Phase diagram of the triangular-lattice Potts antiferromagnet

J. Phys. A: Math. Theor.

et al. 2017

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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.

“…30) which looks very similar to the original action L n = m∈Z ψ 2 −m ψ 1 m+n . We first check by a direct calculation that L br n do satisfy the Virasoro relations:…”

supporting

confidence: 54%

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

Belletête

Gainutdinov

J. Phys. A: Math. Theor.

et al. 2017

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“…To this end we have used the CLN package [59] (again written in C++) that performs floating point operators to any desired numerical precision. We are greatly indebted to Christian R. Scullard who has provided a pure C++ version of Arpack with templates that are compatible with CLN; this is unpublished work, but it has been described in recent publications on a different subject [60][61][62].…”

Section: A6 Ising Modelmentioning

confidence: 99%

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

Saleur

2019

J. High Energ. Phys.

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160

We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional Q-state Potts model conformal field theory. In a recent work [1], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on infinite cylinders. We find that the spectra in the scaling limit are much richer than those proposed in [1]: they involve in particular fields with conformal weight hr,s where r is dense on the real axis.

“…But more generally, any statistical model depending on one (or more) parameters can be studied in the complex plane of the corresponding variable(s). In particular, the chromatic polynomial with Q ∈ C colors has been used as a test bed to develop a range of numerical, analytical and algebraic tools for computing partition function zeros and analyzing their behavior as the (partial) thermodynamic limit is approached [36][37][38][39][40][41][42]. Further information about the physical relevance of studying partition function zeros can be found in [43] and the extensive list of references in [36].…”

Section: Zeros Of Partition Functionsmentioning

confidence: 99%

Cylinder partition function of the 6-vertex model from algebraic geometry

Bajnok

Jiang

et al. 2020

J. High Energ. Phys.

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We compute the exact partition function of the isotropic 6-vertex model on a cylinder geometry with free boundary conditions, for lattices of intermediate size, using Bethe ansatz and algebraic geometry. We perform the computations in both the open and closed channels. We also consider the partial thermodynamic limits, whereby in the open (closed) channel, the open (closed) direction is kept small while the other direction becomes large. We compute the zeros of the partition function in the two partial thermodynamic limits, and compare with the condensation curves.