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