Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands

TL;DR

This study uses exact diagonalization to analyze topological order in fractional quantum Hall systems within topological flat bands, identifying minimal entangled states and extracting universal modular matrices that encode quasi-particle statistics.

Contribution

It introduces a method to identify minimal entangled states and extract modular matrices in topological flat band models, demonstrating their robustness across phase transitions.

Findings

Minimal entangled states form an orthogonal basis for the ground state manifold.

Modular matrices S and U encode quasi-particle statistics and are universal within the topological phase.

These matrices remain robust against perturbations until a quantum phase transition occurs.

Abstract

We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices S (U) which contains the information of mutual (self) statistics, quantum dimensions and fusion rule of quasi-particles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.