arXiv:2505.20234 · Journal of High Energy Physics

Derivations for the MPS overlap formulas of rational spin chains

Tamás Gombor

Published Oct 3, 2025• 1 citation

Abstract

A bstract We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in $$ {\mathfrak{gl}}_N $$ gl N symmetric spin chains. This formula expresses the normalized overlap as a product of a MPS-independent Gaudin-determinant ratio and a MPS-dependent scalar factor constructed from eigenvalues of commuting operators, defined via the K -matrix associated with the MPS. Our proof is fully representation-independent and relies solely on algebraic Bethe Ansatz techniques and the KT -relation. We also propose a generalization of the overlap formula to $$ {\mathfrak{so}}_N $$ so N and $$ {\mathfrak{sp}}_N $$ sp N spin chains, supported by algebra embeddings and low-rank isomorphisms. These results significantly broaden the class of integrable initial states for which exact overlap formulas are available, with implications for quantum quenches and defect CFTs.

DOI: 10.1007/jhep10(2025)035

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

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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.

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

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Jan 1, 20241 citation

Integrable crosscap states in $$ \mathfrak{gl} $$(N) spin chains

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Abstract We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the $$ \mathfrak{gl} $$ gl (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.

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arXiv:2505.20234 · Journal of High Energy Physics

Derivations for the MPS overlap formulas of rational spin chains

Tamás Gombor

Published Oct 3, 2025• 1 citation

arXiv HTML PDF

Abstract

A bstract We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in

{\mathfrak{gl}}_N

gl N symmetric spin chains. This formula expresses the normalized overlap as a product of a MPS-independent Gaudin-determinant ratio and a MPS-dependent scalar factor constructed from eigenvalues of commuting operators, defined via the K -matrix associated with the MPS. Our proof is fully representation-independent and relies solely on algebraic Bethe Ansatz techniques and the KT -relation. We also propose a generalization of the overlap formula to

{\mathfrak{so}}_N

so N and

{\mathfrak{sp}}_N

sp N spin chains, supported by algebra embeddings and low-rank isomorphisms. These results significantly broaden the class of integrable initial states for which exact overlap formulas are available, with implications for quantum quenches and defect CFTs.

DOI: 10.1007/jhep10(2025)035

Related papers

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

Tamás Gombor

arXiv:2410.23282Physical Review LettersSep 19, 20255 citations

On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chainsOn exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains

Tamás Gombor

We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$ -invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the $\mathfrak{gl}(N)$ symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$-invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the $\mathfrak{gl}(N)$ symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.

arXiv:2110.07960Oct 15, 20211 citation

On exact overlaps for gl(N) symmetric spin chains

Tamás Gombor

We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N)-invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the gl(N) symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N)-invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the gl(N) symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.

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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

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

Integrable crosscap states in $\mathfrak{gl}$ (N) spin chainsIntegrable crosscap states in $$ \mathfrak{gl} $$(N) spin chains

Tamás Gombor

Abstract We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the $\mathfrak{gl}$ gl (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.Abstract We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the $$ \mathfrak{gl} $$ gl (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.

arXiv:2207.10598Journal of High Energy PhysicsOct 14, 202213 citations

A bstract We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in

{\mathfrak{gl}}_N

{\mathfrak{so}}_N

so N and

{\mathfrak{sp}}_N

On exact overlaps for

\mathfrak{gl}(N)

symmetric spin chains

We study the integrable two-site states of the quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing

\mathfrak{gl}(N)

-invariant R-matrix. We investigate the overlaps between the integrable two-site states and the wave-functions. To find exact derivations for the factorized overlap formulas for the nested integrable systems is a longstanding unsolved problem. In this paper we give a derivation for a large class of the integrable states of the

\mathfrak{gl}(N)

symmetric spin chain. The first part of the derivation is to calculate recurrence relations for the off-shell overlap that uniquely fix it. Using these recursions we prove that the normalized overlaps of the multi-particle states have factorized forms which contain the products of the one-particle overlaps and the ratio of the Gaudin-like determinants. We also show that the previously proposed overlap formulas agree with our general formula.

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.

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.

Integrable crosscap states in

\mathfrak{gl}

(N) spin chains

Abstract We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the

\mathfrak{gl}

gl (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.