Symmetries of quantum chains, and the Onsager algebra

vendredi 18 janv. 2019, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Institut Camille Jordan)

(based on arXiv:1812.09091)

In this talk I will discuss the symmetries of certain quantum chains associated with two-dimensional statistical models, and, in particular, the so-called Onsager algebra (OA).
The OA has played a central role in Onsager's historical solution of the two-dimensional Ising model (1944). While this model has since then been reinterpreted in terms of free one-dimensional fermions, the OA has appeared in other contexts non-amenable to a free-particle description and raises a number of interesting questions in close relation with integrability. 
I will show here that the OA emerges very naturally as a symmetry of some "clock models", which can be thought of as generalizations of Ising. Such a large symmetry results in rich degeneracies in the spectrum of the associated quantum Hamiltonians or transfer matrices, which, in the Bethe ansatz language, can be understood in terms of particles carrying no energy. Using the Onsager algebra, I will show how insight can be gained into the physics of these particles, and in particular that these have a hidden free-fermionic behaviour despite the genuinely interacting nature of the model.