Gapless to gapless phase transitions in quantum spin chains

We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field $h$. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\omega)$ as a function of $h$. At $h=0$, the envelope of the lowest energy excitations goes soft at {\it two points} $q_1=2\pi/3$ and $q_2=4\pi/3$, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total {\it four} soft modes, and combine to form a new set of excitations that soften at a {\it single} point $q=\pi$ at $h_{c1}\approx 0.94$. Beyond $h_{c1}$ the system enters another gapless B-phase until the transition at $h_{c2}=4$ to the fully polarized phase. We compare the ULS model results with those for the AKLT model as a representative of gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.