SciPost Submission Page
Integrable models on Rydberg atom chains
by Luke Corcoran, Marius de Leeuw, Balázs Pozsgay
Submission summary
| Authors (as registered SciPost users): | Luke Corcoran · Balázs Pozsgay |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2405.15848v3 (pdf) |
| Date submitted: | 2025-03-31 13:01 |
| Submitted by: | Corcoran, Luke |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
We initiate a systematic study of integrable models for spin chains with constrained Hilbert spaces; we focus on spin-1/2 chains with the Rydberg constraint. We extend earlier results for medium-range spin chains to the constrained Hilbert space, and formulate an integrability condition. This enables us to construct new integrable models with fixed interaction ranges. We classify all time- and space-reflection symmetric integrable Rydberg-constrained Hamiltonians of range 3 and 4. At range 3, we find a single family of integrable Hamiltonians: the so-called RSOS quantum chains, which are related to the well-known RSOS models of Andrews, Baxter, and Forrester. At range 4 we find two families of models, the first of which is the constrained XXZ model. We also find a new family of models depending on a single coupling $z$. We provide evidence of two critical points related to the golden ratio $\phi$, at $z=\phi^{-1/2}$ and $z=\phi^{3/2}$. We also perform a partial classification of integrable Hamiltonians for range 5.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We thank the reviewers for the questions and useful comments on our paper.
On the comments of reviewer 1: 1: We did not find many-body chiral symmetry for any of these operators, but it doesn’t mean that it’s not there. Certainly the models do not commute with C= prod_i sigma^z_i 2: We agree that sometimes it can be nice to explicitly see the algebraic equations resulting from the integrability condition, although it is not always helpful. We added an example in the text after equation (110). 3: The method on page 16 is analogous to the method described on page 8, with some adaptations due to the constrained Hilbert space which are discussed in section 4. 4: We kept one subsection to emphasise that there is only a single solution to the integrability condition in this case, and so the different integrable models at each range are clearly listed in the table of contents. 5: We are not aware of the definition of a “disordered” integrable model. If it is possible to derive such models from a solution of the Yang—Baxter equation then our method should apply, but we are not sure.
On the minor changes suggested by reviewer 1: 1: We feel the potential vagueness of the first sentence is clarified in the second sentence. 2: Algebraic origin refers to structures related to the Yang—Baxter equation, which is expanded upon in the second paragraph. 3: We agree there are many terms which are unfamiliar to non integrability experts. However, expanding upon each of these would disturb the flow of the introduction. For example, see “common algebraic framework for solving such models based on the Yang–Baxter equation has since been developed, known as the algebraic Bethe ansatz [5].” The only really important thing to know is that it is some algebraic method which allows for the solution of integrable models. The interested reader can then look at [5] for more details. 4: We joined the references 5: We fixed this 6: We mention in the caption that this dips in the plot correspond to the critical points, we think it is clear now.
On the questions/comments of reviewer 2: 1: The exact `GLL formulation’ was introduced in 2108.02053, we now mention this after equation (3). 2: In this paragraph we are speaking about higher range integrability. The GLL formulation does not make sense for range 2 models, since the Lax operator needs to act diagonally on the first on last site (so it would be trivial in this case). 3: Yes, we fixed this. 4. Yes, we fixed this. 5. One way is to set up a recursion relation for this dimension. On the open chain dim_L = dim_{L-1}+dim_{L-2}, which is solved by the Fibonacci numbers, whose asymptotics are given by powers of the golden ratio. This can be slightly modified on the periodic chain. 6. This follows from the discussion around equation (57) and (58). Since each operator in the checked Lax can be written in a form which commutes with Pi_A, we only need to pay attention to the commutation relations between Pi_A and the permutation operators in the Lax. 7. The indices are correct. First of all, we omitted checks from the R matrices between equations (74)-(77) which we now fixed. Applying the permutation operator P_{AB} leads to a slightly strange index structure on the projectors in the unchecked R matrix. 8. Indeed there are different values of z for which different operators in the double golden chain vanishes. Phase transitions typically occur when the ground state degeneracy of the system changes, this does not occur when the PNP term of the Hamiltonian vanishes. We don’t think these values of z will end up being special.
We thank reviewer 2 for pointing out the typos, which we fixed.
List of changes
- We added an example for the algebraic equations arising from the integrability condition after (110).
- We joined the references in the introduction.
- We now mention in the caption of figure 10 the locations of the critical points.
- We added the reference for the origin of `GLL formulation’.
- The first charge Q_1 for this models is actually a generalised momentum operator, such that exp(Q_1)=U^{r-1}. We fixed this.
- Fixed confusion between L and \check{L}
- Fixed numerous small typos
Current status:
Reports on this Submission
Strengths
1. The paper provides the first systematic approach to construct integrable models in restricted Hilbert spaces.
2. They constructed a new family of integrable models and provided a first investigation on their properties.
3. The paper is well written and well structured, with a clear introduction and an interesting discussion presented in the conclusions. Appropriate references are also provided.
Weaknesses
None that I can see.
Report
The authors replied to all the points raised both by myself and Referee 1.
This is a very original and relevant paper. The changes further improved its clarity and I recommend it to publication in SciPost Physics in its present form.
Requested changes
None
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)