Analytical Bethe ansatz for closed and open gl(\U0001d4a9)-spin chains in any representation

Abstract

We present an ‘algebraic treatment’ of the analytical Bethe ansatz. For this purpose, weintroduce abstract monodromy and transfer matrices which provide an algebraic frameworkfor the analytical Bethe ansatz. It allows us to deal with a generic gl(\U0001d4a9)-spin chain possessing on each site an arbitrary gl(\U0001d4a9)-representation. For open spin chains, we use the classification of the reflection matrices totreat all the diagonal boundary cases.As a result, we obtain the Bethe equations in their full generality for closed and openspin chains. The classifications of finite-dimensional irreducible representationsfor the Yangian (closed spin chains) and for the reflection algebras (open spinchains) are directly linked to the calculation of the transfer matrix eigenvalues.The local Hamiltonian associated with a given integrable spin chain needs to be computedcase by case. A general formula is still lacking at that point. As examples, we recover theusual Hamiltonian and Bethe equations for closed and open spin chains; we treat thealternating spin chains and the closed spin chain with an impurity. We also compute theHamiltonian and Bethe equations for the open spin chain model used in the context oflarge-N QCD.