Bound states and deconfined spinons in the dynamical structure factor of the $J_1 - J_2$ spin-1 chain

TL;DR

This paper investigates the dynamical structure factor of the $J_1 - J_2$ spin-1 chain using DMRG, revealing a transition from magnon modes to fractional spinons and their confinement into bound states.

Contribution

It introduces a detailed analysis of spinon confinement and magnon incommensurability in the $J_1 - J_2$ chain, extending the single-mode approximation to describe spinon dispersion at the transition.

Findings

First-order transition at $J_2=0.76 J_1$ with domain wall-induced continuum.

Observation of spinon confinement into bound states near the transition.

Relation of incommensurability to competing hopping amplitudes.

Abstract

Using a time-dependent density matrix renormalization group approach, we study the dynamical structure factor of the $J_1 - J_2$ spin-1 chain. As $J_2$ increases, the magnon mode develops incommensurability. The system undergoes a first-order transition at $J_2 = 0.76 J_1$, and at that point, domain walls lead to a continuum of fractional quasi-particles or spinons. By studying small variations in $J_2$ around the transition point, we observe the confinement of spinons into bound states in the spectral function and find a smooth evolution of the spectrum into magnon modes away from the phase transition. We employ the single-mode approximation to accurately account for the dispersion of the magnon mode away from the phase transition and describe the associated continua and bound states. We extend the single-mode approximation to describe the dispersion of a spinon at the phase transition point and obtain its dispersion throughout the Brillouin zone. This allows us to relate the incommensurability at and around the transition point to the competition between a negative nearest-neighbour hopping amplitude and a positive next-nearest-neighbour one for the domain wall.