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Classification and Monte Carlo study of symmetric $Z_{2}$ spin liquids on the triangular lattice

We study different ways of symmetry fractionalization in $Z_{2}$ spin liquids on the triangular lattice. Our classification can be used to identify the symmetry fractionalization in the $Z_{2}$ spin liquid reported in recent density-matrix-renormalization-group simulations for $J_{1}-J_{2}$ spin model on the triangular lattice. We find 64 types of symmetry enriched $Z_{2}$ spin liquid states on triangular lattice. Besides 8 states constructed in Schwinger-boson parton wavefunctions, 12 more states can be realized in Abrikosov-fermion parton construction. Within a larger gauge group than SU(2), the rest 44 states are also found in a spin-3/2 system. Among 20 types of Abrikosov-fermion Gutzwillerprojected wavefunctions, No.B5 state is a promising candidate for the $Z_{2}$ spin liquid for $J_{1}-J_{2}$ spin model on the triangular lattice. No.B5 lies close to Dirac spin liquid (DSL). However, variational Monte Carlo simulation find that DSL has a good variational energy and $J_{1}-J_{2}$ spin model cannot open a gap for spinons on top of DSL to stabilize No.B5 state.