We implement the universal wave-function overlap (UWFO) method to extract modular $S$ and $T$ matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular $S$ and $T$ matrices generate a projective representation of $SL(2,\mathbb{Z})$ on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e., the quasiparticle statistics and chiral central charge (up to $E_8$ bosonic quantum Hall states). We use the variational Monte Carlo method to computed the $S$ and $T$ matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and we confirm that the CSL carries the same topological order as the $\nu =\frac{1}{2}$ bosonic Laughlin state. We find that the nonuniversal exponents in the UWFO can be small, and direct numerical computation can be applied on relatively large systems. The UWFO may be a powerful method to calculate the topological order in GPWFs.

• Received 6 October 2014
• Revised 22 February 2015

DOI:https://doi.org/10.1103/PhysRevB.91.125123