## Symmetry of the superconducting gap

First of all, a bit of theory. Superconductivity appears due to the
Cooper paring of two electrons, making non-trivial correlations between
them in space. The correlation is widely known as the gap parameter
$\Delta_{\alpha\beta}\left(\mathbf{k}\right)\propto\left\langle c_{\alpha}\left(\mathbf{k}\right)c_{\beta}\left(-\mathbf{k}\right)\right\rangle $
(the proportionality is merely a convention that will not matter for
us) with $\alpha$ and $\beta$ the spin indices, $\mathbf{k}$ some
wave vector, and $c$ the fermionic destruction operator. $\Delta$
corresponds to the order parameter associated to the general recipe
of second order phase transition proposed by Landau. Physically, $\Delta$
is the energy gap at the Fermi energy created by the Fermi surface
instability responsible for superconductivity.

Since it is a correlation function between two fermions, $\Delta$
has to verify the Pauli exclusion principle, which imposes that $\Delta_{\alpha\beta}\left(\mathbf{k}\right)=-\Delta_{\beta\alpha}\left(-\mathbf{k}\right)$. You can derive this property from the anti-commutation relation of the fermion operator and the definition of $\Delta_{\alpha\beta}\left(\mathbf{k}\right)$ if you wish.
When there is no spin-orbit coupling, both the spin and the momentum
are good quantum numbers (you need an infinite system for the second, but this
is of no importance here), and one can separate $\Delta_{\alpha\beta}\left(\mathbf{k}\right)=\chi_{\alpha\beta}\Delta\left(\mathbf{k}\right)$ with $\chi_{\alpha \beta}$ a spinor matrix and $\Delta\left(\mathbf{k}\right)$ a function.
Then, two possibilities

$\chi_{\alpha\beta}=-\chi_{\beta\alpha}\Leftrightarrow\Delta\left(\mathbf{k}\right)=\Delta\left(-\mathbf{k}\right)$
this situation is referred as the spin-singlet pairing

$\chi_{\alpha\beta}=\chi_{\beta\alpha}\Leftrightarrow\Delta\left(\mathbf{k}\right)=-\Delta\left(-\mathbf{k}\right)$
this situation is referred as the spin-triplet pairing.

Singlet includes $s$-wave, $d$-wave, ... terms, triplet includes
the famous $p$-wave superconductivity (among others, like $f$-wave, ...).

Since the normal situation (say, the historical BCS one) was for singlet
pairing, and because only the second Pauli $\sigma_{2}$ matrix is
antisymmetric, one conventionally writes the order parameter as
$$
\Delta_{\alpha\beta}\left(\mathbf{k}\right)=\left[\Delta_{0}\left(\mathbf{k}\right)+\mathbf{d}\left(\mathbf{k}\right)\boldsymbol{\cdot\sigma}\right]\left(\mathbf{i}\sigma_{2}\right)_{\alpha\beta}
$$
where $\Delta_{0}\left(\mathbf{k}\right)=\Delta_{0}\left(-\mathbf{k}\right)$
encodes the singlet component of $\Delta_{\alpha\beta}\left(\mathbf{k}\right)$
and $\mathbf{d}\left(\mathbf{k}\right)=-\mathbf{d}\left(-\mathbf{k}\right)$
is a vector encoding the triplet state.

Now the main important point: what is the exact $\mathbf{k}$-dependency
of $\Delta_{0}$ or $\mathbf{d}$ ? This is a highly non-trivial question,
to some extend still unanswered. There is a common consensus supposing
that the symmetry of the lattice plays a central role for this question.
I highly encourage you to open the book by Mineev and Samokhin (1998), *Introduction to unconventional superconductivity*, Gordon and
Breach Science Publishers, to have a better idea about that point.

