Integrable long-range spin chains with quantum loop symmetries

Adel Ben Moussa (Institut de Physique Theorique)

Thursday 20th March 15:00-16:00 Maths 311B

Abstract

Quantum spin chains are one-dimensional models of spins. The Heisenberg XXX and XXZ spin chains are particularly successful examples as they are not only experimentally relevant but also theoretically deep. The critical assumption made in these models is that only nearest neigbouring spins interact with each other. However, in real settings, spins that are further apart may interact as well.

I will introduce the q-Haldane-Shastry spin chain, a long-range generalisation of XXZ parameterised by a complex number q, expressed in terms of the Temperley-Lieb algebra and which exhibits quantum loop symmetry (U_q(\hat{sl}_2)). It is remarkably integrable: the corresponding wave functions can be exactly and explicitly given in terms of Macdonald polynomials, all thanks to a remarkable mathematical structure related to the Ruijsenaars-Schneider model. At the singular point q=i, specific features arise and I will explain how we were able to interpret them as a free-fermionic behaviour in a recent work (arXiv:2404.10164).

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