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From the XXZ chain to the integrable Rydbergblockade ladder via noninvertible duality defects
by Luisa Eck, Paul Fendley
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Submission summary
Authors (as registered SciPost users):  Paul Fendley 
Submission information  

Preprint Link:  https://arxiv.org/abs/2302.14081v3 (pdf) 
Date submitted:  20230910 12:39 
Submitted by:  Fendley, Paul 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Strongly interacting models often possess "dualities" subtler than a onetoone mapping of energy levels. The maps can be noninvertible, as apparent in the canonical example of Kramers and Wannier. We analyse an algebraic structure common to the XXZ spin chain and three other models: Rydbergblockade bosons with one particle per square of a ladder, a threestate antiferromagnet, and two Ising chains coupled in a zigzag fashion. The structure yields noninvertible maps between the four models while also guaranteeing all are integrable. We construct these maps explicitly utilising topological defects coming from fusion categories and the lattice version of the orbifold construction, and use them to give explicit conformalfieldtheory partition functions describing their critical regions. The Rydberg and Ising ladders also possess interesting noninvertible symmetries, with the spontaneous breaking of one in the former resulting in an unusual groundstate degeneracy.
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Reports on this Submission
Anonymous Report 3 on 2024130 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.14081v3, delivered 20240130, doi: 10.21468/SciPost.Report.8474
Strengths
1. Interesting
2. Detailed
3. Well written
Report
Report on the paper:
From the XXZ chain to the integrable Rydbergblockade ladder via noninvertible duality defects
Let me start my report by giving my overall assessment of the paper. The novelty, interest and quality of this paper is dual to speed with which this report was produced. This paper clearly satisfies the criteria for publication in SciPost easily, and so I recommend this paper for publication in SciPost. Below, I raise some issues/questions (marked with an *), which the authors can act on, if they deem this appropriate.
The paper studies four integrable models, starting from the XXZ model. They do this by means of an algebraic structure different from the usually employed TemperleyLieb algebra. This different algebraic structure allows the authors study noninvertible duality/topological defect maps between these models, which are closely related to modular tensor categories, as for instance utilised in anyon chains.
In Section 2, the authors introduce and study the algebra underlying this paper. In particular, they introduce two sets of generators, S_j and P_j, in terms of which the hamiltonians are given.
* The operators P_j act like projector operators, as is clear from the second relation in eq. (2.2). For the P_j to be projectors, they should also be hermitian. However, the authors do not commend on the hermiticity of the P_j. It seems that the authors implicitly assume this, because otherwise the hamiltonian obtained would not be hermitian. On the other hand, nonhermitian 'hamiltonians' can be very interesting. Does the construction work for nonhermitian P_j (and/or S_j)?
It is appreciated that the authors explain the graphical presentation of the algebra(s) in detail in section 2.2 and 2.3.
In section 2.4, the authors relate the algebraic structure they utilise to fusion categories, in particular, they use a specific set of Fsymbols.
* The Fsymbols have a large amount of gauge freedom. Using a different gauge can typically change the appearance of the operators/hamiltonian, but does not typically lead to different physics. Is this also the case here?
* A related question, would a (essentially) different solution of the pentagon equation lead to different hamiltonians (provided more than one unitary solution of the pentagon equation exist)?
In section 3, the authors derive four different hamiltonians using their algebraic structure.
* What is the purpose of the sentence following eq. (3.5)?
* In the sentence following eq. (3.6) it is stated that the YBE requires that T(u) and T(u') to commute, but doesn't this follow if R satisfies the YBE? So, should 'requires' be replaced by 'guarantees' ?
* After eq. 3.14, it would be useful to remind the reader that S_j corresponds to the first term in 3.14, P_j to the second.
With the hamiltonians at hand, the authors study the noninvertible mappings between them in great detail. For instance, they derive the products of noninvertible maps as given in Table 1, as well as the relation between the partition functions, such as eq. 4.18.
* The relation between the partition functions of the Rydberg ladder and Ising zigzag chain is not stated in in section 4.4. Perhaps the authors can comment on this omission?
In section 5, the authors study the noninvertible symmetries, and utilise them to study the physics of the models, by relating them to the wellstudied XXZ chain.
* In this context, it is clear that no conventional symmetry relates the three ground states of the Rydberg ladder for large \Delta, as the authors state. However, two of these ground states are related by translational symmetry. Is it possible to utilise this in some way?
Finally, in section 6, the authors use their results to obtain the spectra of their models (at criticality) in the continuum limit. This leads to intriguing relations between the partition functions of the models in terms of various orbifolds, 'in both directions'.
* As a final comment/question, I am wondering if the authors could comment on the applicability of their algebraic approach in different contexts. Is it use restricted to the (type of) models studied in the current (and accompanying) paper, or could this method be applied to other integrable systems, such as S=1 (or even higher spin) integrable chains, or perhaps superintegrable models?
Requested changes
No requested changes, but see the the questions marked by an * in my report.
