Topological defects in lattice models and affine Temperley-Lieb algebra

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 subalgebra 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 mathematical properties that allow to describe representations content of the lattice model with a defect, and to describe its spectrum.Comment: 41

On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

62 pp.The relationship between bulk and boundary properties is one of the founding features of (Rational) Conformal Field Theory. Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley-Lieb algebra, and use the concept of braid translation, which is a natural way to close an open spin chain by adding an interaction between the first and last spins using braiding to bring them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain - the ordinary Temperley-Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic TL algebra. We show that for the Restricted Solid-On-Solid models, which are known to be described by minimal unitary CFTs in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine TL algebra. We then turn to the same kind of analysis for two models whose continuum limits are Logarithmic CFTs - the alternating $gl$(1|1) and $sl$(2|1) spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant TL modules does not give rise to the modules known to be present in the periodic chains. In the $gl$(1|1) case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit

Conformational Studies of Poly(9,9-dialkylfluorene)s in Solution Using NMR Spectroscopy and Density Functional Theory Calculations

Relationships have been obtained between intermonomer torsional angle and NMR chemical shifts (1H and 13C) for isolated chains of two of the most important poly(9,9-dialkylfluorenes), poly[9,9-bis(2-ethylhexyl)fluorene-2,7-diyl] (PF2/6) and the copolymer poly(9,9-dioctylfluorene-co-[2,1,3]benzothiadiazole-4,7-diyl) (F8BT), using DFT calculations. The correlations provide a model for NMR spectral data interpretation and the basis for analysis of conformational changes in poly(9,9-dialkylfluorene-2,7-diyl)s. The correlations obtained for PF2/6 indicate that the 13C chemical shifts of the aromatic carbons close to the intermonomer connection (C1, C2, and C3) have minimum values at planar conformations (0° and 180°) and maximum values at 90° conformations. In contrast, the 1H chemical shifts of the corresponding aromatic ortho protons (Ha and Hb) are greatest for planar conformations, and the minimum values are seen for 90° conformations. For the F8BT copolymer, similar relationships are observed for the 1H (Ha, Hb, and Hc) aromatic shifts. Considering the aromatic carbons of F8BT, the behavior of C2, C4, C5, and C6 is similar to that found for the PF2/6 carbons. However, C1 and C3 of the fluorene moiety behave differently with varying torsion angle. These are in close proximity to the fluorene−benzothiadiazole linkage and are markedly affected by interactions with the thiadiazole unit such that δC1 is a maximum for 180° and a minimum for 0°, whereas δC3 is a maximum for 0° and minimum for 180°. We have studied the 1H and 13C spectra of the two polymers at temperatures between −50 °C and +65 °C. The observed changes to higher or lower frequency in the aromatic resonances were analyzed using these theoretical relationships. Fluorescence studies on PF2/6 in chloroform solution suggest there are no significant interchain interactions under these conditions. This is supported by variable-temperature NMR results. Polymer−solvent and polymer intramolecular interactions were found to be present and influence all of the alkylic and one of the aromatic 1H resonances (Hb). The detailed attribution of the 1H and 13C NMR spectra of the two polymers was made prior to the establishment of the relationships between torsion angle and NMR chemical shifts. This was carried out through DFT calculation of the 1H and 13C shielding constants of the monomers, coupled with distortionless enhancement by polarization transfer and heteronuclear correlation NMR spectra. Several DFT levels of calculation were tested for both optimization of structures and shielding constants calculation. The B3LYP/6-31G(d,p) method was found to perform well in both cases