# What makes a superconductor topological?

I have read a fair bit about topological insulators and proximity induced Majorana bound states when placing a superconductor in proximity to a topological insulator.

I've also read a bit about cuprates being related to topological superconductivity if that helps.

What I cant quite understand is what defines and what is a pure topological superconductor?

Or is this simply not the case and is topological superconductivity something that can only be achieved by means of proximity effect arrangements?

A general description of what one is would probably be most helpful.

In short, what makes a superconductor topological is the nontrivial band structure of the Bogoliubov quasiparticles. Generally one can classify non-interacting gapped fermion systems based on single-particle band structure (as well as symmetry), and the result is the so-called ten-fold way/periodic table. The topological superconductivity mentioned in the question is related to the class D, namely superconductors without any symmetries other than the particle-hole symmetry. The simplest example in 2D is a spinless $p_x+ip_y$ superconductor:

$H=\sum_k c_k^\dagger(\frac{k^2}{2m}-\mu)c_k+ \Delta c_k^\dagger(k_x+ik_y)c_{-k}^\dagger+\text{h.c.}=\sum_k (c_k^\dagger, c_{-k})\left[(k^2/2m-\mu)\tau_z+\Delta k_x\tau_x+\Delta k_y\tau_y\right]\begin{pmatrix}c_k\\ c_{-k}^\dagger\end{pmatrix}$

This Hamiltonian defines a map from the $k$ space (topologically a sphere $S^2$) to a $SU(2)$ matrix $m_k\cdot \sigma$ where $m_k\propto (\Delta k_x, \Delta k_y, \frac{k^2}{2m}-\mu)$ (then normalized), which also lives on a sphere. Therefore such maps are classified by $\pi_2(S^2)=\mathbb{Z}$. If two Hamiltonians belong to the same equivalence class in the homotopy group, it means that one can continuously deform the Hamiltonian from one to another without closing the gap, thus topologically indistinguishable.

The integer, called the Chern number $C$, that classifies the class D topological superconductors can be calculated from the Hamiltonian, and in this case it is $C=1$. This idea can be generalized to other symmetry classes and dimensions, basically one needs to understand the map from the momentum space to the appropriate single-particle "Hamiltonian" space (the general case is much more complicated than the $2\times 2$ Hamiltonian).

This toy model (and its one-dimensional descendants) is behind all recent proposals of realizing topological superconductors in solid state systems. The basic idea is to combine various mundane elements (semiconductors, s-wave superconductor, ferromagnet, etc): since electrons have spin-$1/2$, one needs to have Zeeman field to break the spin degeneracy and get a non-degenerate Fermi surface (thus effectively "spinless" fermions, really spin-polarized). However, in s-wave superconductors electrons with opposite spins are paired. This is why spin-orbit coupling is necessary since it makes the electron spin "winds" around on the Fermi surface, so that at $k$ and $-k$ electrons can pair up. Putting all these together one can realize a topological superconductor.

There are various physical consequences. The general feature is that something peculiar happens on the boundary between superconductors belonging to different topological classes. For example, if the $p_x+ip_y$ superconductor has an edge to the vacuum, there are gapless chiral Majorana fermions localized on the edge. Also if one puts a $hc/2e$ vortex into the superconductor, it traps a zero-energy Majorana bound state.

The question also mentioned cuprates. There are some speculations about the possibility of $d+id$ pairing in cuprates, probably motivated by measurement of Kerr rotations which is a signal of time-reversal symmetry breaking. However this is highly debatable and not very well accepted. Notice that $d+id$ superconductor is the $C=2$ case of the class D family.

• Thanks for writing out the explicit model, and thanks again for the link to this interesting review by Alicea. It's indeed more pedagogical than the original study I cited in my answer. Thanks again. – FraSchelle Mar 13 '15 at 10:05
• Thanks for giving a nice description of the ingredients for topological superconductors, the only part I cant seem to get my head round is the bit on how spin orbit coupling makes the electron spin "winds" around on the fermi surface. Could you please elaborate on that point? – Tom Rylands Mar 14 '15 at 12:50
• @FraSchelle The original references are also very useful to have:) – Meng Cheng Mar 14 '15 at 18:15
• @TomRylands I should have written down the math directly: the spin-orbit coupling is basically $\sigma\cdot p$(sometimes you also see $\sigma\times p$, but the difference between the previous one is just a spin rotation). Naively, to minimize the energy the spin has to be aligned/anti-alighed to momentum. Staying on the Fermi surface, that means the spin direction is rotating as you go around the Fermi surface. In particular, the spin directions at $p$ and $-p$ are opposite. – Meng Cheng Mar 15 '15 at 16:17
• @MengCheng Thanks for your answer. I may have an elementary question. I wonder when the Hamiltonian of $p_x+ip_y$ superconductor as you wrote down is put on torus, then what is the ground state degeneracy? I feel confused at the point that this Hamiltonian has a band, which indicates that it has only one ground state, ie. filling all the states below fermi surface. Am I wrong? What are other ground states? – hehuan0430 Mar 18 '15 at 19:52

A prototypical example of an intrinsic topological superconductor is the so-called $p$-wave superconductor [more details there: What is a $p_x + i p_y$ superconductor? Relation to topological superconductors, also, Meng-Cheng wrote the spinless $p$-wave model in 2D somewhere else on this page, and comment it carefully]. You can also induce topological non-trivial situation in $d$-wave superconductors, since the essential ingredient is the change of sign of the gap. All Cooper-pair based condensate would exhibit a change of sign in the momentum representation of the gap, except the $s$-wave case. The main problem is to transfer this momentum gap-closure into a spatial one.

