Dynamical quadrupole structure factor of frustrated ferromagnetic chain

We investigate the dynamical quadrupole structure factor of a spin-1/2 $J_1$-$J_2$ Heisenberg chain with competing ferromagnetic $J_1$ and antiferromagnetic $J_2$ in a magnetic field by exploiting density-matrix renormalization group techniques. In a field-induced spin nematic regime, we observe gapless excitations at $q=\pi$ according to quasi-long-range antiferro-quadrupole correlations. The gapless excitation mode has a quadratic form at the saturation, while it changes into a linear dispersion as the magnetization decreases.