Aspects of classical and quantum integrable models - Retore, Ana Lúcia (2019)
Aspects of classical and quantum integrability are explored. Gauge transformations play a fundamental role in both cases. Classical integrable hierarchies have an underlying algebraic structure which brings universality for the solutions of all the equations belonging to a hierarchy. Such universality is explored together with the gauge invariance of the zero curvature equation to systematically construct the Bäcklund transformations for the mKdV hierarchy, as well as to relate it with the KdV hierarchy. As a consequence the defect-matrix for the KdV hierarchy is obtained and a few explicit Bäcklund transformations are computed for both Type-I and Type-II. The generalization for super mKdV hierarchy is also explored. We studied symmetries and degeneracies of families of integrable quantum open spin chains with finite length associated to affine Lie algebras \hat{g} = A^{(2)}_{2n} , A^{(2)}_{2n−1}, B^{(1)}_n , C^{(1)}_ n , D^{(1)}_n whose K-matrices depend on a discrete parameter p (p = 0, ...,n). We show that all these transfer matrices have quantum group symmetry corresponding to removing the p^{th} node of the Dynkin diagram of \hat{g}. We also show that the transfer matrices for C^{(1)}_n and D^{(1)}_n also have duality symmetry and the ones for A^{(2)}_{2n−1}, B^{(1)}_n and D^{(1)}_n have Z_2 symmetries that map complex representations into their conjugates. Gauge transformations simplify considerably the proofs by allowing us to work in a way that only unbroken generators appear.The spectrum of the same integrable spin chains with the addition of D(2) n+1 is then determined using analytical Bethe ansatz. We conjecture a generalization for open chains for the Bethe ansatz Reshetikhin’s general formula and propose a formula relating the Dynkin labels of the Bethe states with the number of Bethe root sof each type.