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Integrable Floquet systems related to logarithmic conformal field theory
by Vsevolod I. Yashin, Denis V. Kurlov, Aleksey K. Fedorov, Vladimir Gritsev
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Submission summary
Authors (as registered SciPost users):  Vsevolod Yashin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2206.14277v1 (pdf) 
Date submitted:  20220712 21:51 
Submitted by:  Yashin, Vsevolod 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study an integrable Floquet quantum system related to lattice statistical systems in the universality class of dense polymers. These systems are described by a particular nonunitary representation of the TemperleyLieb algebra. It is found that the Floquet conserved charges form an infinitedimensional loop $\mathfrak{sl}(2)$ algebra, and an explicit operator expression for the Floquet Hamiltonian is given in terms of the charges. This system has a phase transition between local and nonlocal phases of the Floquet Hamiltonian. We provide a strong indication that in the scaling limit this system is described by the logarithmic conformal field theory.
Current status:
Reports on this Submission
Anonymous Report 2 on 202297 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2206.14277v1, delivered 20220907, doi: 10.21468/SciPost.Report.5658
Strengths
New interesting results concerning particular integrable Floquet system
Weaknesses
Misprints, some calculations should be made more clear for general reader, unclear references to LogCFTs
Report
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 nonunitary representation of the TemperleyLieb 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)
Requested changes
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)
Anonymous Report 1 on 2022817 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2206.14277v1, delivered 20220817, doi: 10.21468/SciPost.Report.5549
Strengths
1 The authors provide an interesting new way of constructing conserved charges of the Floquet Hamiltonian of the periodic TemperleyLieb algebra at $\beta = 0$.
2 The paper is well organised.
Weaknesses
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.
Report
This paper investigates the integrable Floquet Hamiltonian related to the periodic TemperleyLieb 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 spellcheck 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 [mathph] 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 twoparameter 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 lefthand 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 noncompactness. As presented, I see no evidence pointing to this.
 In Appendix A, a proof of Property 3 is missing. This proof seems nontrivial 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.