Symmetric and nematic $Z_2$ quantum spin liquids: applications to the $J_1$-$J_2$ Heisenberg model

We classify symmetric and nematic $Z_2$ quantum spin liquid states on the square lattice by analyzing bosonic PSG. We then compute the energies of various symmetric and nematic $Z_2$ spin liquid states for the $J_1$-$J_2$ square Heisenberg model by doing variational Monte Carlo simulations. The connections of our variational Monte Carlo studies with the recent DMRG results on the same model will also be discussed.