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Title:
Instability in tensor network representations of topological phases
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Abstract:
The Tensor Network (TN) representation of many-body quantum states, given by local tensors, provides a promising numerical and conceptual tool for the study of strongly correlated topological phases in two dimension. However, TN representations may be vulnerable to instabilities caused by small variations of the local tensor, especially when the local tensor is not injective. For example, the topological order in TN representations of the toric code ground state has been shown to be unstable: the topological order is stable under variations if and only if the variations respect a $Z_2$ symmetry of the local tensor. In this work, we ask 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 variations. We answer these questions by showing that the tensor network representation of all string-net models are indeed unstable and though enforcing the so-called matrix product operator (MPO) symmetries (which can be thought of as the generalization of the $Z_2$ symmetry of the toric code local tensor) on the local tensors is sufficient for stability, it is not a necessary condition. We show that if a small variation cannot "stand alone" in the tensor network, it also cannot cause instability, even if it breaks the MPO symmetry. In fact, a variation is unstable if and only if it breaks the MPO symmetry and can also stand alone, and this explains all instability behavior observed numerically. The physical reason of instability is found to be the fact that unstable variations induce the condensation of bosonic quasi-particles and destroy the topological order in the system. Therefore, MPO symmetry breaking variations that can stand alone should be forbidden for the encoded topological order to be reliably extracted from the local tensor. On the other hand, if a TN 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.
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