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International Journal of Modern Physics A

Integrable Model of Interacting Fermions Confined by the Morse Potential

V. I. Inozemtsev

Published Jan 10, 1997• 3 citations
Publisher

Abstract

The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (su(n) spins) interacting with an external field with the Morse potential of an arbitrary strength τ2. These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form.

DOI
10.1142/s0217751x9700027x

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The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (su(n) spins) interacting with an external field with the Morse potential of an arbitrary strength τ2. These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form.
The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU( n ) spins) interacting with an external field with the Morse potential of an arbitrary strength τ 2 . These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞ .
It is shown that hyperbolic systems of Sutherland spin 1/2 particles in an external field with the Morse potential characterized by parameter τ 2 have a discrete part of their spectrum in the case of a particular constraint on the τ, parameter of two-particle interaction λ, and the number of particles. The ground state is described by a Jastrow wave function. The results known for systems with an interaction inversely proportional to the squared distance between particles are reproduced in the limit τ → ∞.
We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.
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.
The Sutherland model has a number of variants. One of them is the U( K ) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L , and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U( K ) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U( K B , K F ) Sutherland model [177] consists of bosons having K B possible colors and fermions having K F colors. The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/ r 2 supersymmetric t – J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models. In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.