There are different categories of topological superconductors. Iâ€™m guessing that you are referring to the time-reversal invariant (class DIII) ones, in 2D or 3D. Yes, it is possible to distinguish the surface/edge states of 3D/2D topological superconductors from the bulk. I'm not talking about designing some intricate experimental technique to separate out the contribution from the bulk and the surface; as it turns out, they don't need any separating! The bulk and surface states are distinct in a very fundamental way. That statement is true not just in topological superconductors but also in topological insulators. First allow me to make a few comments on topological insulators case.

Yes, an ideal topological insulator is insulating in the bulk and conducting on the surface/edge. Despite the existence of an extensive list of (3D) **band** topological insulators discovered to date, none of them are insulating in the bulk! The Kondo topological insulator SmB$_6$ is the only one with the properties of an ideal topological insulator; but the Kondo gap opens up only at cryogenic temperatures ($\approx$ 4 K). So this example is not â€śnaturalâ€ť in the sense that it doesnâ€™t exist in a topological phase under ambient conditions.

The reason we can distinguish between the surface (in the 3D case) and bulk states of a topological insulator, even when the bulk is conducting, is because the surface forms the so-called â€śhelical metal.â€ť The conducting electronic states in this helical metal do not hybridize (or mix) with the conducting states in the bulk; picture this with the analogy of two immiscible liquids in a container. The following paper discusses in detail how topological surface states refuse to hybridize with the bulk

Bohm-Jung Yang, Mohammad Saeed Bahramy, and Naoto Nagaosa.
â€śTopological protection of bound states against the hybridization.â€ť
*Nature communications* **4** (2013): 1524. (arXiv version)

These helical states, which form a 2D electronic system on the boundary of a 3D system, have half the degrees of freedom (DOF) of an equivalent purely 2D system; the other half (DOF) are on the other surface (look up the holographic principle). As a result, no time-reversal symmetry preserving perturbation can gap them out. In other words, they are topologically protected (due to symmetry). Another important thing to stress is that just because an insulator is topologically trivial doesnâ€™t mean it cannot have surface states. The only difference is that these states **can** be destroyed.

Now, in the case of topological superconductors, the surface states are even more exotic. The surfaces of 3D time-reversal invariant topological superconductors host Majorana modes. You would observe a topologically protected Dirac cone, just like a 3D topological insulator, on the surface of a topological superconductor. The only difference is that this Dirac cone represents the dispersion of Majorana modes instead of Dirac fermions and has half the degrees of freedom due to particle-hole symmetry. There is a tonne of literature out there which proposes schemes of experimentally detecting these Majorana modes. For the case of 3D time-reversal invariant topological superconductors, however, I am familiar with only two: Scanning Tunneling Microscopy (STM) and Angle-Resolved Photoemission Spectroscopy (ARPES). In a point contact spectroscopy measurement on a topological superconductor you are supposed to see a so-called Zero-Bias Conductance Peak (ZBCP) inside the bulk gap as shown in Fig. 2 of:

Satoshi Sasaki, M. Kriener, Kouji Segawa, Keiji Yada, Yukio Tanaka,
Masatoshi Sato, and Yoichi Ando. â€śTopological Superconductivity in
Cu$_x$Bi$_2$Se$_3$.â€ť *Physical Review Letters* **107**, no. 21 (2011):
217001. (arXiv version)

One small disclaimer: there is still a lot of debate whether the authors of the above paper have really seen the Majorana mode or not. For example check this article which refutes the claims of the above paper:

Niv Levy, Tong Zhang, Jeonghoon Ha, Fred Sharifi, A. Alec Talin, Young
Kuk, and Joseph A. Stroscio. â€śExperimental Evidence for $s$-Wave
Pairing Symmetry in Superconducting Cu$_x$Bi$_2$Se$_3$ Single Crystals
Using a Scanning Tunneling Microscope.â€ť *Physical Review Letters* **110**,
no. 11 (2013): 117001.

(arXiv version)

Alternatively, instead of the momentum-integrated density of states in STM, you can directly â€śseeâ€ť the Dirac cone of Majorana modes using ARPES since you are measuring the bandstructure directly. As matter of fact, the existence of first and second generation of 3D topological insulators was identified using precisely this technique. To my knowledge, no one has so far observed Majorana modes on the surface of a topological superconductor using ARPES. The main reasons are technical: cooling capabilities, energy resolution, lack of stoichiometrically stable (unlike Cu$_x$Bi$_2$Se$_3$) material candidates which can be cleaved to give atomically flat surfaces, etc.

Aside from time-reversal invariant topological superconductors, a broad list of schemes to engineer topological superconductors and detect their Majorana modes can be seen in the following (excellent) review articles:

Jason Alicea. â€śNew directions in the pursuit of Majorana fermions in
solid state systems.â€ť *Reports on Progress in Physics* **75**, no. 7
(2012): 076501. (arXiv version)

C. W. J. Beenakker, â€śSearch for Majorana Fermions in
Superconductors.â€ť *Annual Review of Condensed Matter Physics*
**4**, no. 1 (2013): 113-136. (arXiv version)

**Details** (continued from the comments section below)

As far as the ZBCP goes, I feel that it is still more reliable using STM (compared to transport) for 3D superconductors (class DIII). For the system investigated by Kouwenhoven and others in the 1D Rashba nanowires (class D) the situation is way more ambiguous. Transport is way more finicky; it's (obviously) not as *direct* as STM or ARPES. The discrepancies in the above mentioned STM studies arise (in Yoichi Ando's and my opinion) because Cu$_x$Bi$_2$Se$_3$ is very hard to work with. Since Cu$_x$Bi$_2$Se$_3$ is not stoichiometrically stable, the dopant copper atoms diffuse and form â€śpocketsâ€ť of superconducting and normal (or topological and non-topological) regions (depending on local density of Cu). Ando thinks that the two studies measured different such regions! That being said, I don't know if Cu$_x$Bi$_2$Se$_3$ is topological if the above problem didn't exist. I'm just pointing out the source of the confusion.

