TY - JOUR

T1 - A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain

AU - Adderton, Remy

AU - Batchelor, Murray T.

AU - Wedrich, Paul

N1 - Publisher Copyright:
© 2020 The Author(s). Published by IOP Publishing Ltd.

PY - 2020/9/11

Y1 - 2020/9/11

N2 - The Hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N - 1 types of Temperley-Lieb generators. This generalises a previous result for N = 3 obtained by Fjelstad and M nsson (2012 J. Phys. A: Math. Theor. 45 155208). A pictorial representation of a related coupled algebra is given for the N = 3 case which involves a generalisation of the pictorial presentation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the N = 3 SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight √ 3 and weight 2, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter ρ = e 2πi/3 for the SICP chain and ρ = 1 for the staggered XX chain. These ρ values are derived assuming the Kauffman bracket skein relation.

AB - The Hamiltonian of the N-state superintegrable chiral Potts (SICP) model is written in terms of a coupled algebra defined by N - 1 types of Temperley-Lieb generators. This generalises a previous result for N = 3 obtained by Fjelstad and M nsson (2012 J. Phys. A: Math. Theor. 45 155208). A pictorial representation of a related coupled algebra is given for the N = 3 case which involves a generalisation of the pictorial presentation of the Temperley-Lieb algebra to include a pole around which loops can become entangled. For the two known representations of this algebra, the N = 3 SICP chain and the staggered spin-1/2 XX chain, closed (contractible) loops have weight √ 3 and weight 2, respectively. For both representations closed (non-contractible) loops around the pole have weight zero. The pictorial representation provides a graphical interpretation of the algebraic relations. A key ingredient in the resolution of diagrams is a crossing relation for loops encircling a pole which involves the parameter ρ = e 2πi/3 for the SICP chain and ρ = 1 for the staggered XX chain. These ρ values are derived assuming the Kauffman bracket skein relation.

KW - coupled Temperley-Lieb algebra

KW - staggered XX chain

KW - superintegrable chiral Potts model

UR - http://www.scopus.com/inward/record.url?scp=85091634076&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/aba143

DO - 10.1088/1751-8121/aba143

M3 - Article

SN - 1751-8113

VL - 53

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 36

M1 - 36LT01

ER -