A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain
| Autor: | Murray T. Batchelor, Paul Wedrich, Remy Adderton |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Pure mathematics Statistical Mechanics (cond-mat.stat-mech) 010102 general mathematics Skein relation Zero (complex analysis) General Physics and Astronomy Bracket polynomial FOS: Physical sciences Statistical and Nonlinear Physics Mathematical Physics (math-ph) Temperley–Lieb algebra 01 natural sciences Contractible space Chain (algebraic topology) Modeling and Simulation 0103 physical sciences 0101 mathematics 010306 general physics Hamiltonian (control theory) Condensed Matter - Statistical Mechanics Mathematical Physics Mathematics Resolution (algebra) |
| Popis: | 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 J. F. Fjelstad and T. M\r{a}nsson [J. Phys. A {\bf 45} (2012) 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 $\sqrt{3}$ 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 $\rho= e^{ 2\pi \mathrm{i}/3}$ for the SICP chain and $\rho=1$ for the staggered XX chain. These $\rho$ values are derived assuming the Kauffman bracket skein relation. Comment: 10 pages, 4 figures, further cubic relations added |
| Databáze: | OpenAIRE |
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