I am concerned about the theorem saying that there is no topological order in 1d. According to the seminal paper https://arxiv.org/pdf/1008.3745.pdf, there are no non-trivial topological phases in 1d (we talk about systems with local gapped Hamiltonians).
According to the definition, the topological phase is an equivalence class of ground states under local-unitary transformation. To my intuition, if two ground states belong to the same phase, their parent Hamiltonians lie in the same connected component of parameter space. So, I can continuously deform one Hamiltonian to get another one without closing the gap. Then, we can solve an ODE and find the unitary transformation relating two ground states.
So, what is the algorithm of defining whether two given states in the same phase or not? Let us take 1d spin chain with N sites and two arbitrary gapped ground states. If I consider transformations acting only at one site at once, I am coming to the QI classification of states. There are many different classes under LOCC, and LU-classes are even richer. Should we consider transformations acting on two consecutive sites after that? k-sites transformations? (k << N) Did anybody apply that definition in practice? My question is: for two given states, how we understand, whether an LU-transformation/constant depth quantum circuit connecting them exists or not?
I know that it is more physical and more computationally efficient to look at the fixed points of RG flow. I would appreciate your help.