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arXiv:cond-mat/9710035 · Physical review. B, Condensed matter

Equivalence of the Calogero-Sutherland model to free harmonic oscillators

Nakka Gurappa, Prasanta Kumar Panigrahi

Published Jan 15, 1999• 78 citations
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Abstract

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.

DOI
10.1103/physrevb.59.r2490

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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 W1+\ensuremath∞{W}_{1+\ensuremath{\infty}}W1+\ensuremath∞​ 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 develop an algebraic approach for finding the eigenfunctions of a large class of few and many-body Hamiltonians, in one and higher dimensions, having linear spectra. The method presented enables one to exactly map these interacting Hamiltonians to decoupled oscillators, thereby, giving a precise correspondence between the oscillator eigenspace and the wavefunctions of these quantum systems. The symmetries behind the degeneracy structure and the commuting constants of motion responsible for the quantum integrability of some of these models are made transparent. Apart from analysing a number of well-known dynamical systems like planar oscillators with commensurate frequencies, both with or without singular inverse-square terms and generalized Calogero-Sutherland type models, we also point out a host of other examples where the present approach can be profitably employed. We further study Hamiltonians having Laughlin wavefunction as the ground-state and establish their equivalence to free oscillators. This reveals the underlying linear W1+∞W_{1+\infty}W1+∞​ symmetry algebra unambiguously and establishes the Laughlin wavefunction as the highest weight vector of this algebra.
The present paper is an extended version of another paper [Phys. Rev. B 59, R2490 (1999)], where we have established the equivalence of the Calogero-Sutherland model to decoupled oscillators. Here, we first employ the same approach for finding the eigenstates of a large class of Hamiltonians, dealing with correlated systems. A number of few- and many-body interacting models are studied and the relationship between their respective Hilbert spaces, with that of oscillators, is found. This connection is then used to obtain the spectrum generating algebras for these systems and make an algebraic statement about correlated systems. The procedure to generate solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a method for solving linear differential equations to diagonalize the Sutherland model and establish a precise connection between this correlated system's wave functions, with those of the free particles on a circle. In the process, we obtain an expression for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having Laughlin wave function as the ground state and point out the natural emergence of the underlying linear ${W}_{1+\ensuremath{\infty}}$ symmetry in this approach.
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.
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 investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.