Quantum phase diagram of the three-dimensional subsystem toric code

We study the phase diagram of the Hamiltonian of the three-dimensional subsystem toric code (3d STC), as recently constructed by Kubica and Vasmer [arXiv:2106.02621]. In the sector with no stabilizer violations, we show that the effective Hamiltonian describes two copies of decoupled 3d Z2 lattice gauge theories, where the stabilizer constraints act as local gauge invariance.

Within the lattice gauge theories we find four phases, two of which are 3d toric code phases with deconfined e or m charges in the bulk, and two phases with no deconfined bulk excitations but support a 2d toric code on the boundary.

Due to the presence of deconfined charges, none of these phases is thermally stable when raised to finite temperatures.

Our main conclusions likely extend to the case of gauge color codes.