Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions

TL;DR

This paper introduces a method using universal wave function overlaps to extract modular matrices from Gutzwiller-projected wave functions, enabling characterization of topological order in 2+1D systems.

Contribution

The authors develop and apply the UWFO method with Monte Carlo techniques to compute modular matrices for topological states in GPWFs, demonstrating its effectiveness on large systems.

Findings

Confirmed the CSL has the same topological order as the $ u=1/2$ bosonic Laughlin state.

Found small non-universal exponents in UWFO, allowing for efficient numerical calculations.

Extended UWFO applicability to various lattice geometries in 2D and 3D.

Abstract

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 quasi-particle statistics and chiral central charge (up to $E_8$ bosonic quantum Hall states). We used 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 confirm that the CSL carries the same topological order as the $\nu=\frac{1}{2}$ bosonic Laughlin state. We find that the non-universal exponents in UWFO can be small and direct numerical computation is able to be applied on relatively large systems. We also discuss the UWFO method for GPWFs on other Bravais lattices in two and three dimensions by using the Monte Carlo method. UWFO may be a powerful method to calculate the topological order in GPWFs.