Instability of the Luttinger liquids towards a fractal phase

A novel universality class is found to characterize quantum phase transitions arising from instability of a Luttinger liquid to a  quantum state of matter with highly degenerate ground states.   The latter in turn is  a fractal phase, which may be attributed to an alternative to Landau's spontaneous symmetry breaking, with the number of the Goldstone modes being identified as the fractal dimension. This is achieved through an extensive investigation of an
anisotropic spin-1  biquadratic model.  A peculiar feature of the model is that it enjoys distinct symmetries with varying anisotropic coupling parameters, up to a SU(3) symmetry at the isotropic point.
The ground state phase diagram accommodates three symmetry-protected trivial phases, three exotic quantum states of matter with highly degenerate ground states, and six Luttinger liquids.  In addition, quantum phase transitions between the Luttinger liquids and the symmetry-protected trivial phases are identified to be in the Kosterlitz-Thouless universality class.