Topological Characterization of Non-Abelian Moore-Read State using Density-Matrix Renormailzation Group

TL;DR

This paper uses density-matrix renormalization group calculations to thoroughly characterize the non-Abelian Moore-Read state on a lattice, revealing its topological properties, quasiparticle statistics, and conformal field theory correspondence.

Contribution

It provides the first comprehensive microscopic characterization of the non-Abelian Moore-Read state, including edge spectrum, quasiparticle statistics, and modular matrices, on a lattice model.

Findings

Identified three degenerate ground states with distinct anyonic fluxes.

Confirmed entanglement spectrum matches conformal field theory predictions.

Extracted modular matrices revealing non-Abelian statistics consistent with $SU(2)_2$ Chern-Simons theory.

Abstract

The non-Abelian topological order has attracted a lot of attention for its fundamental importance and exciting prospect of topological quantum computation. However, explicit demonstration or identification of the non-Abelian states and the associated statistics in a microscopic model is very challenging. Here, based on density-matrix renormalization group calculation, we provide a complete characterization of the universal properties of bosonic Moore-Read state on Haldane honeycomb lattice model at filling number $\nu=1$ for larger systems, including both the edge spectrum and the bulk anyonic quasiparticle (QP) statistics. We first demonstrate that there are three degenerating ground states, for each of which there is a definite anyonic flux threading through the cylinder. We identify the nontrivial countings for the entanglement spectrum in accordance with the corresponding conformal field theory. Through inserting the $U(1)$ charge flux, it is found that two of the ground states can be adiabatically connected through a fermionic charge-$\textit{e}$ QP being pumped from one edge to the other, while the ground state in Ising anyon sector evolves back to itself. Furthermore, we calculate the modular matrices $\mathcal{S}$ and $\mathcal{U}$, which contain all the information for the anyonic QPs. In particular, the extracted quantum dimensions, fusion rule and topological spins from modular matrices positively identify the emergence of non-Abelian statistics following the $SU(2)_2$ Chern-Simons theory.