Universal relation between the dispersion curve and the ground-state correlation length in 1D antiferromagnetic quantum spin systems

We discuss an universal relation $\epsilon(i\kappa)=0$ with ${\rm Re} \kappa=1/\xi$ in 1D quantum spin systems with an excitation gap, where $\epsilon(k)$ is the dispersion curve of the low-energy excitation and $\xi$ 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.