Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

T. Gombor

Published Jan 1, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1007/jhep05%282024%29194

Related papers

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:2311.04870•Journal of High Energy Physics•May 16, 2024•13 citations

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.

Nuclear Physics B•Jul 22, 2022•22 citations

On 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.

arXiv:2110.07960•Oct 15, 2021•1 citation

Derivations for the MPS overlap formulas of rational spin chains

Tamás Gombor

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.

arXiv:2505.20234•Journal of High Energy Physics•Oct 3, 2025•1 citation

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:2207.06866•Physics Letters B•Sep 1, 2022•19 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:2410.23282•Physical Review Letters•Sep 19, 2025•5 citations
We find closed formulas for the overlaps of Bethe eigenstates of mathfrakgl(N)\\mathfrak{gl}(N)mathfrakgl(N) symmetric spin chains and integrable boundary states. We derive the general overlap formulas for mathfrakgl(M)oplusmathfrakgl(N−M)\\mathfrak{gl}(M)\\oplus\\mathfrak{gl}(N-M)mathfrakgl(M)oplusmathfrakgl(N−M) symmetric boundary states and give a well-established conjecture for the mathfraksp(N)\\mathfrak{sp}(N)mathfraksp(N) symmetric case. Combining these results with the previously derived mathfrakso(N)\\mathfrak{so}(N)mathfrakso(N) symmetric formula, now we have the overlap functions for all integrable boundary states of the mathfrakgl(N)\\mathfrak{gl}(N)mathfrakgl(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)SO(6)SO(6) and the alternating SU(4)SU(4)SU(4) spin chains which describe the scalar sectors of mathcalN=4\\mathcal{N}=4mathcalN=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 gl \mathfrak{gl} gl(N) symmetric spin chains
A bstract We find closed formulas for the overlaps of Bethe eigenstates of gl \mathfrak{gl} gl gl ( N ) symmetric spin chains and integrable boundary states. We derive the general overlap formulas for gl \mathfrak{gl} gl gl ( M ) ⊕ gl \mathfrak{gl} gl gl ( N − M ) symmetric boundary states and give a well-established conjecture for the sp \mathfrak{sp} sp sp ( N ) symmetric case. Combining these results with the previously derived so \mathfrak{so} so so ( N ) symmetric formula, now we have the overlap functions for all integrable boundary states of the gl \mathfrak{gl} 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 N \mathcal{N} N N = 4 super Yang-Mills and ABJM theories which are important application areas of our results.
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.
On exact overlaps for gl(N)\mathfrak{gl}(N)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 gl(N)\mathfrak{gl}(N)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)\mathfrak{gl}(N)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.
A bstract We derive a universal formula for the overlaps between integrable matrix product states (MPS) and Bethe eigenstates in glN {\mathfrak{gl}}_N glN​ 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 soN {\mathfrak{so}}_N soN​ so N and spN {\mathfrak{sp}}_N spN​ 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.