Let me shortly elaborate -- though not answering (at all? certainly :-) since all these notions are pretty new for me -- about what I know from my pedestrian work on superconductors.

I believe everything is much more complicated when you look at the details, as usual. I may also comment (and would love to be contradicted about that of course) that the Wen's quest for a beautiful definition about topological order somehow lets him going far away from realistic matter contrarieties.

$s$-wave superconductor may be seen as long-ranged topological order / entangled state, as in the review you refer to : Hansson, Oganesyan and Sondhi. But it is *also* symmetry protected. I think that's what you call a *symmetry-enriched-topological-order*. I'm really angry when people supposed the symmetry classification for granted. $s$-wave superconductors exhibit time-reversal symmetry in addition to the particle-hole one, and thus a chiral symmetry for free, so far so good. Suppose you add *one* magnetic impurity. Will the gap close ? Of course not ! The gap is robust up to a given amount of impurities (you can even adopt a handy argument saying that the total energy of the impurities should be of the same order than the energy gap in order to close the gap). You may even end up with *gapless-superconductor*... what's that beast ? Certainly not a topological ordered staff I presume. More details about that in the book by Abrikosv, Gor'kov and Dzyaloshinski. So my first remark would be : *please don't trust too much the classification*, but I think that was part of the message of your question, too. [Since we are dealing with the nasty details, there are also a lot of predicted *re-entrant superconductivity* when the gap close, and then reopens at higher magnetic field (say). Some of them have been seen experimentally. Does it mean the re-entrant pocket is topologically non-trivial ? I've no idea, since we are too far from the beautiful Wen's / east-coast definition I believe. But open -> close -> open gap is usually believe to give a topologically non trivial phase for condensed matter physicists.]

So my understanding about this point is: what does the symmetry do ?

If it *creates* the gap in the bulk an/or the gap closure at the edge, then it's not a good criterion. Most of the symmetry like this are generically written as *chiral* or *sub-lattice symmetry* (the $C\equiv PT$ in the classification). Some topological insulators only have this symmetry (grapheme in particular if I remember correctly), since the particle-hole ($P$) symmetry defines superconductors. According to Wen, this situation does not lead to topological order in the discussion we have previously.

If the symmetry *reenforces* the gap, then it protects you against perturbation. Once again, the time-reversal symmetry of the $s$-wave superconductor protects against any time-reversal perturbation (the Anderson theorem). But it is not responsible for the appearance of the gap ! I confess that's really a condensed-matter physicist point of view, which could be really annoying for those wanting beautiful mathematical description(s). But clearly the $T$-symmetry should change nothing about the long-ranged-entanglement for $s$-wave (according you believe in of course :-)

As for $p+ip$, it's also called *polar phase* in superfluid (recall there is no $p$-wave superconductor in nature at the moment, only in neutral superfluid). As Volovik discuss in his book, this phase is not stable (chapter 7 among others). It is sometimes referred as a *weak topological phase*, which makes no sense. It just corresponds to a fine tuning of the interaction(s) in superfluid. B-phase is robust, and fully gapped. So the pedestrian way is just a way to say that the $p+ip$ is not robust, and that you should have a structural transition (of the order parameter) to the more robust B-phase. NB: I may well confound the names of the subtle phases of superfluid, since it is a real jungle there :-(.

Finally, to try to answer your question: I would disagree. A topological-superconductor (in the condensed matter sense: a $p$-wave say) does not have $T$ symmetry. So it's hard for me to say it is symmetry-enriched the way I used it for $s$-wave. That's nevertheless the symmetry reason why $p$-wave (if it exists in materials) should be really weak with respect to impurities. [Still being in the nasty details: For some reasons I do not appreciate fully, the Majorana may be more robust than the gap itself, even if it's difficult to discuss this point since you need to discuss a dirty semiconductor with strong spin-orbit and paramagnetic effect in proximity with a $s$-wave superconductor, which is amazingly more complicated than a $p$-wave model with impurities, but this latter one should not exist, so...] Maybe I'm totally wrong about that. It sounds that words are not really helpful in topological studies. Better to refer to mathematical description. Say, if the low energy sector is described by Chern-Simons (CS), would we (or not ?) be in a topological order ? Then the next question would be: is this CS induced by symmetry or not ? (I have no answer for this, I'm still looking for mechanism(s) about the CS -- an idea for an other question, too).

*Post-scriptum:* I deeply apologize for this long answer which certainly helps nobody. I nevertheless have the secret hope that you start understanding my point of view about that staff: symmetry may help and have an issue in topological order, but they are certainly not responsible for the big breakthrough Wen did about long-ranged-entanglement. You need a local gauge theory for this, with global properties.

This post imported from StackExchange Physics at 2014-04-05 04:36 (UCT), posted by SE-user FraSchelle