Public Thesis defense - IRMP

Integrable lattice models and supersymmetry by Jean LIENARDY

Pour l’obtention du grade de Docteur en sciences

One of the goals of statistical physics is to understand the magnetic properties of the matter from the atomic properties and quantum interactions. In this dissertation, we are interested in the famous XXZ and XYZ spin-chains, which are spin-1/2 models defined on a one-dimensional lattice.

The XXZ and XYZ spin chains are integrable and therefore possess a rich mathematical structure. Furthermore, for specific values of their parameters, these models exhibit a new feature: the supersymmetry. Precisely, they can be expressed in the framework of supersymmetric quantum mechanics. The supersymmetry implies that the Hamiltonians of these models may possess special ground states called supersymmetry singlets.

The first objective of this thesis is to exploit a relation between the supersymmetry and the theory of (co)homology to detect the existence of supersymmetry singlets and characterise these states. Furthermore, we use the supersymmetry of XYZ spin chains with various boundary conditions to prove the existence of remarkably simple eigenvalues of the transfer matrices of the related supersymmetric eight-vertex models.

The entanglement is a purely quantum property of a system. The second objective of this thesis is to introduce the multipartite fidelity, a measure of the quantum entanglement for systems with a spatial extension, and to compute this quantity for the XXZ spin chain by using the supersymmetry.

A connection between spin chains and enumerative combinatorics appeared at the turn of the century. It was discovered that special components and linear sum rules of XXZ and XYZ spin chains, for specific value of their parameters, were related to combinatorics and in particular to the enumeration of alternating sign matrices. The third objective of this thesis is to use the integrability to gain new insight into the combinatorial properties of the spin-chain ground states. Namely, we find a solution to the boundary quantum Knizhnik-Zamolodchikov equations and use it to prove formulas for particular components of the open XXZ supersymmetry singlet related to alternating sign matrices.

Jury members :

• Prof. Christian Hagendorf (UCLouvain), supervisor
• Prof. Marino Gran (UCLouvain), chairperson
• Prof. Tom Claeys (UCLouvain), secretary
• Prof. Philippe Ruelle (UCLouvain)
• Dr. Miłosz Panfil (University of Warsaw, Poland)
• Prof. Robert Weston (Heriot-Watt University, Scotland)

Pay attention :

The public defense of Jean Liénardy scheduled for Friday 25 September at 16:00 will indeed take place in the form of a video conference Teams too.

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