# One-dimensional Quantum Spin Dynamics of Bethe String States

## TL;DR

This paper investigates the quantum spin dynamics of the antiferromagnetic XXZ chain using algebraic Bethe ansatz, revealing the roles of various excitations across energy scales and providing insights for experimental systems beyond low-energy theories.

## Contribution

It introduces a detailed analysis of multi-particle string excitations in the XXZ chain, extending understanding of high-energy spin dynamics beyond traditional low-energy models.

## Key findings

- Gapless excitations dominate low-energy dynamics at small magnetic polarizations.
- Two- and three-string states characterize intermediate and high-energy spin dynamics.
- Results are relevant for experimental studies in condensed matter and cold atom systems.

## Abstract

Quantum dynamics of strongly correlated systems is a challenging problem. Although the low energy fractional excitations of one dimensional integrable models are often well-understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in a representative one dimensional strongly correlated model, {\it i.e. }, the antiferromagnetic spin-$\frac{1}{2}$ XXZ chain with the Ising anisotropy, via the form-factor formulae. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multi-particle excitations based on the commensurate N\'eel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory. Based on an intuitive physical picture, we speculate that the dynamic feature at high energies due to the multi-particle anti-bound state excitations can be generalized to non-integrable spin systems.