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Critical spin chains and loop models with $U(n)$ symmetry
by Paul Roux, Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur
Submission summary
| Authors (as registered SciPost users): | 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.
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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 $U(n)$.
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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.
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Ask for minor revision