Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow

Authors: Paul Ryan, Dmytro Volin

Preprint number: UUITP-40/20

Abstract: We propose a way to separate variables in a rational integrable gl(n) spin chain with an arbitrary finite-dimensional irreducible representation at each site and with generic twisted periodic boundary conditions. Firstly, we construct a basis that diagonalises a higher-rank version of the Sklyanin B-operator; the construction is based on recursive usage of an embedding of a gl(k) spin chain into a gl(k+1) spin chain which is induced from a Yangian homomorphism and controlled by dual diagonals of Gelfand-Tsetlin patterns. Then, we show that the same basis can be equivalently constructed by action of Bäcklund-transformed fused transfer matricies, whence the Bethe wave functions factorise into a product of ascending Slater determinants in Baxter Q-functions. Finally, we construct raising and lowering operators – the conjugate momenta – as normal-ordered Wronskian expressions in Baxter Q-operators evaluated at zeros of B – the separated variables. It is an immediate consequence of the proposed construction that the Bethe algebra comprises the maximal possible number of mutually commuting charges – a necessary property for Bethe equations to be complete.