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$U(1)\times U(1)$ Symmetry Protected Topological Order in Gutzwiller Wave Functions

Gutzwiller projection is a way to construct many-body wave functions that could carry topological order or symmetry protected topological (SPT) order. However, an important issue is to determine whether or not a given Gutzwiller-projected wave functions (GWF) carries a non-trivial SPT order, and which SPT order is carried by the wavefunction. In this paper, we numerically study the SPT order in a spin $S = 1$ GWF on the Kagome lattice. Using the standard Monte Carlo method, we directly confirm that the GWF has (1) gapped bulk with short-range correlations, (2) a trivial topological order via nondegenerate ground state, and zero topological entanglement entropy, (3) a non-trivial $U(1)\times U(1)$ SPT order via the Hall conductances of the protecting $U(1)\times U(1)$ symmetry, and (4) symmetry protected gapless boundary. This represents numerical evidence of continuous symmetry protected topological order in two-dimensional bosonic lattice systems.