Multipartite entanglement of the topologically ordered state in a perturbed toric code

TL;DR

This paper uses quantum Fisher information to characterize topological phase transitions and stability in a perturbed toric code, revealing how multipartite entanglement signals topological order and its robustness.

Contribution

It introduces a novel multipartite entanglement witness based on QFI to identify topological phases and analyze their stability under perturbations, thermalization, and disorder.

Findings

QFI density relates to Wilson loop expectation values.

Identifies $ ext{Z}_2$ topological order via scaling behavior.

Shows absence of finite-temperature topological order in thermodynamic limit.

Abstract

We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information (QFI), can characterize topological quantum phase transitions in the spin-$\frac{1}{2}$ toric code model on a square lattice with external fields. We show that the QFI density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify $\mathbb{Z}_2$ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilization of topological order after a quantum quench. Moreover, with an upper bound of the QFI, we demonstrate the absence of finite-temperature topological order in the 2D toric code model in the thermodynamic limit. Our results provide insights to topological phases, which are robust against external disturbances, and are candidates for topologically protected quantum computation.