Contributor info: Dr Yacine Ikhlef

Yacine Ikhlef, Alexi Morin-Duchesne

SciPost Phys. 17, 132 (2024) · published 12 November 2024 | Toggle abstract · pdf

We propose a new family $\mathsf{Y}_{k,\ell,x,y,[z,w]}$ of modules over the enlarged periodic Temperley–Lieb algebra $\mathsf{\mathcal EPTL}_N(\beta)$. These modules are built from link states with two marked points, similarly to the modules $\mathsf{X}_{k,\ell,x,y,z}$ that we constructed in a previous paper. They however differ in the way that defects connect pairwise. We analyse the decomposition of $\mathsf{Y}_{k,\ell,x,y,[z,w]}$ over the irreducible standard modules $\mathsf{W}_{k,x}$ for generic values of the parameters $z$ and $w$, and use it to deduce the fusion rules for the fusion $\mathsf{W} × \mathsf{W}$ of standard modules. These turn out to be more symmetric than those obtained previously using the modules $\mathsf{X}_{k,\ell,x,y,z}$. From the work of Graham and Lehrer, it is known that, for $\beta = -q-q^{-1}$ where $q$ is not a root of unity, there exists a set of non-generic values of the twist $y$ for which the standard module $\mathsf{W}_{\ell,y}$ is indecomposable yet reducible with two composition factors: a radical submodule $\mathsf{R}_{\ell,y}$ and a quotient module $\mathsf{Q}_{\ell,y}$. Here, we construct the fusion products $\mathsf{W}×\mathsf{R}$, $\mathsf{W}×\mathsf{Q}$ and $\mathsf{Q} × \mathsf{Q}$, and analyse their decomposition over indecomposable modules. For the fusions involving the quotient modules $\mathsf{Q}$, we find very simple results reminiscent of $\mathfrak{sl}(2)$ fusion rules. This construction with modules $\mathsf{Y}_{k,\ell,x,y,[z,w]}$ is a good lattice regularization of the operator product expansion in the underlying logarithmic bulk conformal field theory. Indeed, it fits with the correspondence between standard modules and connectivity operators, and is useful for the calculation of their correlation functions. Remarkably, we show that the fusion rules $\mathsf{W}×\mathsf{Q}$ and $\mathsf{Q}×\mathsf{Q}$ are consistent with the known fusion rules of degenerate primary fields.

Benoit Estienne, Yacine Ikhlef, Andrei Rotaru

SciPost Phys. 12, 141 (2022) · published 29 April 2022 | Toggle abstract · pdf

We consider the entanglement entropy in critical one-dimensional quantum systems with open boundary conditions. We show that the second Rényi entropy of an interval away from the boundary can be computed exactly, provided the same conformal boundary condition is applied on both sides. The result involves the annulus partition function. We compare our exact result with numerical computations for the critical quantum Ising chain with open boundary conditions. We find excellent agreement, and we analyse in detail the finite-size corrections, which are known to be much larger than for a periodic system.

Yacine Ikhlef, Alexi Morin-Duchesne

SciPost Phys. 12, 030 (2022) · published 20 January 2022 | Toggle abstract · pdf

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.

Benoit Estienne, Yacine Ikhlef, Alexi Morin-Duchesne

SciPost Phys. 10, 054 (2021) · published 4 March 2021 | Toggle abstract · pdf

In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay algebraically with system size, with exponents related to the operators' scaling dimensions. In contrast, in the case of a U(1) symmetry group, the corrections only decay logarithmically with system size, with model-dependent prefactors. We show that the determination of these prefactors boils down to the computation of twisted overlaps.

Thomas Dupic, Benoit Estienne, Yacine Ikhlef

SciPost Phys. 4, 031 (2018) · published 18 June 2018 | Toggle abstract · pdf

We present a new method to compute R\'enyi entropies in one-dimensional critical systems. The null-vector conditions on the twist fields in the cyclic orbifold allow us to derive a differential equation for their correlation functions. The latter are then determined by standard bootstrap techniques. We apply this method to the calculation of various R\'enyi entropies in the non-unitary Yang-Lee model.