Establishing non-Abelian topological order in Gutzwiller projected Chern insulators via Entanglement Entropy and Modular S-matrix

TL;DR

This paper demonstrates non-Abelian topological order in Gutzwiller projected Chern insulators using entanglement entropy and modular S-matrix calculations, revealing non-Abelian braiding statistics of Majorana quasiparticles.

Contribution

It establishes a method to identify non-Abelian topological order in projected Chern insulators via entanglement signatures and modular S-matrix, linking wavefunctions to Chern-Simons theories.

Findings

Gutzwiller projected Chern insulator wavefunctions exhibit non-Abelian topological order.

The modular S-matrix confirms non-Abelian braiding statistics of Majorana fermions.

Microscopic evidence shows that powers of Chern number C wavefunctions realize SU(N)_C topological order.

Abstract

We use entanglement entropy signatures to establish non-Abelian topological order in projected Chern-insulator wavefunctions. The simplest instance is obtained by Gutzwiller projecting a filled band with Chern number C=2, whose wavefunction may also be viewed as the square of the Slater determinant of a band insulator. We demonstrate that this wavefunction is captured by the $SU(2)_2$ Chern Simons theory coupled to fermions. This is established most persuasively by calculating the modular S-matrix from the candidate ground state wavefunctions, following a recent entanglement entropy based approach. This directly demonstrates the peculiar non-Abelian braiding statistics of Majorana fermion quasiparticles in this state. We also provide microscopic evidence for the field theoretic generalization, that the Nth power of a Chern number C Slater determinant realizes the topological order of the $SU(N)_C$ Chern Simons theory coupled to fermions, by studying the $SU(2)_3$ (Read-Rezayi type state) and the $SU(3)_2$ wavefunctions. An advantage of our projected Chern insulator wavefunctions is the relative ease with which physical properties, such as entanglement entropy and modular S-matrix can be numerically calculated using Monte Carlo techniques.