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

by Luisa Eck, Paul Fendley

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.

Current status:
In refereeing

Submission & Refereeing History

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Submission 2302.14081v3 on 10 September 2023

Reports on this Submission

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


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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