# Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices

## TL;DR

This paper develops a nested algebraic Bethe ansatz method to solve an open GL(N) spin chain with non-diagonal boundary conditions, demonstrating integrability and providing a complete solution for general N and M.

## Contribution

It introduces a novel approach using projected K-matrices to establish integrability and solve the model via nested algebraic Bethe ansatz for non-diagonal boundary conditions.

## Key findings

- Constructed commuting transfer matrix for the model.
- Solved the model for general N and M using nested algebraic Bethe ansatz.
- Numerical evidence supports the completeness of the solution.

## Abstract

We consider an open spin chain model with GL(N) bulk symmetry that is broken to GL(M) x GL(N-M) by the boundary, which is a generalization of a model arising in string/gauge theory. We prove the integrability of this model by constructing the corresponding commuting transfer matrix. This construction uses operator-valued "projected" K-matrices. We solve this model for general values of N and M using the nested algebraic Bethe ansatz approach, despite the fact that the K-matrices are not diagonal. The key to obtaining this solution is an identity based on a certain factorization property of the reduced K-matrices into products of R-matrices. Numerical evidence suggests that the solution is complete.