Design local spin models for Gutzwiller-projected parton wave functions

We introduce a method to design a local spin Hamiltonian to realize a Gutzwiller-projected parton wave functions (GPWF) as its ground state. For example, the Dirac spin liquid (DSL) state is quite close to the true ground state of the spin-1/2 Heisenberg model on kagome lattice. We examine what kind of perturbations we should add in order to drive the DSL to more stable chiral spin liquid (CSL), valence bond solid (VBS) or Gutzwiller-projected spin Hall (GSH) states. We compute the two-body reduced-density-matrices (2-RDM) of GPWFs for those target states, and compare them to the 2-RDM of the DSL. This allows us to design local spin models with only two-body interactions that may realize those interesting target states. Our results agree very well with recent numerical calculations for CSL on kagome lattice. We also study spin-1 systems on kagome lattice, and design local spin models that may realize CSL, VBS and symmetry-protected topological (SPT) states. Our work establishes a directional guide for further numerical simulations.