Gapless fermions and quantum order

Using a two-dimensional quantum spin-1/2 model as a concrete example, we studied the relation between gapless fermionic excitations (spinons) and quantum orders in some spin liquid states. Using a winding number, we find that the projective symmetry group characterizing the quantum order can directly determine the pattern of Fermi points in the Brillouin zone. Thus quantum orders provide an origin for gapless fermionic excitations. Gapless fermions can appear even from a pure boson system as long as the boson ground state has a proper quantum order.