## SciPost Submission Page

# Logarithmic correlation functions for critical dense polymers on the cylinder

### by Alexi Morin-Duchesne, Jesper Lykke Jacobsen

### Submission summary

As Contributors: | Alexi Morin-Duchesne |

Preprint link: | scipost_201908_00001v1 |

Date submitted: | 2019-08-01 |

Submitted by: | Morin-Duchesne, Alexi |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Condensed Matter Physics - Theory |

### Abstract

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter $n$. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity $\alpha \in (0,\infty)$. These correlators are defined as ratios $Z(x)/Z_0$ of partition functions, where $Z_0$ is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For $Z(x)$, the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions $1$ and $x$ where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite $n$ using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as $n\to \infty$ are expressed as simple integrals that depend on the scaling parameter $\tau = \frac {x-1} n \in (0,1)$. For small $\tau$, the leading behaviours are proportional to $\tau^{1/4}$, $\tau^{1/4}\log \tau$, $\log \tau$ and $\log^2 \tau$. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge $c=-2$ and conformal dimensions $\Delta = -\frac18$ or $\Delta = 0$. With these assumptions, we obtain differential equations of order two and three satisfied by the conformal correlation functions, solve these equations in terms of hypergeometric functions, and find a perfect agreement with the lattice results. We use the lattice results to compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-8-17 Invited Report

### Strengths

1. Exact calculation of correlation functions of dense polymer model

2. Used to derive conclusions about 2-d log conformal field theory

### Weaknesses

none noticed.

### Report

This is a very nice and impressive paper, in which the authors have studied correlation functions in the dense polymer problem, and used its relation to the logarithmic conformal field theory, to get a better understanding of the latter. The exact calculation of correlations in the dense polymer problem, is a tour de force, and the authors have demostrated formidable mathematical skills to express the results in terms of hypergeometric functions etc. .

As emphasized in the intoduction, the log-conformal field theories are of special interest, as they represent the case where the primary field do not undergo a simple multiplicative scale change under dilation. They turn up in the studies of many interesting models like the sandpile model,

dense polymers, and critical percolation. The results are exact, interesting and nontrivial.

I have not checked the algebra. That would be a very demanding job. The authors have summarized the results clearly, and omitting the algebaric details. Going by the past reputation of the authors, i would expect these to be correct, and recommend publication of the paper in its present form.