Edge Spin fractionalization in one-dimensional spin-$S$ quantum antiferromagnets

TL;DR

This paper demonstrates that gapped one-dimensional spin-$S$ antiferromagnetic chains with U(1) symmetry host exponentially localized fractional edge modes, confirmed through analytical solutions and numerical methods, and robust against disorder.

Contribution

It reveals the existence and robustness of fractional $rac{S}{2}$ edge modes in spin-$S$ chains, including integrable and non-integrable models, using analytical and numerical approaches.

Findings

Fractional edge modes are exponentially localized in spin-$S$ chains.

Edge modes are robust to disorder coupling to Néel order.

Fractional spins are confirmed as stable quantum observables in the thermodynamic limit.

Abstract

We show that a gapped spin-$S$ chain with antiferromagnetic (AFM) order exhibits in the thermodynamic limit exponentially localized fractional $\pm \frac{S}{2}$ edge modes when the system possesses U(1) symmetry. We show this for integrable and non integrable spin chains both analytically and numerically. Through exact analytical solutions, we show that an AFM spin-$\frac{1}{2}$ chain with {\it explicitly} broken $\mathbb{Z}_2$ symmetry and an integrable AFM spin-$1$ chain with {\it spontaneously} broken $\mathbb{Z}_2$ symmetry have $\pm \frac{1}{4}$ and $\pm \frac{1}{2}$ fractionalized edge modes, respectively. Furthermore, employing the density matrix renormalization group technique, we extend this analysis to {\it generic} $XXZ-S$ chains with $S\leq 3$ and demonstrate that these fractional spins are robust quantum observables, substantiated by the observation of a variance of the associated fractional spin operators that is consistent with a vanishing functional form in the thermodynamic limit. Moreover, we find that the edge modes are robust to disorder that couples to the N\'eel order parameter.