# Integrable Matrix Product States from boundary integrability

## TL;DR

This paper establishes a new linear intertwiner relation called the 'square root relation' for integrable Matrix Product States in spin chains, linking them to twisted Boundary Yang-Baxter equations and providing explicit solutions with applications in AdS/CFT.

## Contribution

It introduces the 'square root relation' as a novel integrability condition for MPS and constructs explicit solutions related to twisted Yangians with symmetry group pairs.

## Key findings

- The 'square root relation' leads to full Boundary Yang-Baxter equations.
- Explicit solutions are provided for symmetric pairs like (SU(N),SO(N)).
- Applications include computing one-point functions in defect AdS/CFT.

## Abstract

We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to "operator valued" solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation. We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the "square root relation", because it involves half of the steps of the reflection equation. It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations. We provide explicit solutions in a number of cases characterized by special symmetries. These correspond to the "symmetric pairs" $(SU(N),SO(N))$ and $(SO(N),SO(D)\otimes SO(N-D))$, where in each pair the first and second elements are the symmetry groups of the spin chain and the integrable state, respectively. These solutions can be considered as explicit representations of the corresponding twisted Yangians, that are new in a number of cases. Examples include certain concrete MPS relevant for the computation of one-point functions in defect AdS/CFT.