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Cambridge University Press eBooks

Nested Bethe ansatz for spin-½ fermions with σ-interactions

Ladislav Šamaj, Zoltán Bajnok

Published May 16, 2013• 0 citations
Publisher
DOI
10.1017/cbo9781139343480.009

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We study the magnetization process of the S=1 XXZ chain with biquadratic exchange interaction at zero temperature. This system can be regarded as a quantum lattice gas composed of "particles" with Sz=0, ±1. We introduce an approximate nested Bethe ansatz (ANBA) to examine the possibility of phase transition induced by the magnetic field. In the high-magnetization region σ≥σ0 for certain σ0, the ANBA gives the exact ground state energies. For σ<σ0 the ANBA gives approximate solutions. The analysis shows that a phase transition occurs at σ0 across which the magnetic susceptibility changes discontinuously. Comparing with the finite-size diagonalization, we argue that σ0 is the exact critical magnetization.
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.