SciPost Phys. 7, 040 (2019) ·
published 30 September 2019

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We compute lattice correlation functions for the model of critical dense polymers on a semiinfinite cylinder of perimeter $n$. In the lattice loop model, contractible loops have a vanishing fugacity whereas noncontractible 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 halfarcs are attached to the boundary of the cylinder. For $Z(x)$, the boundary of the cylinder is also decorated with simple halfarcs, 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 TemperleyLieb algebra in the XX spin chain. The resulting asymptotics as $n\to \infty$ are expressed as simple integrals that depend on the scaling parameter $\tau = \frac {x1} n \in (0,1)$. 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 nonabelian.
SciPost Phys. 4, 034 (2018) ·
published 19 June 2018

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We investigate six types of twopoint 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
CardyPeschel formula in the case where $x$ and $y$ are set to the corners. For
type b, we find a $\logxy$ 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.