SciPost Phys. 14, 092 (2023) ·
published 3 May 2023
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We determine the spaces of states of the two-dimensional $O(n)$ and $Q$-state Potts models with generic parameters $n,Q\in \mathbb{C}$ as representations of their known symmetry algebras. While the relevant representations of the conformal algebra were recently worked out, it remained to determine the action of the global symmetry groups: the orthogonal group for the $O(n)$ model, and the symmetric group $S_Q$ for the $Q$-state Potts model. We do this by two independent methods. First we compute the twisted torus partition functions of the models at criticality. The twist in question is the insertion of a group element along one cycle of the torus: this breaks modular invariance, but allows the partition function to have a unique decomposition into characters of irreducible representations of the global symmetry group. Our second method reduces the problem to determining branching rules of certain diagram algebras. For the $O(n)$ model, we decompose representations of the Brauer algebra into representations of its unoriented Jones-Temperley-Lieb subalgebra. For the $Q$-state Potts model, we decompose representations of the partition algebra into representations of the appropriate subalgebra. We find explicit expressions for these decompositions as sums over certain sets of diagrams, and over standard Young tableaux. We check that both methods agree in many cases. Moreover, our spaces of states are consistent with recent bootstrap results on four-point functions of the corresponding CFTs.
SciPost Phys. 12, 147 (2022) ·
published 6 May 2022
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We define the two-dimensional $O(n)$ conformal field theory as a theory that
includes the critical dilute and dense $O(n)$ models as special cases, and
depends analytically on the central charge. For generic values of
$n\in\mathbb{C}$, we write a conjecture for the decomposition of the spectrum
into irreducible representations of $O(n)$.
We then explain how to numerically bootstrap arbitrary four-point functions
of primary fields in the presence of the global $O(n)$ symmetry. We determine
the needed conformal blocks, including logarithmic blocks, including in
singular cases. We argue that $O(n)$ representation theory provides upper
bounds on the number of solutions of crossing symmetry for any given four-point
function.
We study some of the simplest correlation functions in detail, and determine
a few fusion rules. We count the solutions of crossing symmetry for the $30$
simplest four-point functions. The number of solutions varies from $2$ to $6$,
and saturates the bound from $O(n)$ representation theory in $21$ out of $30$
cases.
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 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.
SciPost Phys. 4, 034 (2018) ·
published 19 June 2018
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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.
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 spin-1 $su(2)_k$ 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 spin-1 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
finite-size corrections to arbitrary order. For the strip we provide exact
expressions that have been verified using high-precision 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.
