What does it mean for a topological phase to be "symmetry protected"? I have seen some very nice and enlightening awnsers to questions related to topological order and insulators, such as here, or here. However, I'm still puzzled by the concept of "symmetry protection" of phases.
Topological phases are said to be robust again any local perturbations. In other words, is the topological order protected by some sort of "unbreakable" internal symmetry that prevents the system to go to another phase ?
Sorry if my question seems a bit vague, I lack understanding on the topic and would very much appreciate a general awnser.
 A: I think the first helpful fact to clarify is that there are two different kinds of topological phases: there are so-called Symmetry Protected Topological (SPT) Phases (displaying 'symmetry protected topological order') and there are (intrinsic) Topological Phases (displaying '(intrinsic) topological order'). As some quick examples: topological insulators and topological superconductors are example of the former, whereas the quantum Hall states and various spin liquids are examples of the latter.
There are no clear-cut one-line definitions of either that everyone would agree with, but both have clear characteristics. But let me first make a different point, to clarify both the similarity and difference with the more usual spontaneous symmetry breaking phases: all phases characterized by spontaneous symmetry breaking are also symmetry protected! Take for example the 1D quantum Ising model $$H = \sum_n (- \sigma^z_n \sigma^z_{n+1} + g \sigma^x_n)$$
This spin hamiltonian has a $\mathbb Z_2$ symmetry defined by flipping the spin in the $z$-direction, and as is well-known the ground state spontaneously breaks this symmetry for $g <1$, in this case called a ferromagnetic phase. Now, this phase is stable under arbitrary perturbations, on the condition that these perturbations respect this $\mathbb Z_2$ symmetry. As soon as you add for example a term $h \sigma^z$, then the ferromagnetic phase is no longer well-defined, or put differently: it is no longer protected, meaning you can continuously deform the Hamiltonian and ground state to become a trivial paramagnet without encountering any phase transitions. So in a sense whenever we have spontaneous symmetry breaking, we should call it Symmetry Protected Spontaneous Symmetry Breaking. That would however be a redundancy in description, since any spontaneous symmetry breaking needs such symmetry protection.
That is different for topological phases: some phases are, as you say, stable under arbitrary perturbations. That is, you cannot connect the phase to a trivial(*) phase without encountering a phase transition. These are called (intrinsic) topological phases. So this is a whole different game, and in a way it is remarkable that such phases exist at all. The intuitive reason is that these phases have long range entanglement in the ground state (meaning that far-away sites of the system can be strongly entangled with one another), and you cannot undo such entanglement by mere local perturbations. This long range entanglement is also the cause for many interesting physical consequences. In particular such intrinsic topological phases have very interesting low energy excitations (the concept of 'fractionalization' comes up here: at low energy the system behaves as if there are exotic particles, more complex than the actual particles that make up the system on a more fundamental level). There are two conceptual reasons for why such phases are called 'topological': (1) in the mathematical treatment of these phases, the mathematical field of topology often pops up (for example in the context of the low energy theory being a topological quantum field theory, or by the order parameter of the phase being a certain topological invariant); (2) on a more mundane level, the system behaves differently if you put it on spaces with different topology: the system for example behaves differently if you put it on a open slab (e.g. a rectangle) versus on at torus (i.e. no boundaries). Both (1) and (2) are associated to the fact that such topological phases have interesting phenomena on their physical boundaries.
The previous paragraph focused more on such phases with intrinsic topological order, where symmetry is not that important (although there are interesting formulations of topological order where symmetry comes up, but that would be too much of a digression). As mentioned before, there are also symmetry protected topological phases: these have more short range entanglement, and like symmetry breaking phases the phases are only well-defined if the Hamiltonian has a certain symmetry. However the ground state does not break this symmetry spontaneously. Instead the system behaves 'topologically' in the above sense: often topology comes up in its mathematical description, and there peculiar effects going on around the boundary (like the gapless edges of topological insulators). The main physical difference between the intrinsic and the symmetry protected topological phases, is that the former hosts fascinating behaviour in the bulk (i.e. away from the boundaries), in the form of exotic/fractional quasi-particles, whereas the latter (being the symmetry protected topological phases) doesn't have much going on in the bulk, and the mainly interesting physics is on the boundary (however it does have some interesting structure in the entanglement of the bulk --despite the entanglement being short range-- but this is quite difficult to measure physically).
I hope this gives a bit of the general overview that you were asking for. Let me know if I can clarify further.
(*) One can wonder how one defines the notion of 'trivial phase'. A good rule of thumb is: a phase is trivial if one cannot say anything interesting about it.
