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This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley–Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as non-unitary models like percolation or self-avoiding walks. Our approach is essentially algebraic and focusses on the defects from two points of view: the "crossed channel" where the defect is seen as an operator acting on the Hilbert space of the models, and the "direct channel" where it corresponds to a modification of the basic Hamiltonian with some sort of impurity. Algebraic characterizations and constructions are proposed in both points of view. In the crossed channel, this leads us to new results about the center of the affine Temperley–Lieb algebra; in particular we find there a special basis with non-negative integer structure constants that are interpreted as fusion rules of defects. In the direct channel, meanwhile, this leads to the introduction of fusion products and fusion quotients, with interesting algebraic properties that allow to describe representations content of the lattice model with a defect, and to describe its spectrum. [ABSTRACT FROM AUTHOR]

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $${\mathfrak{gl}(1|1)}$$ spin-chain and its continuum limit-the $${c=-2}$$ symplectic fermions theory-and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, ) and Gainutdinov et al. (Nucl Phys B 871:289-329, ). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL, goes over in the continuum limit to a bigger algebra than $${\boldsymbol{\mathcal{V}}}$$ , the product of the left and right Virasoro algebras. This algebra, $${\mathcal{S}}$$ -which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field $${S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}$$ , with a symmetric form $${S_{\alpha\beta}}$$ and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL in the $${\mathfrak{gl}(1|1)}$$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of $${\mathfrak{sp}_{N-2}}$$ . The semi-simple part of JTL is represented by $${U \mathfrak{sp}_{N-2}}$$ , providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL image represented in the spin-chain. On the continuum side, simple modules over $${\mathcal{S}}$$ are identified with 'fundamental' representations of $${\mathfrak{sp}_\infty}$$ . [ABSTRACT FROM AUTHOR]

Logarithmic conformal field theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self-avoiding walks, etc), or of critical points in several classes of disordered systems (transition between plateaux in the integer and spin quantum Hall effects). Much progress in their understanding has been obtained by studying algebraic features of their lattice regularizations. For reasons which are not entirely understood, the non-semi-simple associative algebras underlying these lattice models--such as the Temperley-Lieb algebra or the blob algebra--indeed exhibit, in finite size, properties that are in full correspondence with those of their continuum limits. This applies not only to the structure of indecomposable modules, but also to fusion rules, and provides an 'experimental' way of measuring couplings, such as the 'number b' quantifying the logarithmic coupling of the stress-energy tensor with its partner. Most results obtained so far have concerned boundary LCFTs and the associated indecomposability in the chiral sector. While the bulk case is considerably more involved (mixing in general left and right moving sectors), progress has also recently been made in this direction, uncovering fascinating structures. This study provides a short general review of our work in this area. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop a method to describe perturbatively the entanglement entropy in a simple impurity model, the Interacting Resonant Level Model (IRLM), at low energy (i.e. in the strong coupling regime). We use integrability results for the Kondo model to describe the infrared fixed point, conformal field theory techniques initially developed by Cardy and Calabrese and a quantization scheme that allows one to compute exactly Renyi entropies at arbitrary order in in principle, even when the system size or the temperature is finite. We show that those universal quantities at arbitrary interaction parameter in the strong coupling regime are very well approximated by the same quantities in the free fermion system in the case of attractive Coulomb interaction, whereas a strong dependence on the interaction appears in the case of repulsive interaction. [Copyright &y& Elsevier]

Abstract: This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the identification of the space of states as a bimodule over the Temperley–Lieb (TL) algebra and the quantum group . The extension of this analysis in the bulk case involves considerable difficulties, since the symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones–Temperley–Lieb — JTL — algebra). Even the simplest case of the spin chain — corresponding to the symplectic fermions theory in the continuum limit — presents very rich aspects, which we will discuss in several papers. In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the fundamental representation and its dual. We prove that this centralizer is only a subalgebra of at that we dub . We then begin the analysis of the continuum limit of the JTL algebra: using general arguments about the regularization of the stress–energy tensor, we identify families of JTL elements going over to the Virasoro generators in the continuum limit. We then discuss the symmetry of the (continuum limit) symplectic fermions theory from the lattice and JTL point of view. The analysis of the spin chain as a bimodule over and is discussed in the second paper of this series. [Copyright &y& Elsevier]

Abstract: This paper is the second in a series devoted to the study of periodic super-spin chains. In our first paper (Gainutdinov et al., 2013 [3]), we have studied the symmetry algebra of the periodic spin chain. In technical terms, this spin chain is built out of the alternating product of the fundamental representation and its dual. The local energy densities — the nearest neighbour Heisenberg-like couplings — provide a representation of the Jones–Temperley–Lieb (JTL) algebra . The symmetry algebra is then the centralizer of , and turns out to be smaller than for the open chain, since it is now only a subalgebra of at — dubbed in Gainutdinov et al. (2013) [3]. A crucial step in our associative algebraic approach to bulk logarithmic conformal field theory (LCFT) is then the analysis of the spin chain as a bimodule over and . While our ultimate goal is to use this bimodule to deduce properties of the LCFT in the continuum limit, its derivation is sufficiently involved to be the sole subject of this paper. We describe representation theory of the centralizer and then use it to find a decomposition of the periodic spin chain over for any even N and ultimately a corresponding bimodule structure. Applications of our results to the analysis of the bulk LCFT will then be discussed in the third part of this series. [Copyright &y& Elsevier]