Incommensuration in quantum antiferromagnetic chain
  A dimerized quantum Heisenberg or XY antiferromagnetic chain has a gap in the spectrum. We show that a weak incommensurate modulation around a dimerized chain produces a zero temperature quantum critical point. As the incommensuration wavelength is varied, there is a transition to a modulated gapless state. The critical behaviour is in the universality class of the classical commensurate-incommensurate (Pokrovsky-Talapov) transition. An analogous metal-insulator transition can also take place for an incommensurate chain. 
Somendra M. Bhattacharjee and Sutapa Mukherji