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Journal of the Physical Society of Japan

Completeness of Bethe Ansatz States in One-Dimensional δ-Function Fermions

Shuichi Murakami, Miki Wadati

Published Dec 15, 1995• 1 citation
Publisher

Abstract

We study the one-dimensional repulsive δ-function fermion system on an infinite interval. We construct a basis set of the Bethe ansatz eigenstates and show the completeness and the orthonormality of this basis set. We discuss the correspondence between these Bethe ansatz states and the states made by the quantum inverse scattering method. It justifies the method to study thermodynamics by the quantum Gel'fand-Levitan equations. In addition, we evaluate thermodynamic quantities by using the complete set of the Bethe ansatz states and obtain the first two terms of the virial expansions.

DOI
10.1143/jpsj.64.4571

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We study the one-dimensional repulsive δ-function fermion system on an infinite interval. We construct a basis set of the Bethe ansatz eigenstates and show the completeness and the orthonormality of this basis set. We discuss the correspondence between these Bethe ansatz states and the states made by the quantum inverse scattering method. It justifies the method to study thermodynamics by the quantum Gel'fand-Levitan equations. In addition, we evaluate thermodynamic quantities by using the complete set of the Bethe ansatz states and obtain the first two terms of the virial expansions.
The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.
The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.
Abstract This chapter discusses how the Bethe ansatz solution of the one-dimensional Bose gas with repulsive δ-function interaction is extended to finite temperatures, the thermody- namic Bethe ansatz. The excitations of this system consist of particle and hole excitations, which can be described by the corresponding densities of Bethe ansatz roots. It shows how these Bethe ansatz root densities are used to define an appropriate expression for the entropy of the system of Bose particles, which is the main ingredient for the extension of the Bethe ansatz method to finite temperature.