Exact results of the quantum phase transition for the topological order

TL;DR

This paper uses duality between models to analyze a quantum phase transition from a topologically ordered state to a spin-polarized phase, revealing non-local order parameters and characterizing the transition without symmetry breaking.

Contribution

It establishes an exact duality-based framework to understand the quantum phase transition in topological order, introducing non-local string order parameters.

Findings

The transition is not driven by spontaneous symmetry breaking.

String-like non-local order parameters characterize the QPT.

Duality relates open-string and closed-string behaviors at the transition.

Abstract

In this paper a duality between the d=2 Wen-plaquette model in a transverse field and the d=1 Ising model in a transverse field is used to learn the nature of the quantum phase transition (QPT) between a spin-polarized phase and a topological ordered state with the string-net condensation. The QPT is not induced by spontaneous symmetry breaking and there are no conventional Landau-type local order parameters. Instead the string-like non-local order parameters are introduced to describe the QPT. In particular, the duality character between the open-string and closed string for the QPT are explored.