Dipartimento di Matematica, Sapienza Università di Roma, Sala di Consiglio Giovedì 9 Febbraio 2023, ore 16:00 *Speaker:* Daniel Ueltschi, University of Warwick *Title*: Quantum spin chains, loop representations, dimerisation *Abstract:* In contrast to their classical counterparts, one-dimensional quantum spin systems are interesting, they have intriguing behaviour, and they are difficult to study. I will describe a family of systems with nearest-neighbour interactions and O(n) symmetry. Its ground state phase diagram is expected to have a rich structure. For large spins it is possible to prove the occurrence of dimerisation in an open domain. The proof involves a loop representation and the method of cluster expansions. This is joint work with Jakob Bjornberg, Peter Muhlbacher, and Bruno Nachtergaele.