# Boson condensation and instability in the tensor network representation of topological states

**Abstract**:The tensor network representation of many-body quantum states, given by local tensors provides a promising numerical tool for the study of strongly correlated topological phases in two dimension. However, the topological order in tensor network representations of the Toric code ground state has been shown to be unstable under certain small variations of the local tensor, if these small variations does not obey the local Z2 symmetry of the local tensor. In this work we ask the questions of whether other types of topological orders suffer from similar kinds of instability and if so, whether we can protect the order by enforcing certain symmetry on the tensor. We answer these questions by showing that the tensor network representation of all string-net models are indeed unstable, but the matrix product operator (MPO) symmetry identified by Burak *et al.* can help to protect the order. We find that, `stand-alone' variations that break MPO symmetry lead to instability because they induce the condensation of bosonic quasi-particles and destroy the topological order in the system. Therefore, such variations must be forbidden for the encoded topological order to be reliably extracted from the local tensor. On the other hand, if a tensor network algorithm is used to simulate the phase transition due to boson condensation, then such variation directions must be allowed in order to access the continuous phase transition process correctly.

**Series:**IQIM Postdoctoral and Graduate Student Seminar Series