Onsager algebra and Ising-type structures in root-of-unity six-vertex models

Eric Vernier

LPSM

Mon, Jan. 23rd 2023, 11:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

I will start by reviewing a surprising connection between the six vertex model (or its higher spin generalizations) and the Onsager algebra, an infinite-dimensional Lie algebra which appeared in the solution of the two-dimensional Ising model. Using Kramers-Wannier duality, a family of N-states integrable vertex models/quantum spin chains are constructed having the Onsager algebra as a symmetry algebra. Those are then identified as the six-vertex model and its higher-spin descendents, at specific "root-of-unity" values of the anisotropy parameter. While the integrability of six-vertex models is famously related to an underlying quantum group structure, the enlarged Onsager symmetry could similarly be related to exotic quantum group representations occuring at root of unity. However, this leaves certain aspects such as duality somewhat hidden in the six-vertex/quantum group formulation.
I will therefore revert the logic and show that the (higher spin) root-of-unity six-vertex models can be re-expressed more simply in terms of Ising (clock) spins with products of 2-spins interactions only. The Onsager algebra symmetry emerges naturally in this framework, and the quantum-group related structures and Yang-Baxter equations of the vertex models can be traced back to simpler star-triangle equations in the spin formulation.
This is based on E. Vernier, E. O'Brien, P. Fendley, JSTAT (2019), and some work in preparation.

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