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Journal of Physics A Mathematical and General

Algebraic Bethe ansatz for the nonlinear Schrodinger model. II. Mixed fermion and boson fields

Yu-Sen Zhou

Published May 21, 1988• 7 citations
Publisher

Abstract

For pt.I see ibid., vol.21, p.2391, (1988). The Bethe ansatz equations for a mixture of species of fermions and bosons in one dimension, interacting with a repulsive delta potential, are obtained. It is shown that the matrix periodic boundary conditions of the system can be obtained from those of a fermion or boson system. Finally, the extension of these results to the nonlinear Schrodinger model with a supermatrix is made.

DOI
10.1088/0305-4470/21/10/017

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For pt.I see ibid., vol.21, p.2391, (1988). The Bethe ansatz equations for a mixture of species of fermions and bosons in one dimension, interacting with a repulsive delta potential, are obtained. It is shown that the matrix periodic boundary conditions of the system can be obtained from those of a fermion or boson system. Finally, the extension of these results to the nonlinear Schrodinger model with a supermatrix is made.
The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.