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

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

Automated summary

The paper examines the sensitivity of topological phases of matter to the underlying topology, showing that phase transitions are accompanied by symmetry breaking in the excitation space. Various signatures like ground state degeneracy and topological entanglement entropy are analyzed, while the introduction of the open-loop operator effectively captures the phase transition. Furthermore, the analysis is extended to an open quantum setup by defining effective collapse operators that cool the system to distinct steady states, both of which are topologically ordered, with the phase transition between them effectively captured by the expectation value of the open-loop operator.