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

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.