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.
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- 1 Budapesti Műszaki és Gazdaságtudományi Egyetem / Budapest University of Technology and Economics [BUTE]
- 2 Max-Planck-Institut für Quantenoptik / Max Planck Institute of Quantum Optics [MPQ]
- 3 Rudolf Peierls Centre for Theoretical Physics, University of Oxford
- Emberi Eroforrások Minisztériuma (through Organization: Emberi Erőforrások Minisztérium / Ministry of Human Capacities [Emmi])
- Engineering and Physical Sciences Research Council [EPSRC]
- Nemzeti Kutatási, Fejlesztési és Innovációs Hivatal / National Research, Development and Innovation Office [NKFIH]