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Journal of Physics A Mathematical and General

Multi-particle scattering at the same position in the Bethe ansatz for two-band electrons with spin-exchange interaction

Zu‐Jian Ying, You‐Quan Li, Shi-Jian Gu

Published Apr 27, 2001• 2 citations
Publisher

Abstract

The same-position scattering (SPS) of more than two electrons in a one-dimensional model of two-band electrons with spin-exchange interaction is discussed. The boundary conditions of three- and four-particle SPS are given. It is shown that the conditions can be fulfilled by the two-particle boundary conditions for the Bethe ansatz (BA) wavefunction. Consequently, the definition of the BA wavefunction can be extended to those cases of more than two particles occupying the same position. Therefore, unlike the case in lattice models in which configurations with more than two particles at one site are excluded in applying the approach, the BA is valid without the exclusion of multi-particle SPS in the spin-exchange model. A relation between the SU (2) × SU (2) symmetry and the BA equation is also indicated.

DOI
10.1088/0305-4470/34/18/318

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The same-position scattering (SPS) of more than two electrons in a one-dimensional model of two-band electrons with spin-exchange interaction is discussed. The boundary conditions of three- and four-particle SPS are given. It is shown that the conditions can be fulfilled by the two-particle boundary conditions for the Bethe ansatz (BA) wavefunction. Consequently, the definition of the BA wavefunction can be extended to those cases of more than two particles occupying the same position. Therefore, unlike the case in lattice models in which configurations with more than two particles at one site are excluded in applying the approach, the BA is valid without the exclusion of multi-particle SPS in the spin-exchange model. A relation between the SU (2) × SU (2) symmetry and the BA equation is also indicated.
We consider the problem of consistence between the Bethe ansatz (BA) wave function and the multiparticle (more than two) scattering in one-dimensional δ-function interacting SU(4) fermions, which the approach of BA does not explicitly take into account. We find the scattering conditions of three and four particles located at the same position and show that the conditions can be fulfilled by the two-particle connection conditions of the BA wave function. So the definition of the BA wave function can be exactly extended to those cases with multiple occupancies. The inconsistence between the BA and multiparticle interacting on a same site in the degenerate Hubbard model, which makes the BA fail for the model, is shown to vanish in the limit of small site spacing. A correspondence relation of the BA equation and SU(4) symmetry of the system is also indicated for the fermions. The degeneracy of state with BA eigenenergy is given. Singlet lies in the case when there are equal numbers of particles in each inner component.
A lattice model consisting of a single narrow band is introduced to describe some aspects of heavy electrons. The model excludes double occupancy of the sites and electrons on nearest-neighbor sites interact via a charge interaction and spin exchange. The model is integrable in one dimension for some special values of the coupling constants. These cases are related to the SU(3) invariance. The Bethe-ansatz equations are obtained and ground-state and thermodynamic properties are discussed and solved in some limiting cases.
A lattice model consisting of a single narrow band is introduced to describe some aspects of heavy electrons. The model excludes double occupancy of the sites and electrons on nearest-neighbor sites interact via a charge interaction and spin exchange. The model is integrable in one dimension for some special values of the coupling constants. These cases are related to the SU(3) invariance. The Bethe-ansatz equations are obtained and ground-state and thermodynamic properties are discussed and solved in some limiting cases.
We consider a one-dimensional two-band model of electrons on a lattice with nearest-neighbor hopping t constrained by the excluded multiple occupancy of the sites, an interband splitting, \ensuremath{\Delta}, and intraband and interband exchange J between neighboring sites. This model is integrable at the supersymmetric point and reduces to the SU(4)-invariant supersymmetric t-J model with crystal field splitting. We present the exact solution via Bethe's ansatz and discuss ground-state properties and the excitation spectrum as a function of \ensuremath{\Delta}.
We discuss the one-dimensional Hubbard model for a finite chain of atoms, that is within the single band approximation, in the context of the transfer matrix. We show two ways of constructing the finite Hilbert space, one with the basis of electron configurations, and the second created as the antisymmetric part of the orbital and spin function tensor product. We classify the set of all electron configurations according to the action of the symmetric group. We present the Bethe Ansatz (BA) solution for the example of three electrons moving on a ring with N = 6 nodes. After that, we concentrate on giving the motivation for exploring the properties of the transfer matrix. We discuss the scattering and the winding operators in the tensor product space, resulting from the physical and the auxiliary spaces. We consider the transfer matrix as the partial trace over the auxiliary space from the winding operator for the auxiliary particle. After that, we derive the explicit expressions for some elements of the transfer matrix in the basis of the spin electron configurations.
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.