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

Abstract

The relationship between bulk and boundary properties is one of the founding features of (rational) conformal field theory (CFT). 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, in physical terms, 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 Temperley-Lieb algebra. We show that for the Restricted Solid-On-Solid (RSOS) models, which are known to be described by minimal unitary CFTs (with central charge c < 1) in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed (identical) 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 Temperley-Lieb (ATL) algebra. We then turn to the same kind of analysis for two models whose continuum limits are logarithmic CFTs (LCFTs) -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 (in that case, indecomposable but not irreducible) modules of the TL algebra 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. The situation is similar for the sl(2|1) case. The problem of relating bulk and boundary lattice models for LCFTs thus remains open.

A fusion for the periodic Temperley-Lieb algebra and its continuum limit

“…The study of infinite-dimensional modules of ATL and the possible associated lattice models appears like one of the most interesting areas for future study. An obvious application of our results concerns fusion for the blob algebra modules combining with idea of "braid translation" from [26]. Institute of Melbourne University and to the organizers of the conference Integrability in lowdimensional Quantum Systems in Creswick in 2017 where a part of this work was done.…”

Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

“…For the reader's convenience we summarise here some of the notations used in this paper. They are, as far as possible, the same as those used in [27,28,[37][38][39][40][41][42].…”

Phase diagram of the triangular-lattice Potts antiferromagnet

“…In addition, (38) must be supplemented by further relations to make the algebra finite-dimensional, giving rise to the so-called affine TL algebra. We refer to [92] for further discussion.…”