Tensor network approach to phase transitions of a non-Abelian topological phase

TL;DR

This paper introduces a tensor network method to study phase transitions from a non-Abelian Fibonacci topological phase, revealing its enclosure by three distinct non-topological phases and analyzing their critical properties.

Contribution

It develops a tensor network approach to map the phase diagram of a Fibonacci topological phase and identifies the nature of phase transitions and surrounding phases.

Findings

Fibonacci topological phase is enclosed by three non-topological phases.

Phase transitions are characterized by critical properties.

The norm of the wavefunction maps to a two-coupled Potts model partition function.

Abstract

The non-abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wavefunction with two tuning parameters dual with each other, and the norm can be exactly mapped into a partition function of the two-coupled $\phi^{2}$-state Potts models, where $\phi =(\sqrt{5}+1)/2$ is the golden ratio. By developing the tensor network representation of this wavefunction on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-abelian Fibonacci topological phase is enclosed by three distinct non-topological phases and their dual phases of a single $\phi^{2}$-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those non-topological phases.