Competing topological orders in three dimensions: X-cube versus toric code

We study the competition between two different topological orders in three dimensions by considering the X-cube model and the three-dimensional toric code. The corresponding Hamiltonian can be decomposed into two commuting parts, one of which displays a self-dual spectrum. To determine the phase diagram, we compute the high-order series expansions of the ground-state energy in all limiting cases. Apart from the topological order related to the toric code and the fractonic order related to the X-cube model, we found two new phases which are adiabatically connected to classical limits with nontrivial sub-extensive degeneracies. All phase transitions are found to be first order.

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