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

We discuss a universal relation $\ensuremath{\varepsilon}(i\ensuremath{\kappa})=0$ with $\mathrm{Re}\ensuremath{\kappa}=1/\ensuremath{\xi}$ in 1D quantum spin systems with an excitation gap, where $\ensuremath{\varepsilon}(k)$ is the dispersion curve of the low-energy excitation and $\ensuremath{\xi}$ 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 $\mathrm{XYZ}$ 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 the $S=1/2$ zigzag spin ladder numerically.