Topologically ordered states are fundamentally important in theoretical physics, which are also suggested as promising candidates to build fault-tolerant quantum devices. However, it is still elusive how topological orders can be affected or detected under noises. In this work, we find a quantity, termed as the ring degeneracy $\mathcal{D}$, which is robust under pure noise to detect both trivial and intrinsic topological orders. The ring degeneracy is defined as the degeneracy of the solutions of the self-consistent equations that encode the contraction of the corresponding tensor network (TN). For the $Z_N$ orders, we find that the ring degeneracy satisfies a simple relation $\mathcal{D}=(N+1)/2+d$, with $d=0$ for odd $N$ and $d=1/2$ for even $N$. Simulations on several nontrivial states (two-dimensional Ising model, $Z_N$ topological states, and resonating valence bond states) show that the ring degeneracy can tolerate noises up to a strength associated to the gap of the TN boundary theory.

• Received 9 October 2018
• Revised 25 March 2019

DOI:https://doi.org/10.1103/PhysRevB.99.195101