# Two-point boundary correlation functions of dense loop models

### Submission summary

 As Contributors: Jesper Lykke Jacobsen · Alexi Morin-Duchesne Arxiv Link: http://arxiv.org/abs/1712.08657v2 (pdf) Date accepted: 2018-03-14 Date submitted: 2018-03-05 01:00 Submitted by: Morin-Duchesne, Alexi Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Mathematical Physics 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.

### Ontology / Topics

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Published as SciPost Phys. 4, 034 (2018)

Dear editor,

We thank the referees for their reports on our article. Please find below a list of comments and changes made in response to the second referee’s report.

All the best,
Alexi Morin-Duchesne and Jesper Jacobsen

### List of changes

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0. We have adapted the spelling to the best of our knowledge so that British English is used throughout.

1. 2. 3. We have made the changes requested by the referee.

4. Our claim was indeed incorrect. We have changed the text to remove this false statement.

5. The sentence just below (2.22) has been fixed.

6. The parentheses have been removed.

7. As explained above (2.21), for generic values of q and v a linear combination of link states, <v| is obtained by taking the dagger of |v>, with q mapped to q, and not to 1/q even if in the end we consider q on the unit circle. In section 3, this applies to the state v_0: <v_0| is then the transpose of |v_0>, with no complex conjugation. We indeed have <v_0|v_0> = 0, but because the entries of |v_0> are complex-valued, this does not imply that |v_0> = 0. No changes were made to the text.

8. 9. We have corrected the two typos.

10. Experience from other lattice models indicates that changes in boundary conditions often correspond to primary fields in CFT. Comparing our lattice result with the correlation functions of primary fields is indeed the first natural things to do. But in some cases, such changes in the boundary instead correspond to logarithmic fields or to compositions of primary fields, for instance.

To justify the type of correlator that arises from the lattice computation, one must dig deeper in the CFT arguments. This is precisely the goal pursued in Sections 4.1 and 4.2: to give the detailed CFT argument that explains why (3.45) and (3.68) must respectively be compared with (2.4) and (2.5). We have added a sentence at the beginning of Section 4 to make this clearer.

11. 12. 13. These typos have been fixed.

We have also performed these two extra changes:

(i) We have added a sentence in Section 4.2 about projective modules, announcing that these will be discussed further in the Conclusion.

(ii) We have added two new references in Section 3.1.

### Submission & Refereeing History

Resubmission 1712.08657v2 on 5 March 2018
Submission 1712.08657v1 on 24 January 2018