I know of several definitions of phases of matter:
- The first is the "old" one, Landau theory and symmetry breaking. In this definition we pick a local order parameter $m$ (as far as I can tell this is quite vaguely defined). If the state had the same symmetry of the Hamiltonian, we would have $\langle m\rangle=0$, but as some parameter is varied and crosses a critical value, we observe spontaneous symmetry breaking and $\langle m \rangle\neq 0$. We know this definition does not describe all phases as it doesn't include topological phases.
- Let $H_0$ and $H_1$ be two Hamiltonians. The two Hamiltonians (or their ground states) are in the same phase if there exists a continuous gapless path $\gamma \mapsto H_\gamma$ interpolating them. If we include the requirement that $H_\gamma$ must have some symmetry for all $\gamma$, we encounter the notion of symmetry protected topological order.
- One of the keywords in Landau's definition is local order parameter. We can find new phase transitions by considering non-local order parameters, such as the string order parameter in the Haldane phase.
Now: the Haldane phase is the prototypical example of both spontaneous symmetry breaking signalled by a non-local order parameter and symmetry protected topological order. Since in SPTO we require the whole phase to have a symmetry, by definition the symmetry must be broken to exit the phase. Since we call this "topological" order, I guess this symmetry breaking cannot be detected by a local order parameter, so, my questions are the following:
Is the symmetry breaking between SPTO phases always signalled by a non-local order parameter?
Conversely, is symmetry breaking with a non-local order parameter always SPTO (there exists a gapless path...)
In general, what is the relation between non-local order parameters and topological phases?