U(1) x 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 function (GWF) carries a nontrivial SPT order, and which SPT order is carried by the wave function. 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 a nondegenerate ground state, and zero topological entanglement entropy, (3) a nontrivial U(1) x U(1) SPT order via the Hall conductances of the protecting U(1) x U(1) symmetry, and (4) a symmetry-protected gapless boundary. This represents numerical evidence of continuous symmetry-protected topological order in two-dimensional bosonic lattice systems.