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Two-point boundary correlation functions of dense loop models
by Alexi Morin-Duchesne, Jesper Lykke Jacobsen
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Submission summary
| Authors (as registered SciPost users): | Jesper Lykke Jacobsen · Alexi Morin-Duchesne |
| Submission information | |
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| Preprint Link: | http://arxiv.org/abs/1712.08657v1 (pdf) |
| Date submitted: | 2018-01-24 01:00 |
| Submitted by: | Morin-Duchesne, Alexi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate six types of two-point boundary correlation functions in the dense loop model. These are defined as ratios $Z/Z^0$ of partition functions on the $m\times n$ square lattice, with the boundary condition for $Z$ depending on two points $x$ and $y$. We consider: the insertion of an isolated defect (a) and a pair of defects (b) in a Dirichlet boundary condition, the transition (c) between Dirichlet and Neumann boundary conditions, and the connectivity of clusters (d), loops (e) and boundary segments (f) in a Neumann boundary condition. For the model of critical dense polymers, corresponding to a vanishing loop weight ($\beta = 0$), we find determinant and pfaffian expressions for these correlators. We extract the conformal weights of the underlying conformal fields and find $\Delta = -\frac18$, $0$, $-\frac3{32}$, $\frac38$, $1$, $\tfrac \theta \pi (1+\tfrac{2\theta}\pi)$, where $\theta$ encodes the weight of one class of loops for the correlator of type f. These results are obtained by analysing the asymptotics of the exact expressions, and by using the Cardy-Peschel formula in the case where $x$ and $y$ are set to the corners. For type b, we find a $\log|x-y|$ dependence from the asymptotics, and a $\ln (\ln n)$ term in the corner free energy. This is consistent with the interpretation of the boundary condition of type b as the insertion of a logarithmic field belonging to a rank two Jordan cell. For the other values of $\beta = 2 \cos \lambda$, we use the hypothesis of conformal invariance to predict the conformal weights and find $\Delta = \Delta_{1,2}$, $\Delta_{1,3}$, $\Delta_{0,\frac12}$, $\Delta_{1,0}$, $\Delta_{1,-1}$ and $\Delta_{\frac{2\theta}\lambda+1,\frac{2\theta}\lambda+1}$, extending the results of critical dense polymers. With the results for type f, we reproduce a Coulomb gas prediction for the valence bond entanglement entropy of Jacobsen and Saleur.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2018-2-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1712.08657v1, delivered 2018-02-21, doi: 10.21468/SciPost.Report.359
Strengths
1. The authors make the aim of their paper clear from the very beginning.
2. They set up the problem in a clear fashion and give some simple examples to illustrate mathematical objects they will be using.
3. Despite the technical calculations, Sections 3 and 4 were still quite readable. The authors allowed the reader to get a sense of the methods used without having to understand all the finer points.
Weaknesses
1. I felt that the conclusion of the paper itself introduced new ideas that could maybe be referred to earlier in the paper (eg. the relation between the projective modules (5.1)).
Report
In this paper the authors calculate correlation functions for critical dense polymers for six different types of boundary conditions and use these results to determine the conformal dimensions in the corresponding conformal field theory. They also find logarithmic behaviour in the correlation functions, as expected for these types of loop models. The authors then proceed to use knowledge of the characters in the conformal field theory to find results for the more general class of dense loop models. Overall the paper is well-written and clear in its set up and results. I recommend this paper for publication after the authors address the points listed under "requested changes."
Requested changes
Below I list some points the authors may wish to check.
0. Authors may wish to fix one convention of either US or British English (eg. page 19 contains both "behaviour" and "behavior").
1. Page 5, 1st sentence: rectangular lattice -> square lattice (as in the abstract)
2. Page 5, 2nd sentence: 2mn -> mn tiles (since there are mn tiles on the mxn lattice)
3. Page 5, equation (2.9): I think the first relation should be $(e_j)^2 = \beta e_j$
4. Page 6, 2nd last sentence "We note that for $\gamma=0$, $v\dot v'$ is zero unless $v$ and $v'$ have the same number of defects": It seems that (2.14) has a factor of $\gamma^d$ in the case $d'=d$, which seems to indicate that the product is zero also in that case.
5. Pages 6-8: I'm not sure why in equation (2.22) the authors flip the ket v' and attach it to v, but in (2.14) they do it the other way around. It seems an arbitrary choice, so I don't understand the motivation for having them different in the two cases. It also seems that in (2.29) the convention is that the ket is on the top (unflipped) and the bra is flipped, which corresponds to the description in (2.14) not (2.22).
6. Page 11, after equation (2.44): have either $q^{1/2}+q^{-1/2}$ with parenthesis both or neither times.
7. Section 3. Since the authors take $\beta=0$, it seems that $\ket{v_0}$ should be the zero vector, since $\bra{v_0}\ket{v_0}=0$. This can't be true, however, since then the righthand side of (3.17) would be zero.
8. Page 14, equation (3.17). Should it be $O((\Lambda_1/\Lambda_0)^{m/2})$?
9. Page 16, below (3.42): $x=rx'$ is repeated; I guess one should be $y=ry'$.
10. Section 3.3-3.4: how do the authors know to compare equation (3.45) with (2.4), but (3.68) with (2.5)? I suppose it is because (2.4) and (2.5) are the only options for the form of the correlation function and so the approach is to match it with whichever is of the correct form. If that is the case (or otherwise), some extra explanation could be beneficial here.
11. Page 28, 3rd sentence: Section 4.2 -> Section 4.1.
12. Page 31, 2nd sentence after (4.33): the first "as" can be removed.
13. Pages 43-44, equations (D.6) and (D.13): I think the relation $b_1 e_1 b_1 = \beta_1 b_1$ should be $e_1 b_1 e_1 = \beta_1 e_1$, similarly for the second relation in (D.13).
Report #1 by Anonymous (Referee 1) on 2018-2-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1712.08657v1, delivered 2018-02-21, doi: 10.21468/SciPost.Report.356
Strengths
1. Largely analytic.
2. The CFT arguments are quite general.
Weaknesses
None.
Report
The authors present a wealth of results for two-point boundary correlation functions of dense loop models. In particular, using a mapping onto the XX model and fermion techniques, they obtain expressions for 6 different types (a through f) of two-point correlators reproducing (either analytically or analytically with a final numerical evaluation) the known conformal weights -1/8, 0, -3/32, 3/8, 1 of critical dense polymers. They demonstrate that these results are entirely consistent with the predictions of logarithmic CFT including (i) the asymptotic growth (involving logarithms) of two point correlations due to nonunitarity (type a) and (ii) the compatability with the existence of Jordan cells and a Jordan partner field (type b). I believe these are the first such results obtained exactly starting with the lattice. This is a wonderful paper! I have no hesitation in recommending that it is accepted by SciPost in its current form.
Requested changes
None.