How SU(2)$_4$ Anyons are Z$_3$ Parafermions

1 Interesting, well written, to the point paper

1 Only minor issues, see report

Report on 'How $su(2)_4$ anyons are $Z_3$ parafermions'.

This paper discusses the relation between the anyons of
$su(2)_4$ and $Z_3$ parafermionic zero-modes in detail, in
particular for the braid group representation of the
l=1/2 representation of $su(2)_4$.

The authors start by reviewing the braid group representation
associated with n l=1/2 anyons of $su(2)_4$, giving the explicit
action of the braid generators on the Hilbertspace of n l=1/2
anyons. To make the connection with the $Z_3$ parafermionic
zero modes, the authors show explicitly that these braid
generators can be constructed in terms of the $Z_3$ parafermions
(or, equivalently, in terms of the 'clock basis').

As the authors state, establishing this relation is an important
result, because it gives a 'simple' operator algebra 'implementation'
of the $su(2)_4$ non-abelian anyons.

The paper is well written, and for the most easy to follow. One point
which is slightly unclear concerns the Hilbertspace in the clock basis,
and 'equations' (32-34). Here, it seems that which alternative is
relevant, depends on which operators are chosen at the bonds with
labels not equal to j. Is it true that all of these states are by construction
orthonormal, or are they declared to be orthonormal? This issue is
related to a question I have about the proof of the odd cases, starting
on page 5. In the more complicated case, it seems the authors are merely
matching the coefficients a and b in eq. 44, while it should be possible
to derive them explicitly by evaluating the lhs of eq. 44. In any case,
to establish the result, one should show that the normalization of the two
states in the 'clock Hilbertspace' are the same, and independent of the
choices of the operators with labels different from j-1 and j (even though
this seems simple to show, I think it should be mentioned).

I appreciated app. B, in which the authors construct the result, which in
the main text is posed, and checked for consistency.

Some final comments/questions.

The construction seems to hinge on the fact that the quantum dimension of
the l=1/2 anyon is a simple root. This is not the case for the l=1/2 anyon
in $su(2)_{10}$ (which has central charge c=5/2), even though they take a simple
form. Does this exclude a similar construction for $su(2)_{10}$?

I suggest, for obvious reasons, that the authors swap reference one and two.

In conclusion, this is an interesting, and well written paper. I recommend it
for publication in SciPost, but ask the authors to consider the points I brought
up.

See report

Very explicit and physical illustration of a general theory and accessible to physicists.

Some standard confusion as to the difference between anyons and their various different avatars.

The paper is a very interesting and useful addition to the literature on a very interesting generalization of Majoranas to parafermions. I recommend its publications with the following requested changes.

1. Clarify, as already pointed out, the differences between anyons and defects.
2. Add the reference of Jones, V. Jones, Braid groups, Hecke algebras and type II1 factors, in Geometric Methods in
Operator Algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser. 123, Longman Sci. Tech.,
Harlow, 1986, 242–273. Harlow, 1986, where the main result appeared in a slightly different context.
3. The citation of 14 for universality with additional operations is inaccurate in two aspects: a) ref 14 discussed a slightly different theory--the Jones version, b) the universality for SU(2)_4 is first proved in a different reference arXiv:1405.7778
Universal Quantum Computation with Metaplectic Anyons, Shawn X. Cui, Zhenghan Wang

1. explicitly establishing a detailed relation at the level of braid group representation between the braiding statistics of SU(2)_4 anyons and the algebra of Z_3 parafermionic operators

1. the use of terminology in this paper is confusing and potentially misleading

The value of this paper is in explicitly establishing a detailed relation at the level of braid group representation between the braiding statistics of SU(2)_4 anyons and the algebra of Z_3 parafermionic operators. I think this work deserves to be published.

However, I consider the use of terminology in this paper to be confusing and potentially misleading and think it is necessary to correct this before publication. Throughout this paper, the authors should make it unambiguously clear when they are referring to parafermionic operators, which are mathematical objects that act in Hilbert space, as opposed to “parafermions,” which is a somewhat ambiguous term, but usually refers to physical objects that are (quasi)particle-like with root of unity exchange statistics (i.e. Abelian anyons), and both of these should also be clearly distinguished from parafermion zero modes, which are physical objects that are defects (into which parafermions can tunnel without changing the energy of the defect) with non-Abelian exchange statistics. It is, of course, the case that parafermionic operators can be used to provide a representation of the braiding statistics of parafermion zero modes, but one should not equate the two and, in fact, the authors specifically are utilizing the mathematical objects (parafermionic operators) with no concern to the notions of stability or physical realization associated with the parafermionic zero mode defects. (This is similar to the situation with Majorana fermionic operators, Majorana fermions, and Majorana zero modes.) This problem is highlighted by the title “How SU(2)$_4$ Anyons are Z$_3$ Parafermions,“ which is a bad title seeing as how it is incorrect to say these two things are the same, for any usage/misusage of the term “Parafermions.” A better title would be something along the lines of “A braid group representation of SU(2)$_4$ Anyons from the algebra of Z$_3$ Parafermion Operators.”

I would also like to point out that the relationship between SU(2)_4 Anyons and Z$_3$ Parafermion zero modes (which, as noted above, is related to the algebra of parafermionic operators) has previously been made much more concrete than suggested in the introduction and in any of the cited references. Specifically, the paper arXiv:1410.4540 has fully described the algebraic theory of symmetry defects in topological phases (which describes Z_3 parafermion zero modes) in terms of G-crossed category theory and the relation between the defect theory and the theory obtained by gauging the symmetry (which describes SU(2)_4 anyons) has been explained in detail. Moreover, the basic data has been worked out in this paper for the example of Z_N parafermionic zero modes, and their relation to SO(N)_2 anyons noted, making the relation between the braiding statistics of these theories fairly explicit (much more so than previous references, though somewhat less explicitly in terms of braid group representations than what has been worked out in the authors’ paper).

See report