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

We discuss a universal relation $\varepsilon \left(i\kappa \right)=0\mathrm{}$ with $\mathrm{Re}\kappa =1/\xi$ in 1D quantum spin systems with an excitation gap, where $\varepsilon \mathrm{}\left(k\right)\mathrm{}$ 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 $\mathrm{XYZ}$ 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 $S=1\mathrm{}$ bilinear-biquadratic spin chain and the $S=1/2\mathrm{}$ zigzag spin ladder numerically.