Unfortunately, there is no known example of $p$-wave superconductors in nature. $p$-wave superfluids exist, and recent experiments aim at demonstrating the Majorana physics there.

Nevertheless, Gor'kov and Rashba in Phys. Rev. Lett. 87, 37004 (2001) shown that a conventional superconductor ($s$-wave) with spin-orbit interaction would lead to a mixture of both $s$- and $p$-wave correlations (*). Carefully selecting the spinless $p$-wave structure by means of a Zeeman effect may make the Majorana physics emerging, hence the proposals by a few peoples, see e.g.

associating $s$-wave superconductors in the proximity with spin-orbit systems under strong exchange interaction. Such a proposal is actually under experimental exploration in several groups around the world.

• (*): Note that there are a lot of papers by Edelstein studying similar effects all along the 80's and 90's but these papers are not as clear as the one by Gor'kov and Rashba to my taste

In order to maintain the gap -- an essential ingredient in the topological business as you already know from topological insulator -- it seems preferable to be in proximity, since bulk systems are not perfectly understood yet (role of impurities, exact gap-symmetry, multiple phase-transitions between different gap-symmetries, ... are still under debate, and pretty difficult to answer experimentally) and might well be less robust. About proximity-induced stable topological system in nano-wires, see e.g.

but clearly the subject of proximity and/or bulk is still vivid. In addition, there are a lot of different proposals to realise Majorana physics now, as e.g. spatially organised ferromagnetic macro-molecules on top of a superconductor, quantum-dots arrays, ... I do not enter into much details. My understanding about all these proposals is that they try to reproduce the same toy-model Hamiltonian as discussed in the above cited papers (and kindly wrote by @MengCheng in her answer somewhere else on this page). For an pedagogical review about toy-model of Majorana wires, please see J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys. 7, 412 (2011) or arXiv:1006.4395

Do not hesitate to ask further questions in this or separate post.

• The Lutchyn et. al. and Oreg et. al. papers are actually about proximity-induced topological superconductivity in semiconductor nanowires, whose proposals have been actively pursued by experimentalists. – Meng Cheng Mar 13 '15 at 2:03
• @MengCheng Thanks for this comment. This was clear in my head, not in the post indeed :-) – FraSchelle Mar 13 '15 at 9:53

I'd like to point out a different sense in which superconductors are topological, which was hinted at by Meng Cheng in his comment and also discussed in the reference he provided. The above discussions all primarily focused on superconductors where the fluctuations of the electromagnetic gauge field can be ignored. In this case, it is appropriate to use the BdG framework and the topological superconductors are examples of topological phases $\it{without}$ topological order i.e., there are no fractionalized excitations (or anyons) in the system.

However, it was pointed out quite some time ago in

https://arxiv.org/abs/cond-mat/0404327

that if the dynamics of the gauge field are taken into account, then even an $s$-wave superconductor is topologically ordered and has anyonic excitations that braid non-trivially with each other. Indeed, they showed that in 2+1d, an $s$-wave superconductor has $\mathbb{Z}_2$ topological order, same as the Toric Code. A related paper extended this to other ($d$-wave etc) superconductors and also discussed symmetries: https://arxiv.org/abs/1606.03462

I'm not certain how realistic these proposals are since they seem to require confining electromagnetism to 2 spatial dimensions, but in principle it's been known since Hansson et al. that superconductors are intrinsically topological.

• Though this answer is perfectly true, it is an other type of topology thing. It was clear enough (at least for me) that the question was about emergence of Majorana modes in a superconductor, which are then called topological superconductor. Majorana modes appear as zero-energy modes in a gapped system having certain symmetries, see the topological classification. In particular, you need a particle-hole symmetry to get a "superconductor", though you can only discuss first quantised Hamiltonian with this method (integrable models without interaction are of this type). – FraSchelle Jan 27 '17 at 5:12
• In a sense, a Majorana mode is nothing but a sub-gap solution of a given class of Hamiltonian, in the same way topological insulators have sub-gap states (usually called edge mode in this context). What you refer about is the topological order. A system is said to be in a topological phase when its low energy ground state can be described by a topological quantum field theory, as the BF theory in the paper you cite. So it is an intrinsic quantum field construction. – FraSchelle Jan 27 '17 at 5:15
• I hope this comment makes the difference clear. In particular, I hope it is clear that the fractional excitations you mentioned are not necessarily Majorana modes I mentioned. – FraSchelle Jan 27 '17 at 5:15

In simplest terms, the presence of sub-gap zero energy localized modes (Majorana modes) makes a superconductor topological. A superconducting ground state is just a bunch of Cooper pairs and the BdG Hamiltonian describes excitations above the ground state. If the excitation spectrum has these localized modes then it is a topological superconductor otherwise it is a non-topological superconductor. These zero energy states are topologically protected and can not be removed by applying a perturbation and the only way to get rid of these states is through a topological phase transition where the gap has to close. Closing the gap brings a continuum of states and the zero energy modes can then be removed. These zero energy modes are quasiparticle excitations and are very are special as they exhibit non-abelian statistics unlike a fundamental Majorana particle (neutrino for example, though it is still not resolved that neutrino is a Majorana fermion or not).

• This is wrong ! A sub-gap mode is necessarily localised in space, true. But it is not necessarily a Majorana mode. For $s$-wave superconductor, zero-energy states will come in Kramers pairs, which are not Majorana per definition (the spin degree of freedom avoids to make $\gamma^{\dagger}=\gamma$). Exemples are everywhere ($\pi$-phase shift, zero-energy modes around magnetic defects, trivial states in vortex, ...), and that's the difficulties to identify Majorana modes in experiments. – FraSchelle Jan 27 '17 at 5:04