Motions of the String Solutions in the XXZ Spin Chain under a Varying Twist

TL;DR

This paper analyzes how the roots of the Bethe ansatz equations in the XXZ spin chain evolve under a changing twist, revealing finite size effects and root motions that influence the system's phase and quasiparticle statistics.

Contribution

It provides a detailed analytic and numerical study of root motions in the XXZ spin chain under twist, including finite size corrections and their implications for phase and statistics.

Findings

Root motions depend on string type and finite size effects.

Roots collide with branch points or fluctuate around string centers.

Results connect regimes of different anisotropy parameters and relate Berry phase to quasiparticle statistics.

Abstract

We determine the motions of the roots of the Bethe ansatz equation for the ground state in the XXZ spin chain under a varying twist angle. This is done by analytic as well as numerical study at a finite size system. In the attractive critical regime $ 0< \Delta <1 $, we reveal intriguing motions of strings due to the finite size corrections to the length of the strings: in the case of two-strings, the roots collide into the branch points perpendicularly to the imaginary axis, while in the case of three-strings, they fluctuate around the center of the string. These are successfully generalized to the case of $n$-string. These results are used to determine the final configuration of the momenta as well as that of the phase shift functions. We obtain these as well as the period and the Berry phase also in the regime $ \Delta \leq -1$, establishing the continuity of the previous results at $ -1 < \Delta < 0 $ to this regime. We argue that the Berry phase can be used as a measure of the statistics of the quasiparticle ( or the bound state) involved in the process.