Integrable Floquet systems related to logarithmic conformal field theory

New interesting results concerning particular integrable Floquet system

Misprints, some calculations should be made more clear for general reader, unclear references to LogCFTs

Report on ``Integrable Floquet systems related to logarithmic conformal field theory''

The paper ``Integrable Floquet systems related to logarithmic conformal field theory'' is devoted to the study of integrable Floquet quantum system described by a non-unitary representation of the Temperley-Lieb algebra. Authors construct an infinite number of charges that commute with this Hamiltonian and explicit operator expression for the Floquet Hamiltonian. The results are interesting and worth to be published in SciPost. However, I have some comments to be addressed, before the final decision could be made. Namely:

1. When mentioning Floquet CFT it is neccesary to give the reference on the first paper on this topic (arXiv:1805.00031); Also, it is generally unclear what new information this study provides for understanding LogCFT (although the authors repeatedly mention their connection in the text).
2. There are number of small misprints like in equation (20) (missing part on the left) and (22) (seems to be miss tilde). Also there are grammar typos in the text, so the spellcheck is required.
3. I think it would be helpful to add some details on derivation of (16) and (18) (and probably other particular calculations which could be unclear for reader)

1. When mentioning Floquet CFT it is neccesary to give the reference on the first paper on this topic (arXiv:1805.00031); Also, it is generally unclear what new information this study provides for understanding LogCFT (although the authors repeatedly mention their connection in the text).
2. There are number of small misprints like in equation (20) (missing part on the left) and (22) (seems to be miss tilde). Also there are grammar typos in the text, so the spellcheck is required.
3. I think it would be helpful to add some details on derivation of (16) and (18) (and probably other particular calculations which could be unclear for reader)

1- The authors provide an interesting new way of constructing conserved charges of the Floquet Hamiltonian of the periodic Temperley-Lieb algebra at $\beta = 0$.
2- The paper is well organised.

1- Some claims are unproved.
2- There are a number of typos in the formulas.
3- There are a number of typos in the English.

This paper investigates the integrable Floquet Hamiltonian related to the periodic Temperley-Lieb algebra in the special case where the loop weight is equal to zero. The authors nicely use Lie algebra techniques to construct an infinite number of charges that commute with this Hamiltonian. They also investigate the representation of this algebra related to the $g\ell(1,1)$ spin chain, or equivalently to the XX Hamiltonian. They express the conserved charges in terms of the symplectic fermions and argue that a phase transition arises in this model.

I find that the results are new and interesting and should eventually deserve a publication in Scipost. However, having gone over the calculations carefully, I find that there are a number of imprecisions and typos in the math which should be fixed before this paper is accepted. This paper also has a number of elementary typos in the English (for instance therms $\to$ terms, reson $\to$ reason, reions $\to$ regions, and many more), so I would ask that the authors do a proper spell-check before resubmitting their manuscript.

Here is my list of comments:

- Equation (9) and the geometry presented in Figure 1 remind me very much of the construction used in arXiv:0812.2746 [math-ph] and in ref [31], with transfer matrices $T(u)$ built out of two successive rows of $R$-matrices shifted by one site. In fact, the Floquet evoluation operator $U_F(z) $ is precisely equal to this transfer matrix (with the correct parameterisation $z=z(u)$). In the above articles, it is shown that $T(u)$ is part of a two-parameter family of commuting transfer matrices, say $T(u,v)$, which commute only when the second parameter is varied. Thus performing an expansion of $T(u,v)$ along $v$, one obtains elements of the algebra that commute with $T(u)$. (These papers consider transfer matrices with a boundary, but the arguments also work for the periodic case.) It would be interesting to see if these conserved charges coincide with those of Section 3.3 for $\beta = 0$. Moreover, the construction of $T(u,v)$ and of the conserved charges also works away from $\beta = 0$, which could give some insight for the problem raised by the authors in the conclusion.

- There seems to be a mistake with the very last equation in (16), at the bottom right. I find
$[H_e, q_-^2] = -2 q_e^3 - 4 q_e^1$
$[H_o, q_-^2] = 2 q_e^3 + 4 q_e^1 $
and something slightly different for $q_-^{2n}$ with $n>1$.

- The meaning of $tl_\pm$ is not clear. I had understood from the text above (13) that these are two separate algebras: $tl_+$ and $tl_-$, respectively generated by $q^m_+$ and $q^m_-$. But the text above (21) now appears to imply that $tl_\pm$ is a unique algebra.

- A proof of equations (18) appears non trivial and is missing.

- Clearly $[H_n,H_m]$ is missing as the left-hand side of the first equation in (20).

- The sentence above (21) says that the Lie algebra is embedded into the $s\ell(2)$ loop algebra. But isn’t it the other way around?

- The sentence starting 3.3 states that a complete set of conserved charges is constructed later in this section. This is an unsubstantiated claim: the authors indeed constructs many such charges in this section, but do not address the question of whether a full set of charges is indeed obtained.

- Should the right side of (22) read $\tilde q^m_+$ instead of $q^m_+$?

- I believe there is en error in the bottom equation of (49), where $i \sin$ should instead be replaced by $\cos$.

- For (51), it would be useful to repeat the range of $s$, as I believe it starts at $s=1$ for the first equation and $s=0$ for the second.

- The second member of (52) should be identical to (36), but it is not. On one hand, the powers of $z$ differ by one, and on another hand the first term has $q_-$ instead of $q_+$.

- The reasoning behind the division into sections $R_1$ and $R_2$ and the resulting phase transition in section 4.3.2 is not sufficiently clear. My understanding is that these two regions describe different intervals of $p$ over which the sum is performed. So for instance (64) is incorrect: when $H_F$ contains terms of type $R_2$, it also has terms of type $R_1$. Then the transition is in fact one between $|z|<1$, which has only terms of type $R_1$, and $|z|>1$, which has terms of types $R_1$ and $R_2$. The authors should discuss these questions in greater detail.

- In the abstract, the authors claim that they provide strong indication that their Floquet system is described by a logarithmic CFT. While the systems that they study are known to be related to dense polymers and symplectic fermions, both of which are related to $c=-2$ logarithmic CFTs, I don’t see any new elements provided by the authors’ calculation that give further information about the status of this model as a logarithmic CFT. The paragraph on the same topic in the conclusion is vague and lacks a proper argument that would convince me of this claim.

- The authors should also explain why they believe that the phase transition has something to do with compactness vs non-compactness. As presented, I see no evidence pointing to this.

- In Appendix A, a proof of Property 3 is missing. This proof seems non-trivial to me.

- I believe the last equation in (75) to be incorrect: the second and last terms in the parenthesis should have the prefactor $(-1)^{m+1}$.

- I don’t understand the relevance of Appendix B to this paper.