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We study quantum quenches in the two-dimensional Kitaev toric code model andcompute exactly the time-dependent entanglement entropy of the non-equilibriumwave-function evolving from a paramagnetic initial state with the toric codeHamiltonian. We find that the area law survives at all times. Adding disorderto the toric code couplings makes the entanglement entropy per unit boundarylength saturate to disorder-independent values at long times and in thethermodynamic limit. There are order-one corrections to the area law from thecorners in the subsystem boundary but the topological entropy remains zero atall times. We argue that breaking the integrability with a small magnetic fieldcould change the area law to a volume scaling as expected of thermalized statesbut is not sufficient for forming topological entanglement due to the presenceof an excess energy and a finite density of defects.

Exact results on the quench dynamics of the entanglement entropy in the toric code