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arXiv:1802.01598 · Physics Letters B

Scalar one-point functions and matrix product states of AdS/dCFT

Marius de Leeuw, Charlotte Kristjansen, Γεώργιος Λιναρδόπουλος

Published Apr 4, 2018• 90 citations
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Abstract

We determine in a closed form all scalar one-point functions of the defect CFT dual to the D3–D5 probe brane system with k units of flux which amounts to calculating the overlap between a Bethe eigenstate of the integrable SO(6) spin chain and a certain matrix product state of bond dimension k. In particular, we show that the matrix product state is annihilated by all the parity odd charges of the spin chain which has recently been suggested as the criterion for such a state to correspond to an integrable initial state. Finally, we discuss the properties of the analogous matrix product state for the SO(5) symmetric D3–D7 probe brane set-up.

DOI
10.1016/j.physletb.2018.03.083

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We determine in a closed form all scalar one-point functions of the defect CFT dual to the D3–D5 probe brane system with k units of flux which amounts to calculating the overlap between a Bethe eigenstate of the integrable SO(6) spin chain and a certain matrix product state of bond dimension k. In particular, we show that the matrix product state is annihilated by all the parity odd charges of the spin chain which has recently been suggested as the criterion for such a state to correspond to an integrable initial state. Finally, we discuss the properties of the analogous matrix product state for the SO(5) symmetric D3–D7 probe brane set-up.
We study the matrix product state which appears as the boundary state of the AdS/dCFT set-up where a probe D7 brane wraps two two-spheres stabilized by fluxes. The matrix product state plays a dual role, on one hand acting as a tool for computing one-point functions in a domain wall version of N=4 SYM and on the other hand acting as the initial state in the study of quantum quenches of the Heisenberg spin chain. We derive a number of selection rules for the overlaps between the matrix product state and the eigenstates of the Heisenberg spin chain and in particular demonstrate that the matrix product state does not fulfil a recently proposed integrability criterion. Accordingly, we find that the overlaps can not be expressed in the usual factorized determinant form. Nevertheless, we derive some exact results for one-point functions of simple operators and present a closed formula for one-point functions of more general operators in the limit of large spin-chain length.
A bstract Invoking a quantum dressing procedure as well as the representation theory of twisted Yangians we derive a number of summation formulas for the overlap between integrable matrix product states and Bethe eigenstates which involve only eigenvalues of fused transfer matrices and which are valid in the presence of inhomogeneities as well as twists. Although the method is general we specialize to the SO(6) spin chain for which integrable matrix product states corresponding to evaluation representations of the twisted Yangian Y + (4) encode the information about one-point functions of the D3-D5 domain wall version of N \mathcal{N} N N = 4 SYM. Considering the untwisted and homogeneous limit of our summation formulas we finally fill the last gap in the analytical understanding of the overlap formula for the SO(6) sector of the D3-D5 domain wall system.
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.