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From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects

by Luisa Eck, Paul Fendley

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Submission summary

Authors (as registered SciPost users): Paul Fendley
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-09-10 12:39
Submitted by: Fendley, Paul
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


Strongly interacting models often possess "dualities" subtler than a one-to-one mapping of energy levels. The maps can be non-invertible, 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: Rydberg-blockade bosons with one particle per square of a ladder, a three-state antiferromagnet, and two Ising chains coupled in a zigzag fashion. The structure yields non-invertible 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 conformal-field-theory partition functions describing their critical regions. The Rydberg and Ising ladders also possess interesting non-invertible symmetries, with the spontaneous breaking of one in the former resulting in an unusual ground-state degeneracy.

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Reports on this Submission

Anonymous Report 3 on 2024-1-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2302.14081v3, delivered 2024-01-30, doi: 10.21468/SciPost.Report.8474


1. Interesting
2. Detailed
3. Well written


Report on the paper:

From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible 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 Temperley-Lieb algebra. This different algebraic structure allows the authors study non-invertible 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, non-hermitian 'hamiltonians' can be very interesting. Does the construction work for non-hermitian 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 F-symbols.

* The F-symbols 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 non-invertible mappings between them in great detail. For instance, they derive the products of non-invertible 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 non-invertible symmetries, and utilise them to study the physics of the models, by relating them to the well-studied 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 super-integrable models?

Requested changes

No requested changes, but see the the questions marked by an * in my report.

  • validity: top
  • significance: top
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 2 on 2024-1-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2302.14081v3, delivered 2024-01-26, doi: 10.21468/SciPost.Report.8459


- uncovers an algebra underlying four different integrable spin chains
- carefully written, self-contained
- there is no doubt that the paper is correct


- 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


The authors introduce an algebra defined by eqs. (2.1)-(2.2), which is similar in spirit to a Temperley-Lieb 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 3-state Potts chain, and two RSOS-type 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 self-contained, 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 re-organizing slightly sections. 2 and 3, the manuscript would improve.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Report 1 by Linnea Grans-Samuelsson on 2023-10-20 (Invited Report)

  • Cite as: Linnea Grans-Samuelsson, Report on arXiv:2302.14081v3, delivered 2023-10-20, doi: 10.21468/SciPost.Report.7892


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 non-invertible mappings and symmetries, and provide an excellent introduction to section 4. More generally, the paper is well structured.


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.


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 Temperley-Lieb 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 3-state antiferromagnet, the integrable Rydberg-blockade 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 Temperley-Lieb approach, such as the parameter $\Delta$ no longer appearing in the algebra. This is helpful, as the Temperley-Lieb approach may be very familiar to readers of this paper.

The authors carefully work out non-invertible 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 Rydberg-blockade 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 non-invertible 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?

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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