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, tensor network
representations may be vulnerable to instabilities caused by small
perturbations of the local tensor, especially when the local tensor is not
injective. For example, 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 do not obey a
local Z2 symmetry of the tensor. In this paper, we ask the questions of
whether other types of topological orders suffer from similar kinds of
instability and if so, what is the underlying physical mechanism and whether we
can protect the order by enforcing certain symmetries 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)
symmetries of the local tensor can help to protect the order. We find that,
`stand-alone' variations that break the MPO symmetries 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 based variational 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.Comment: 44 pages, 85 figures, comments welcom
