Symmetry protected gapless $Z_2$ spin liquids

Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical $Z_2$ gauge fields. Requiring $U(1)$ spin conservation, time reversal and certain space group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin one-half systems on square, triangular and kagome lattices, we show that all gapped symmetric $Z_2$ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped $Z_2$ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.