Anonymous Report 2 on 2024126 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.14081v3, delivered 20240126, doi: 10.21468/SciPost.Report.8459
Strengths
 uncovers an algebra underlying four different integrable spin chains
 carefully written, selfcontained
 there is no doubt that the paper is correct
Weaknesses
 a clearer separation between new models, new methods, new results on the one hand, and review of existing literature on the other hand, would be nice
Report
The authors introduce an algebra defined by eqs. (2.1)(2.2), which is similar in spirit to a TemperleyLieb algebra or some of its cousins (chromatic algebra, BMW algebra etc.) at some specific value of the free parameter ($Q=3$ or $q=e^{i \pi/3}$ ).
They introduce the Baxterization of that algebra in eq. (3.4).
Then they show that this algebra occurs in four different quantum spin chains: the XXZ spin chain intringuingly, for any value of the anisotropy $\Delta$, not related to $q$ here, the 3state Potts chain, and two RSOStype chains (`Rydberg blockade' chain and Ising ladder).
Then they exploit this algebraic understanding to carefully analyze mappings between these four different models, both in their lattice formulation (chapters 4 and 5) and in the continuum limit (chapter 6).
The paper is very carefully written. I think it could be published as it is. My only criticism is that, because the paper is selfcontained, it lacks conciseness. It contains a lot of standard material, which is not always clearly separated from the new results. For some readers, it might be difficult to appreciate what is new here. If the authors could clarify this, for instance by reorganizing slightly sections. 2 and 3, the manuscript would improve.
Report 1 by Linnea GransSamuelsson on 20231020 (Invited Report)
 Cite as: Linnea GransSamuelsson, Report on arXiv:2302.14081v3, delivered 20231020, doi: 10.21468/SciPost.Report.7892
Strengths
1. Provides a useful, unified setting for dealing with four different lattice models, and goes into a good amount of detail about the mappings between them. Also exploits the mappings to find interesting properties of the models.
2. Pedagogical and clear.
3. Diagram 4.1 and table 1 nicely summarize the main results about noninvertible mappings and symmetries, and provide an excellent introduction to section 4. More generally, the paper is well structured.
Weaknesses
1. (Minor point) Much care is taken in Section 2 to show graphical presentations of the algebra defined in eqs (2.1) and (2.2), and fusion categories are also introduced from the perspective of labelled graphs. However, in latter sections there is very little graphical interpretation given (even though topological defects often do lend themselves well to a graphical description), making the paper feel a bit disconnected between the first and second parts.
Report
A common approach to relating the integrable XXZ spin chain to other integrable lattice models (such as RSOS) is through expressing the Hamiltonians in terms of the TemperleyLieb algebra. Here, the authors use instead another algebra, as well as the more general setting of fusion categories. The relevant algebras and categories are pedagogically introduced to the reader.
The authors show that apart from the XXZ spin chain, three other models have Hamiltonians that can be written in terms of this other algebra: the 3state antiferromagnet, the integrable Rydbergblockade ladder and the Ising zigzag ladder. As they are described by the same algebra, there must be relations between these models; detailing and exploiting these relations is the main theme of the paper. The authors take care to repeatedly remind the reader of the difference from the TemperleyLieb approach, such as the parameter $\Delta$ no longer appearing in the algebra. This is helpful, as the TemperleyLieb approach may be very familiar to readers of this paper.
The authors carefully work out noninvertible mappings and symmetries between and within the four models using the formalism of topological defects. These mappings are first helpfully summarized in diagram 4.1 and table 1, giving the reader a convenient overview, after which each mapping is presented in detail. These mappings are used to relate the partition functions to each other, and to derive properties of the models for different regimes of $\Delta$. By using in particular the relation between the XXZ spin chain (whose CFT description in the critical regime is very well understood) and the other three models, the exact CFT spectrum of the other three models in the critical regime is computed. The framework of orbifolding used for the CFT partition functions is pedagogically presented.
I recommend that this paper is published in SciPost Physics after minor revisions (see requested changes). The models under consideration are interesting models, and these mappings between and within them are very useful.
Requested changes
1. Each of subsections 4.1, 4.2 and 4.3 finish with a relation between the partition functions of the models discussed in that subsection. However, in subsection 4.4 there is no such final expression between the integrable Rydbergblockade ladder and the zigzag Ising ladder, leaving the reader hanging. If the ultimate goal is to always express partition functions in terms of $Z_{XXZ}$ (as described in the introduction), and the authors do not intend to spell out any relation directly between $Z_{IRL}$ and $Z_{zig}$, that should be (re)stated explicitly both at the very end of subsection 4.4 and around the beginning of section 4, to remind the reader of this. This would increase the clarity of section 4.
2. (Minor fix) In the heading of subsection 5.1, the first $\mathcal{Q}$ should have a subscript "zig"  especially since there's another $Q$ that is otherwise only distinguished by font choice.
3. (Optional suggestion) At the beginning of subsection 4.3, a graphical way of presenting the noninvertible maps and the defect commutation relations is shown, and it is noted that it can be used for all maps that are discussed. Perhaps it could be shown in some form already at the beginning of section 4?