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.

Automated summary

We investigate the competition between two different topological orders in three dimensions by studying the X-cube model and the three-dimensional toric code. By decomposing the corresponding Hamiltonian and calculating high-order series expansions of the ground-state energy, we determine the phase diagram and the nature of phase transitions.