Integrable boundary conditions in the antiferromagnetic Potts model
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Springer
Received: March 31, 2020 Accepted: May 9, 2020 Published: May 27, 2020
Niall F. Robertson,a Michal Pawelkiewicz,a Jesper Lykke Jacobsena,b,c and Hubert Saleura,d a
Universit´e Paris Saclay, CNRS, CEA, Institut de Physique Th´eorique, F-91191 Gif-sur-Yvette, France b ´ Laboratoire de Physique de l’Ecole Normale Sup´erieure, ENS, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France c ´ Sorbonne Universit´e, Ecole Normale Sup´erieure, CNRS, Laboratoire de Physique (LPENS), F-75005 Paris, France d Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, U.S.A.
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine D22 Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K-matrices of the D22 model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12] and [13]) are discussed. Keywords: Bethe Ansatz, Lattice Integrable Models, Conformal Field Theory ArXiv ePrint: 2003.03261
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)144
JHEP05(2020)144
Integrable boundary conditions in the antiferromagnetic Potts model
Contents 1 Introduction
1
staggered six-vertex model and the D22 model Background Review of the staggered six-vertex model Mapping between the two models: general strategy Deriving the Boltzmann weights 2.4.1 Vertices 1 to 6 2.4.2 Vertices 7 to 30 2.4.3 Vertices 31 to 38 2.4.4 Sign differences
3 3 3 6 7 8 9 10 12
3 The 3.1 3.2 3.3 3.4
open D22 model Hamiltonian limit Additional symmetries The γ → 0 limit Non-zero γ
14 15 17 18 19
4 The Bethe Ansatz solution 4.1 The XXZ subset 4.1.1 Even number of Bethe roots 4.1.2 Odd number of Bethe roots 4.2 Other solutions of Bethe Ansatz equations 4.2.1 The n = 2 sector 4.2.2 The n = 1 sector 4.2.3 The n = 0 sector
19 21 21 24 24 25 27 29
5 Other Temperley-Lieb representations 5.1 Loop representation 5.2 RSOS representation
30 30 30
6 Discussion
32
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Introduction
The critical antiferromagnetic Potts model has been the subject of intense study for many years [1–3]. A striking feature is that the conformal field theory describing its continuum limit is “non-compact”, leading to the observation of a continuum of critical exponents [4, 5]. Due to this unusual feature, this model has subsequently become the subject of many
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JHEP05(2020)144
2 The 2.1 2.2 2.3 2.4
• Wj — standard modules over TLN
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