The Bethe Ansatz Equation

Yupeng Wang, Wen‐Li Yang, Junpeng Cao +1 more

Johnson Leow

In the present thesis, we will review various Bethe ansatz techniques such as the (nested)\nalgebraic Bethe ansatz and the (nested) coordinate Bethe ansatz using well known models\nlike the XXX model and the Hubbard model. Subsequently, the Bethe ansatz techniques\nare applied to obtain Bethe ansatz equations and Bethe vectors of the su(2|2)-invariant S -\nmatrix. Finally, these results are used to obtain the full S -matrix and Bethe equations of the\nstring sigma model.

Kiyoshi Sogo

Hans-Peter Eckle

Abstract This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.

Bill Sutherland

H. Babujian, M. Karowski, A. Zapletal

A system of SU(N)-matrix difference equations is solved by means of a nested version of a generalized Bethe Ansatz, also called "off shell" Bethe Ansatz. The highest weight property of the solutions is proved. (Part I of a series of articles on the generalized nested Bethe Ansatz and difference equations.)