Novel Order Parameter to Characterize Valence-Bond-Solid States
  We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the configuration of the valence bonds. This allows us to determine the phase transition point in between different VBS states accurately. We demonstrate this theory in the bond-alternating Heisenberg chain and in the frustrated spin ladder. 
Masaaki Nakamura and Synge Todo