Parameterized Family of Toric Code Phase

The generalization of the geometric phase of quantum states, known as the Berry phase, can be extended to many-body quantum systems. While geometric phases for short-range entangled states are well understood, those for topologically ordered states remain less clear, especially in terms of concrete lattice realizations.

In this presentation, we discuss parametrized families of topologically ordered states within the toric code phase and provide explicit lattice model examples, including families that pump the em-exchange defect and those that pump a boundary one-parameter family.