arXiv:2607.04203

Solving for the integrable boundary states of the ABJM spin chain from $KT$-relations

Nan Bai, Hui Yang, M Shao

Published Jul 5, 2026• 0 citations

Abstract

We study integrable boundary states of the alternating SU(4) spin chain arising in ABJM theory. Starting from the $KT$-relation, we directly solve the integrability constraints for states with $n$-site translational invariance. For odd and even $n$, these constraints are reduced respectively to state equations and operator equations for the elementary $n$-site block and the matrix $K(u)$. We analyze chiral and achiral cases for $n\leq4$. In the 1-site case we allow general operator-valued integrable pairs with an internal space, while in the remaining cases we focus on $c$-number solutions.

DOI: 10.48550/arxiv.2607.04203

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arXiv:2607.04203

Solving for the integrable boundary states of the ABJM spin chain from $KT$ -relationsSolving for the integrable boundary states of the ABJM spin chain from $KT$-relations

Nan Bai, Hui Yang, M Shao

Published Jul 5, 2026• 0 citations

arXiv HTML PDF

Abstract

We study integrable boundary states of the alternating SU(4) spin chain arising in ABJM theory. Starting from the

KT

-relation, we directly solve the integrability constraints for states with

n

-site translational invariance. For odd and even

n

, these constraints are reduced respectively to state equations and operator equations for the elementary

n

-site block and the matrix

K(u)

. We analyze chiral and achiral cases for

n\leq4

. In the 1-site case we allow general operator-valued integrable pairs with an internal space, while in the remaining cases we focus on

c

-number solutions.

DOI: 10.48550/arxiv.2607.04203

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arXiv:1812.11094SciPost PhysicsMay 27, 201949 citations

Integrable Matrix Product States from boundary integrability

Balázs Pozsgay, Piroli, L., Vernier, E.

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.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.

May 27, 201950 citations

Solving for the integrable boundary states of the ABJM spin chain from

KT

-relations

We study integrable boundary states of the alternating SU(4) spin chain arising in ABJM theory. Starting from the

KT

-relation, we directly solve the integrability constraints for states with

n

-site translational invariance. For odd and even

n

, these constraints are reduced respectively to state equations and operator equations for the elementary

n

-site block and the matrix

K(u)

. We analyze chiral and achiral cases for

n\leq4

. In the 1-site case we allow general operator-valued integrable pairs with an internal space, while in the remaining cases we focus on

c

-number solutions.

Abstract We develop a general framework for constructing

2n

-site chiral integrable matrix product states in Aharony-Bergman-Jafferis-Maldacena spin chain, based on reflection equations and the fusion procedure. For four-site chiral integrable product states, we propose their exact overlap formulas with Bethe states. We also investigate the chiral integrable subspaces numerically.

We find closed formulas for the overlaps of Bethe eigenstates of

\\mathfrak{gl}(N)

symmetric spin chains and integrable boundary states. We derive the general overlap formulas for

\\mathfrak{gl}(M)\\oplus\\mathfrak{gl}(N-M)

symmetric boundary states and give a well-established conjecture for the

\\mathfrak{sp}(N)

symmetric case. Combining these results with the previously derived

\\mathfrak{so}(N)

symmetric formula, now we have the overlap functions for all integrable boundary states of the

\\mathfrak{gl}(N)

spin chains which are built from two-site states. The calculations are independent from the representations of the quantum space therefore our formulas can be applied for the

SO(6)

and the alternating

SU(4)

spin chains which describe the scalar sectors of

\\mathcal{N}=4

super Yang-Mills and ABJM theories which are important application areas of our results.

Exact overlaps for all integrable two-site boundary states of

\mathfrak{gl}

(N) symmetric spin chains

A bstract We find closed formulas for the overlaps of Bethe eigenstates of

\mathfrak{gl}

gl ( N ) symmetric spin chains and integrable boundary states. We derive the general overlap formulas for

\mathfrak{gl}

gl ( M ) ⊕

\mathfrak{gl}

gl ( N − M ) symmetric boundary states and give a well-established conjecture for the

\mathfrak{sp}

sp ( N ) symmetric case. Combining these results with the previously derived

\mathfrak{so}

so ( N ) symmetric formula, now we have the overlap functions for all integrable boundary states of the

\mathfrak{gl}

gl ( N ) spin chains which are built from two-site states. The calculations are independent from the representations of the quantum space therefore our formulas can be applied for the SO(6) and the alternating SU(4) spin chains which describe the scalar sectors of

\mathcal{N}

N = 4 super Yang-Mills and ABJM theories which are important application areas of our results.

(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.