Affiliation:
1. Université catholique de Louvain
2. L'Institut de physique théorique
3. Sorbonne University
4. École Normale Supérieure
Abstract
We compute lattice correlation functions for the model of critical
dense polymers on a semi-infinite cylinder of perimeter
nn.
In the lattice loop model, contractible loops have a vanishing fugacity
whereas non-contractible loops have a fugacity
\alpha \in (0,\infty)α∈(0,∞).
These correlators are defined as ratios Z(x)/Z_0Z(x)/Z0
of partition functions, where Z_0Z0
is a reference partition function wherein only simple half-arcs are
attached to the boundary of the cylinder. For
Z(x)Z(x),
the boundary of the cylinder is also decorated with simple half-arcs,
but it also has two special positions 11
and xx
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
nn
using the representation of the enlarged periodic Temperley-Lieb algebra
in the XX spin chain. The resulting asymptotics as
n\to \inftyn→∞
are expressed as simple integrals that depend on the scaling parameter
\tau = \frac {x-1} n \in (0,1)τ=x−1n∈(0,1).
For small \tauτ,
the leading behaviours are proportional to
\tau^{1/4}τ1/4,
\tau^{1/4}\log \tauτ1/4logτ,
\log \taulogτ
and \log^2 \taulog2τ.
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=-2c=−2
and conformal dimensions \Delta = -\frac18Δ=−18
or \Delta = 0Δ=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.
Subject
General Physics and Astronomy
Cited by
1 articles.
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1. Bipartite fidelity for models with periodic boundary conditions;Journal of Statistical Mechanics: Theory and Experiment;2021-02-01