Topological line in frustrated toric code models

Typical topological systems undergo a topological phase transition in the presence of a strong enough perturbation. We propose an adjustable frustrated toric code with a ``topological line'' at which no phase transition happens in the system and the topological order is robust against a nonlinear perturbation of arbitrary strength. This important result is a consequence of the interplay between frustration and nonlinearity in our system, which also causes the emergence of other interesting phenomena such as reentrant topological phases and survival of the topological order under local projection operations. Our study opens a window towards more robust topological quantum codes which are the cornerstones of large-scale quantum computing.