Quantum phase transitions in the K-layer Ising toric code

We investigate the quantum phase diagram of the K-layer Ising toric code corresponding to K layers of twodimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays 1K2 topological order originating from the toric codes in each layer, the system shows 12 topological order in the high-Ising limit. The latter is demonstrated for general K by deriving an effective low-energy model in Kth-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases K = 3 and K = 4 we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both K separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.