Symmetry protected gapless $Z_2$ spin liquids

Abstract

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.