Few comments on ARPES: this is the real deal! You can indeed have a smoking gun detection using ARPES; after all you see the Majorana mode dispersion directly. The reason people haven't seen this yet, as I mentioned, is mainly technical. First of all, the state-of-the-art ARPES systems can go to into the $\approx$ 10 K range. Most topological superconductor candidates have $T_c$â€™s way below that (e.g. half-Heusler compounds and heavy-fermion superconductors). Some (laser-based) systems can go down to 1.5 K. It might seem that Cu$_x$Bi$_2$Se$_3$ (with $T_c = 3.8$ K) might be perfect. But it's not enough to go *just* below $T_c$. You have to go *way* below in temperature to see the (superconducting) gap features clearly (i.e. when the gap is big enough); the gap size roughly follows a power law like $\approx 1/T^2$ below $T_c$. Also the gaps of many candidate materials are very small (compared to topological insulator band gaps). Conventionally synchrotron-based ARPES is the state-of-the-art tool for doing ARPES experiments. However, recently laser-based ARPES is becoming more commonplace in labs. Apart from the compactness and significantly less resource consumption (both monetary and operating manpower) laser-based ARPES is becoming popular because itâ€™s â€śclean.â€ť Check out this image to see what I mean by that:

The synchrotron-based ARPES systems have more â€śjuiceâ€ť (high photon energies in the tens of eV) as opposed to laser-based ones (6-7 eV), but they have very poor resolution. Good resolution is what we really need if we want to see the features of the Majorana modes inside the *tiny* superconducting gap.

Few comments on potential topological superconductor candidates: in addition to being very good topological insulator candidates, some of the half-Heusler compounds become superconducting at very low temperatures (1-2 K). For example LaPtBi, YPtBi, and LuPtBi have $T_c$â€™s of 0.9 K, 0.77 K, 1 K respectively. Unfortunately the samples of these materials that have been grown so far are â€śrock-likeâ€ť (as opposed to crystalline) and itâ€™s very hard to cleave them get an atomically flat surface to do ARPES on. After all the Majorana modes live on the surface!

Long story short, ARPES can give us smoking gun signatures Majorana modes. Iâ€™m sure there are other clever (indirect) ways of getting smoking gun signatures as well.

Let me first answer your question "is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?"

(1) Topological superconductors, by definition, are free fermion states that have time-reversal symmetry but no U(1) symmetry (just like topological insulator always have time-reversal and U(1) symmetries by definition). Topological superconductor are not p+ip superconductors in 2+1D. But it can be p-wave superconductors in 1+1D.

(2) 1+1D topological superconductor is a SET state with a Majorana-zero-mode at the chain end. But time reversal symmetry is not important. Even if we break the time reversal symmetry, the Majorana-zero-mode still appear at chain end. In higher dimensions, topological superconductors have no topological order. So they cannot be SET states.

(3) In higher dimensions, topological superconductors are SPT states.

The terminology is very confusing in literature:

(1) Topological insulator has trivial topological order, while topological superconductors have topological order in 1+1D and no topological order in higher dimensions.

(2) 3+1D s-wave superconductors (or text-book s-wave superconductors which do not have dynamical U(1) gauge field) have no topological order, while 3+1D real-life s-wave superconductors with dynamical U(1) gauge field have a Z2 topological order. So 3+1D real-life topological superconductors (with dynamical U(1) gauge field and time reversal symmetry) are SET states.

(3) p+ip BCS superconductor in 2+1D (without dynamical U(1) gauge field) has a non-trivial topological order (ie LRE) as defined by local unitary (LU) transformations. Even nu=1 IQH state has a non-trivial topological order (LRE) as defined by LU transformations. Majorana chain is also LRE (ie topologically ordered). Kitaev does not use LU transformation to define LRE, which leads to different definition of LRE.

## Best Answer

The something is the superconducting order parameter, which is

loosely$\Delta_{\alpha\beta}(r-r')=\langle\psi_\alpha(r)\psi_\beta(r')\rangle$ where $\psi_{\uparrow(\downarrow)}$ is the operator that annihilates a spin up (spin down) electron. Now $\Delta$ must transform under the symmetry group of the crystal. So the terms $s, p, d$ and all their ilk refer to the possible representations.Except that $s, p, d$ and so forth label representations of $SO(3)$, whereas $\Delta$ should transform under the point group $G$ of the crystal, which is a discrete subgroup of $SO(3)$. So a single irreducible representation of $SO(3)$ like spin 2 (aka $d$) actually may break up into multiple irreps of $G$. So the usual language is somewhat confusing, but any lengthy paper should define precisely what it means.