Background: There is a single qubit quantum gate of the form $$ \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \\ 1 & i \end{pmatrix}. $$ As far as I know, this was originally introduced as the $T$ gate in Gottesman's thesis https://arxiv.org/abs/quant-ph/9705052 (see equation A.5). In https://arxiv.org/abs/quant-ph/0403025 the authors call a similar version of this gate the $T$ gate.
However, as we know, this eventually fell out of favor since the $T$ gate almost universally means the $\pi/8$ rotation gate (i.e., $T = S^{1/2} = Z^{1/4}$).
Question: Since $T$ is off limits, what should we call this gate? Or does this gate already have a very standard name that isn't $T$?
Things to consider:
In https://arxiv.org/abs/1702.06990 the authors call this the $M_3$ gate (they seem to claim Gottesman calls it that in his thesis but I am not sure I can find where he calls it anything other than $T$?).
In our recent paper https://arxiv.org/abs/2305.07023 we call it the $M$ gate but admittedly I am asking this question specifically because I think that is a poor name (for starters it literally doesn't stand for anything related to any of its properties).
In https://threeplusone.com/pubs/on_gates.pdf the author cites Craig Gidney's stim python package and calls this (and the ones like it) the "axis-cycling gates". The name used is $C$ but I personally am not a fan since this usually denotes the Clifford group or Clifford hierarchy (or even the two qubit canonical gates).
There is a twitter post https://twitter.com/CraigGidney/status/1393069730785857536 where some other choices are listed. One choice therein seems to be: ``face" gates denoted by $F$ since this gate always maps the faces of a octahedron to another face.
My thoughts: I personally think ``axis cycling gates" above is pretty descriptive since there are $16$ of these in total and each is defined by their conjugation action: cycling the axes up to a factor of $-1$. The gate above is sort of the canonical version as it acts as $X \to Y$, $Y \to Z$, and $Z \to X$. Maybe $A$ would be a good name?
Another property we point out in our above paper is this gate's importance to the binary tetrahedral group $2T$. In particular, $2T$ contains the the $8$ element Pauli group (from $SU(2)$) and the 16 axis cycling gates and that's it. In other words, the (special) Paulis plus the above gate generate $2T$. However, the obvious choice of name $T$ is out but maybe there is some other letter/notation related to this property.
Edit (7/26/23) In the 3rd version of our paper, we ended up going with $F$ for this gate and calling it the "facet" gate. This was inspired by what Mark said below, however, "face" and "phase" are near-homophones and so we thought facet made more sense (besides it is technically more accurate). This coincides with a comment made by Markus Heinrich in this question. Compare with figure 3 here.
