The research works presented in this thesis belong to the domain of quantum integrable models defined on a one-dimensional lattice. Their aim is the development of a new quantum separation of variables (SoV) scheme for these models. The goal of this method is to characterize completely the spectrum of the transfer matrix, an operator generating a family of commuting quantum conserved charges, making such models integrable. It is achieved by constructing a so-called separate basis of the Hilbert space in which the N-variables coupled spectral problem, N arbitrary large, reduces to N independent spectral problems in one variable. In such a basis the N variables eigen-wave functions separate into the product of N wave functions in one variable. A recent breakthrough gives the construction of such bases from the transfer matrices themselves, the spectrum being given through their fusion relations stemming from the representation theory of the underlying quantum symmetry algebra.
We have obtained such a construction for gl(m,n) super-symmetric quantum integrable lattice models. For the gl(1,2) case with certain quasi-periodic boundary conditions, it has enabled us to get a complete characterization of the transfer matrix spectrum in terms of solution to a system of finite-difference cubic equations and equivalently, through an associated functional equation, the so-called quantum spectral curve.
Within this new SoV method we have also computed the scalar products of the separate states (having factorized wave functions) for higher rank models, hence determining the spectral measure for these separate bases. For the gl(3) case, this measure is pseudo-orthogonal, and we have characterized it completely. Alongside the transfer matrix operator, we have also introduced a new set of quantum conserved charges leading to orthogonal SoV bases and hence to diagonal measure. It paves the way for the computation of form factors and correlation functions for these higher rank models.
Gratuit
Disciplines