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 nonintegrable spin chains, both analytically and numerically. Through exact analytical solutions, we show that an AFM spin-$\frac{1}{2}$ chain with explicitly broken ${\mathbb{Z}}_2$ symmetry and an integrable AFM spin-1 chain with 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 generic $XXZ\text{−}S$ chains with $S\le 3$ and demonstrate that these fractional spins are robust quantum observables, substantiated by the observation of 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 a disorder that couples to the Néel order parameter.

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• Magnetism
• 1-dimensional spin chains
• Bethe ansatz
• Bosonization
• Density matrix renormalization group