## The $p_{x}+\mathbf{i}p_{y}$ superconductivity

For what bothers you, **the $p_{x}+\mathbf{i}p_{y}$ superconductivity
is the superconducting theory based on the following "choice"**
$\Delta_{0}=0$, $\mathbf{d}=\left(k_{x}+\mathbf{i}k_{y},\mathbf{i}\left(k_{x}+\mathbf{i}k_{y}\right),0\right)$
such that one has
$$
\Delta_{\alpha\beta}\left(\mathbf{k}\right)\propto\left(\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right)\left(k_{x}+\mathbf{i}k_{y}\right)\equiv\left(k_{x}+\mathbf{i}k_{y}\right)\left|\uparrow\uparrow\right\rangle
$$
which is essentially a phase term (when $k_{x}=k\cos\theta$ and $k_{y}=k\sin\theta$)
on top of a spin-polarized electron pair. This phase
accumulates around a vortex, and has non-trivial properties then.

Note that the notation $\left|\uparrow\uparrow\right\rangle $ refers
to the spins of the electrons forming the Cooper pair. A singlet state
would have something like $\left|\uparrow\downarrow\right\rangle -\left|\downarrow\uparrow\right\rangle $, and for $s$-wave $\Delta_0$ is $\mathbf{k}$ independent, whereas $\mathbf{d}=0$.

- Note that the $p$-wave also refers to the angular momentum $\ell=1$
as you mentioned in your question. Then, in complete analogy
with conventional composition of angular momentum (here it's for two
electrons only), the magnetic moment can be $m=0,\;\pm1$. The natural
spherical harmonic for these states are then $Y_{\ell,m}$ with $Y_{1,\pm1}\propto k_{x}\pm\mathbf{i}k_{y}$
and $Y_{1,0}\propto k_{z}$, so it should be rather natural to find
the above mentioned "choice" for $\mathbf{d}\left(\mathbf{k}\right)$.
I nevertheless say a "choice" since this is not a real choice:
the symmetry of the gap should be imposed by the material you consider,
even if it is not yet satisfactorily understood.
- Note also that only the state $m=+1$ appears in the $p_{x}+\mathbf{i}p_{y}$ superconductivity. You might wonder about the other magnetic momentum contribution... Well, they are discarded, being less favourable (having a lower transition temperature for instance) under specific conditions that you have to know / specify for a given material. Here you may argue about the Zeeman effect for instance, which polarises the Cooper pair. [NB: I'm not sure about the validity of this last remark.]

## Relation between $p_{x}+\mathbf{i}p_{y}$ superconductivity and emergent unpaired Majorana modes

Now, quickly, I'll try to answer your second question: **why is this
state important for emergent unpaired Majorana fermions in the vortices excitations
?** To understand that, one has to remember that the emergent unpaired
Majorana modes in superconductors are non-degenerate particle-hole
protected states at zero-energy (in the middle of the gap if you prefer).
Particle-hole symmetry comes along with superconductivity, so we already
validate one point of our check list. To make non-degenerate mode,
one has to fight against the Kramers degeneracy. That's the reason
why we need spin-triplet state. If you would have a singlet state
Cooper pair stuck in the vortex, it would have been degenerate, and
you would have been unable to separate the Majorana modes, see also Basic questions in Majorana fermions
for more details about the difference between Majorana modes and
unpaired Majorana modes in condensed matter.

A more elaborate treatment about the topological aspect of $p$-wave
superconductivity can be found in the book by Volovik, G. E. (2003),
*Universe in a Helium Droplet*, Oxford University Press, available
freely from the author's website http://ltl.tkk.fi/wiki/Grigori_Volovik.

- Note that Volovik mainly discuss superfluids, for which $p$-wave has been observed in $^{3}$He. The $p_{x}+\mathbf{i}p_{y}$ superfluidity is also called the $A_{1}$-phase [Volovik, section 7.4.8]. There is no known $p$-wave superconductor to date.
- Note also that the two above mentionned books (Samokhin and Mineev, Volovik) are
not strictly speaking introductory materials for the topic of superconductivity.
More basics are in Gennes, Tinkham or Schrieffer books (they are all named
*blabla... superconductivity blabla...*).

This post imported from StackExchange Physics at 2017-09-27 09:38 (UTC), posted by SE-user FraSchelle