2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99253We discuss an universal relation $ε(iκ)=0$ with ${\rm Re} κ=1/ξ$ in 1D quantum spin systems with an excitation gap, where $ε(k)$ is the dispersion curve of the low-energy excitation and $ξ$ is the correlation length of the ground-state. We first discuss this relation for integrable models such as the Ising model in a transverse filed and the XYZ model. We secondly make a derivation of the relation for general cases, in connection with the equilibrium crystal shape in the corresponding 2D classical system. We finally verify the relation for the S=1 bilinear-biquadratic spin chain and $S=1/2$ zigzag spin ladder numerically.6 pages, 2 figures, PRB acceptedStatistical MechanicsStrongly Correlated Electronstext