carry a fraction of the quantum numbers of the fundamental particles. The wave functions are simple enough to allow the computation of physical quantities such as the dynamical correlation functions. Recently, it has become clear that the spin version of this model is closely related to the known spin chains such as the XXX chain. The wave functions of the spin models can be obtained as simply as those of the Calogero-Sutherland model by diagonalising simple differential operators known as the Dunkl operators [2]. In these lectures, I restrict to the
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