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I read about the unitary and non-unitary order parameter states here https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.75.657 and https://arxiv.org/abs/1512.01151

The form of the order parameter depends on whether the condensate state is unitary or non-unitary where unitary states are described by $\vec{d}\times \vec{d}^\star=0$. And there are many such states that can be written down for spin-triplet pairing.

According to the answer here, What is a $p_x + i p_y$ superconductor? Relation to topological superconductors

it seems the all the components of d-vector have the same momentum dependence i.e. $d_i({\bf{k}})= d_i (k_x+ik_y)$. In this case, the order parameter can be diagonalized such that only $d_x$ and $d_y$ are non-zero and off-diagonal elements $d_z$ can be set to 0.

It seems from Table IV in the first reference, that this is an example of a non-unitary state. On the other hand, http://www.icmr.ucsb.edu/programs/documents/Kallin.pdf a state with $d_x,d_y=0$ and $d_z=\triangle_0 (k_x+ik_y)$ is also a description for a chiral p-wave superconductor. This is a unitary state though and is also the so-called equal spin pairing state where up and down spin sectors have same component in the order parameter.

My questions are- 1) What is the correct form for the d-vector for a $p_x+ip_y$ superconductor? 2) Why is the $p_x+ip_y$ superconductor described by such a non-unitary state or unitary state as above?

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