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We investigate the quantum phase diagram of the $K$-layer Ising toric codecorresponding to $K$ layers of two-dimensional toric codes coupled by Isinginteractions. While for small Ising interactions the system displays$\mathbb{Z}_2^K$ topological order originating from the toric codes in eachlayer, the system shows $\mathbb{Z}_2$ topological order in the high-Isinglimit. The latter is demonstrated for general $K$ by deriving an effectivelow-energy model in $K^{\rm th}$-order degenerate perturbation theory, which isgiven as an effective anisotropic single-layer toric code in terms ofcollective pseudo-spins 1/2 refering to the two ground states of isolated Isingchain segments. For the specific cases $K=3$ and $K=4$ we apply high-orderseries expansions to determine the gap series in the low- and high-Ising limit.Extrapolation of the elementary energy gaps gives convincing evidence that theground-state phase diagram consists of a single quantum critical point in the3d Ising* universality class for both $K$ separating both types of topologicalorder, which is consistent with former findings for the bilayer Ising toriccode.

Quantum phase transitions in the $K$-layer Ising toric code