Analytical Bethe Ansatz for closed and open gl(n)-spin chains in any representation

TL;DR

This paper develops an algebraic framework for the analytical Bethe Ansatz applicable to both closed and open gl(n)-spin chains with arbitrary representations, providing a unified approach and explicit Bethe equations.

Contribution

It introduces an algebraic method using monodromy and transfer matrices to derive Bethe equations for general gl(n)-spin chains with various boundary conditions and representations.

Findings

Derived Bethe equations for all diagonal boundary cases.

Unified algebraic framework for closed and open chains.

Included examples like alternating chains and impurity models.

Abstract

We present an "algebraic treatment" of the analytical Bethe Ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe Ansatz. It allows us to deal with a generic gl(n)-spin chain possessing on each site an arbitrary gl(n)-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. As examples, we recover the usual closed and open spin chains, we treat the alternating spin chains and the closed spin chain with impurity.