arXiv:2602.01697 · Communications in Theoretical Physics

Chiral Integrable Boundary States of ABJM Spin Chain from Reflection Equations

Yang Liu, Nan Bai, M Shao, Jun-Bao Wu

Published Sep 30, 2026• 0 citations

Abstract

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.

DOI: 10.1088/1572-9494/aeae10

Related papers

A general fusion procedure for open $\mathfrak{gl}(N)$ spin chains: Application to the ABJM spin chain

Nan Bai

arXiv:2507.19394Jul 25, 20250 citations

Exact overlaps for all integrable two-site boundary states of gl(N) symmetric spin chains

T. Gombor

Jan 1, 20241 citation

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

Nan Bai, Hui Yang, M Shao

arXiv:2607.04203Jul 5, 20260 citations

Exact overlaps for all integrable two-site boundary states of $$ \mathfrak{gl} $$(N) symmetric spin chains

Tamás Gombor

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.

arXiv:2311.04870Journal of High Energy PhysicsMay 16, 202413 citations

Overlaps for matrix product states of arbitrary bond dimension in ABJM theory

Tamás Gombor, Charlotte Kristjansen

We find a closed formula for the overlap of Bethe eigenstates of an alternating SU(4) spin chain, describing the scalar sector of ABJM theory, and matrix product states of any bond dimension representing 1/2 BPS co-dimension one domain walls in the field theory. One point functions of the defect CFTs involved, being directly expressible in terms of these overlaps, are hence completely determined.

arXiv:2207.06866Physics Letters BSep 1, 202219 citations

Exact Overlaps for All Integrable Matrix Product States of Rational Spin Chains

Tamás Gombor

The overlaps between integrable matrix product states (MPSs) and Bethe states are important in both the nonequilibrium statistical physics and the AdS/CFT duality. We present the general MPS overlap formula. The result is a product of a ratio of Gaudin determinants and a prefactor. The Gaudin determinants depend on the spin chain but not on the MPS. The MPS dependent prefactor is given for all integrable MPSs of the gl_{N}, o_{N}, and sp_{N} symmetric spin chains with arbitrary representations.

arXiv:2410.23282Physical Review LettersSep 19, 20255 citations

arXiv:2602.01697 · Communications in Theoretical Physics

Chiral Integrable Boundary States of ABJM Spin Chain from Reflection Equations

Yang Liu, Nan Bai, M Shao, Jun-Bao Wu

Published Sep 30, 2026• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1088/1572-9494/aeae10

Related papers

A general fusion procedure for open $\mathfrak{gl}(N)$ spin chains: Application to the ABJM spin chainA general fusion procedure for open $\mathfrak{gl}(N)$ spin chains: Application to the ABJM spin chain

Nan Bai

We formulate a general fusion procedure for open $\mathfrak{gl}(N)$ spin chains. We construct the fused boundary reflection matrices and the corresponding fused reflection equations. By using the intertwining relation between the fused reflection matrices and the fusion operator, we identify the invariant subspace of the fused reflection matrices carrying the irreducible representations of $\mathfrak{gl}(N)$ . We also construct the fused transfer matrix and evaluate it explicitly in the total tensor product space and the invariant subspaces. Finally, we demonstrate that the ABJM spin chain model originates from such fusion procedure and derive three classes of boundary reflection matrices solutions on the anti-fundamental representation space of $\mathfrak{su}(4)$ .We formulate a general fusion procedure for open $\mathfrak{gl}(N)$ spin chains. We construct the fused boundary reflection matrices and the corresponding fused reflection equations. By using the intertwining relation between the fused reflection matrices and the fusion operator, we identify the invariant subspace of the fused reflection matrices carrying the irreducible representations of $\mathfrak{gl}(N)$. We also construct the fused transfer matrix and evaluate it explicitly in the total tensor product space and the invariant subspaces. Finally, we demonstrate that the ABJM spin chain model originates from such fusion procedure and derive three classes of boundary reflection matrices solutions on the anti-fundamental representation space of $\mathfrak{su}(4)$.

arXiv:2507.19394Jul 25, 20250 citations

Exact overlaps for all integrable two-site boundary states of gl(N) symmetric spin chains

T. Gombor

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.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.

Jan 1, 20241 citation

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

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.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.

arXiv:2607.04203Jul 5, 20260 citations

Exact overlaps for all integrable two-site boundary states of $\mathfrak{gl}$ (N) symmetric spin chainsExact overlaps for all integrable two-site boundary states of $$ \mathfrak{gl} $$(N) symmetric spin chains

Tamás Gombor

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.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.

arXiv:2311.04870Journal of High Energy PhysicsMay 16, 202413 citations

Overlaps for matrix product states of arbitrary bond dimension in ABJM theory

Tamás Gombor, Charlotte Kristjansen

arXiv:2207.06866Physics Letters BSep 1, 202219 citations

Exact Overlaps for All Integrable Matrix Product States of Rational Spin Chains

Tamás Gombor

arXiv:2410.23282Physical Review LettersSep 19, 20255 citations

Abstract We develop a general framework for constructing

2n

A general fusion procedure for open

\mathfrak{gl}(N)

spin chains: Application to the ABJM spin chain

We formulate a general fusion procedure for open

\mathfrak{gl}(N)

spin chains. We construct the fused boundary reflection matrices and the corresponding fused reflection equations. By using the intertwining relation between the fused reflection matrices and the fusion operator, we identify the invariant subspace of the fused reflection matrices carrying the irreducible representations of

\mathfrak{gl}(N)

. We also construct the fused transfer matrix and evaluate it explicitly in the total tensor product space and the invariant subspaces. Finally, we demonstrate that the ABJM spin chain model originates from such fusion procedure and derive three classes of boundary reflection matrices solutions on the anti-fundamental representation space of

\mathfrak{su}(4)

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.

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.

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.