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Integrable Matrix Product States from boundary integrability

by Balázs Pozsgay, Lorenzo Piroli, Eric Vernier

Submission summary

As Contributors: Balázs Pozsgay
Arxiv Link: https://arxiv.org/abs/1812.11094v4
Date accepted: 2019-05-17
Date submitted: 2019-05-14
Submitted by: Pozsgay, Balázs
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Mathematical Physics

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.

Current status:

Ontology / Topics

See full Ontology or Topics database.

AdS/CFT correspondence Integrable boundary conditions Matrix product states (MPS) Reflection algebra

Author comments upon resubmission

We have made the requested modifications. Instead of the tensor product notation at that specific point asked by the referee 1 we introduced the $\cdot$ notation. It is important that this is just a short-hand notation, and this is explained in the text. The ,,square root relation'' and the ,,Boundary Yang-Baxter equation'' for the objects $\omega$ and $\psi(u)$ are to be understood with all their indices spelled out, but we believe that sometimes it is useful to write down simplified notations too. We hope that the present form is clear.

List of changes

-We corrected the typos.
-We replaced the tensor product notation with $\cdot$ at the places requested.

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