SciPost Submission Page
Critical spin chains and loop models with $U(n)$ symmetry
by Paul Roux, Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Jesper Lykke Jacobsen · Sylvain Ribault · Paul Roux |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2404.01935v1 (pdf) |
| Code repository: | https://gitlab.com/s.g.ribault/representation-theory.git |
| Date submitted: | 2024-06-12 11:24 |
| Submitted by: | Roux, Paul |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the $O(n)$ model (symmetry group $O(n)$) and the Potts model (symmetry group $S_Q$). Both models make sense for $n,Q\in \mathbb{C}$ and not just $n,Q\in \mathbb{N}$, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the unitary group $U(n)$. We focus on the two-dimensional case, where the models can be described either as gases of non-intersecting orientable loops, or as alternating spin chains. This allows us to determine their spectra either by computing a twisted torus partition function, or by studying representations of the walled Brauer algebra. In the critical limit, our models give rise to a CFT with global $U(n)$ symmetry, which exists for any $n\in\mathbb{C}$. Its spectrum is similar to those of the $O(n)$ and Potts CFTs, but a bit simpler. We conjecture that the $O(n)$ CFT is a $\mathbb{Z}_2$ orbifold of the $U(n)$ CFT, where $\mathbb{Z}_2$ acts as complex conjugation.
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
Current status:
Reports on this Submission
Report #3 by Bernard Nienhuis (Referee 3) on 2024-10-16 (Invited Report)
Strengths
1) The paper discusses a very natural class of models that require no further motivation.
2) It thoroughly discusses the necessary mathematics.
3) The reference list contains convenient back references to the referring text.
Weaknesses
1) The technical parts are difficult to follow.
Report
The paper Critical Spin models with U(n) symmetry, by Roux, Jacobsen, Ribault and Saleur, addresses the natural question how spin models with U(n) symmetry are similar to or different from those with U(n) symmetry. In my view it is a very useful contribution to the public knowledge about critical phenomena and their relation with symmetry groups and their representations.
As I was studying the paper I was very pleased to see that the references have a link back to the pages where they are mentioned. This greatly improves the experience of reading the paper. Perhaps this is used widely, but I did not notice it before.
My recommendation is that the paper be published in SciPost Physics. I trust that the authors will give due attention to the following remarks.
1) In the abstract the third paragraph states that something exists for all complex n. On first reading I took that to be the U(n) symmetry (group), which I believe does not generally exist. Now that I am compiling my notes to a report, I think it is meant that the CFT with U(n) symmetry exists. Perhaps this ambiguity can be removed.
2) Page 4 line 13 from the bottom. "entirely similar" sounds strange, I think a better choice is "globally similar".
3) I do not follow the reasoning excluding the diagram (3.9). I do not see how allowing this diagram as one of the terms in the Hamiltonian results in the redundancy of a site.
4) I am puzzled by the claim just before (3.11a) that the physics of the dilute and the completely packed model is equivalent. I would say that the physics of the dilute models includes that of the completely packed ones.
5) The text following (3.15) is not correct. In the second sentence the word 'fugacities' is better replaced by 'weights'. The models with the weights (3.15) are integrable only for a special value of x. Therefore, the form for the square lattice weights can not be determined by demanding that the model is integrable, and this is certainly nog argued in ref. [27]
6) The references 18, 24, 27, 28, 34 are incomplete. (and in general the doi codes are printed in an unpleasantly small font.)
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report
The authors build a CFT arising from a loop model. The claim is that this CFT has $U(n)$ symmetry. I see some issues with this claim, related to the number of currents in general and in special cases.
1. $U(n)$ has $n^2$ generators, therefore we should expect $n^2$ currents in the holomorphic sector, but the partition function (4.1) only shows $n^2-1$ currents. This is claimed to be related to the fact that the adjoint representation of $U(n)$ is irreducible around eq. (2.3). I disagree with the line of reasoning: one way to see we should still expect $n^2$ currents is that the algebra $u(n) = su(n) ⊕ u(1)$. So we expect $n^2-1$ currents from the $su(n)$ and 1 current from the $u(1)$. It appears to me that the authors disregard the latter.
2. Specifically, for $U(2)$ we should have 4 currents in the holomorphic sector. This is not the case for the partition function at hand, and in (6.2) it's shown that, for $n=2$, the CFT reduces to a $SU(2)$ model, which does not have $U(2)$ symmetry.
3. For $n=1$, one should recover some $U(1)$ symmetric model, which is known to have have one conserved current in the holomorphic sector (e.g. the compactified free boson at generic radius). However, the multiplicity of the currents from the partition function (4.1) vanishes here. One way out for this particular issue would be that there's some extra operator that has the dimension of a current for $n=1$, but this seems at odds with the philosophy of obtaining the spectrum of this theory by analytic continuation.
The authors should clarify what the actual symmetry of the CFT described by (4.1) actually is, because currently it looks like it's something smaller than $U(n)$.
Recommendation
Ask for major revision
Report
This is an interesting paper which presents new results on U(n) spin chains and loop models. I found the paper a bit difficult to read, though, and I have a number of comments and suggestions of stylistic nature that might improve the presentation and help the reader, see the attached report. The paper will eventually be a very valuable contribution to SciPost.
Recommendation
Ask for minor revision