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Soutenance de Louis Vignoli

Quantum separation of variables for higher rank and supersymmetric integrable systems
When Oct 29, 2021
from 02:00 to 04:00
Where Salle des thèses
Contact Name Louis Vignoli
Attendees Louis Vignoli
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The work of this thesis belongs to the studies of quantum integrable models on the one-dimensional lattice. It focuses on the development of a new quantum separation of variables method for these particular models. This approach aims to characterise the transfer matrix spectrum—which contains all the conserved quantities responsible for the integrability of such systems—by constructing a so-called separate basis of the Hilbert space of the system, enabling the reduction of its spectral problem with N coupled variables, N big or even infinite, in N independent one-variable spectral problems. The wave functions of the eigenstates are then products of N one-variable wave functions. A recent advance in this context was the construction of such bases from the transfer matrix itself, and the characterisation of the spectrum by the fusion equations stemming from the representation theory of the underlying symmetry algebra.

Here, we obtain this construction for supersymmetric gl(m|n) integrable models. For the gl(1|2) case with specific quasi-periodic boundary conditions, it allows us to characterize completely the spectrum in terms of solutions of a discrete system of finite difference cubic equations, but also and equivalently with a functional equation called the quantum spectral curve.

We also compute scalar products between separate states in higher rank integrable models, determining the measure associated to separate bases. For the gl(3) model, this measure is pseudo-orthogonal and we characterize it completely. We also put forward a family of conserved quantities that construct a basis and its dual, both separate and orthogonal from each others, simplifying the computation of scalar products between separate states. This paves the ways towards the computation of form factors and correlation functions of such systems in the quantum separation of variables settings.