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

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.

5 pages with 3 figures

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• Chiral Spin Liquid and Quantum Criticality in Extended $S=1/2$ Heisenberg Models on the Triangular Lattice
• Machine Learning ${\mathbb{Z}}_2$ Quantum Spin Liquids with Quasi-particle Statistics
• Nature of chiral spin liquids on the kagome lattice
• Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields
• Experimentally Probing Topological Order and Its Breakdown via Modular Matrices
• Abelian SU$(N{)}_1$ Chiral Spin Liquids on the Square Lattice
• Entanglement entropy of (3+1)D topological orders with excitations
• Intrinsic sign problem in fermionic and bosonic chiral topological matter
• Intrinsic sign problems in topological quantum field theories
• Time-reversal symmetry breaking Abelian chiral spin liquid in Mott phases of three-component fermions on the triangular lattice
• Gapped domain walls between 2+1D topologically ordered states
• Classification and emergence of quantum spin liquids in chiral Rydberg models
• Anyon Quantum Dimensions from an Arbitrary Ground State Wave Function
• Detecting topological order from modular transformations of ground states on the torus
• Geometric and computational aspects of chiral topological quantum matter