arXiv:2207.10598 · Journal of High Energy Physics

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

Tamás Gombor

Published Oct 14, 2022• 13 citations

Abstract

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.

DOI: 10.1007/jhep10(2022)096

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arXiv:2207.10598 · Journal of High Energy Physics

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

Tamás Gombor

Published Oct 14, 2022• 13 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1007/jhep10(2022)096

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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.

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On exact overlaps for gl(N) symmetric spin chains

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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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Abstract The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.Abstract The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

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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.

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Tamás Gombor

arXiv:2410.23282Physical Review LettersSep 19, 20255 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 chains

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

\mathfrak{gl}

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.