Topological phase transitions induced by varying topology and boundaries in the toric code

One of the important characteristics topological phases matter is topology underlying manifold on which they are defined. In this paper, we present sensitivity such to topology, by studying phase transitions induced due change in boundary conditions. We claim that these accompanied broken symmetries excitation space and gain further insight analyze various signatures like ground state degeneracy, entanglement entropy while introducing open-loop operator whose expectation value effectively captures transition. Further, extend analysis an open quantum setup defining effective collapse operators, dynamics cool system different topologically ordered steady states. show transition between states captured operator.