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.
Eric Vernier, Jesper Lykke Jacobsen, Hubert Saleur
SciPost Phys. 2, 004 (2017) ·
published 21 February 2017

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We revisit the phase diagram of spin1 $su(2)_k$ anyonic chains, originally
studied by Gils {\it et. al.} [Phys. Rev. B, {\bf 87} (23) (2013)]. These
chains possess several integrable points, which were overlooked (or only
briefly considered) so far.
Exploiting integrability through a combination of algebraic techniques and
exact Bethe ansatz results, we establish in particular the presence of new
first order phase transitions, a new critical point described by a $Z_k$
parafermionic CFT, and of even more phases than originally conjectured. Our
results leave room for yet more progress in the understanding of spin1 anyonic
chains.
Jan de Gier, Jesper Lykke Jacobsen, Anita Ponsaing
SciPost Phys. 1, 012 (2016) ·
published 19 December 2016

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We compute the boundary entropy for bond percolation on the square lattice in
the presence of a boundary loop weight, and prove explicit and exact
expressions on a strip and on a cylinder of size $L$. For the cylinder we
provide a rigorous asymptotic analysis which allows for the computation of
finitesize corrections to arbitrary order. For the strip we provide exact
expressions that have been verified using highprecision numerical analysis.
Our rigorous and exact results corroborate an argument based on conformal field
theory, in particular concerning universal logarithmic corrections for the case
of the strip due to the presence of corners in the geometry. We furthermore
observe a crossover at a special value of the boundary loop weight.