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

We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information, can characterize topological quantum phase transitions in the spin-21 toric code model (TCM) on a square lattice with external fields. We show that the quantum Fisher information 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 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 quantum Fisher information, we demonstrate the absence of finite-temperature topological order in the two-dimensional TCM 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.

Automated summary

We demonstrate that multipartite entanglement witnessed by the quantum Fisher information can characterize topological quantum phase transitions in the spin-21 toric code model. We identify the topological order by its scaling behavior using the ground state's quantum Fisher information density. We also investigate the thermalization and disorder-assisted stabilization of topological order using the multipartite entanglement witness.