Quantum critical phase transition between two topologically-ordered phases in the Ising toric code bilayer

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly-coupled \(\mathbb{Z}_2\times\mathbb{Z}_2\) and the strongly-coupled \(\mathbb{Z}_2\) topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the 3d Ising\(^*\) universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local \(\mathbb{Z}_2\) conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a non-local order parameter.