Abstract
We investigate the quantum phase diagram of the K-layer Ising toric code corresponding to K layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays Z₂^K topological order originating from the toric codes in each layer, the system shows Z₂ topological order in the high-Ising limit. The latter is demonstrated for general K by deriving an effective low-energy model in K^ th-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudo-spins 1/2 refering 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 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.