Abstract
One-point functions of certain non-protected scalar operators in the defect CFT dual to the D3-D5 probe brane system with k units of world volume flux can be expressed as overlaps between Bethe eigenstates of the Heisenberg spin chain and a matrix product state. We present a closed expression of determinant form for these one-point functions, valid for any value of k. The determinant formula factorizes into the k=2 result times a k-dependent prefactor. Making use of the transfer matrix of the Heisenberg spin chain we recursively relate the matrix product state for higher even and odd k to the matrix product state for k=2 and k=3 respectively. We furthermore find evidence that the matrix product states for k=2 and k=3 are related via a ratio of Baxter's Q-operators. The general k formula has an interesting thermodynamical limit involving a non-trivial scaling of k, which indicates that the match between string and field theory one-point functions found for chiral primaries might be tested for non-protected operators as well. We revisit the string computation for chiral primaries and discuss how it can be extended to non-protected operators.
Highlights
The non-vanishing flux on the D5-brane represents k D3 banes dissolved in its worldvolume, while in the field-theory language the symmetry-breaking is described by a non-zero vacuum expectation value of scalar fields, that form a k-dimensional unitary representation of su(2) [12,13,14]
One-point functions of certain non-protected scalar operators in the defect CFT dual to the D3-D5 probe brane system with k units of world volume flux can be expressed as overlaps between Bethe eigenstates of the Heisenberg spin chain and a matrix product state
The general k formula has an interesting thermodynamical limit involving a non-trivial scaling of k, which indicates that the match between string and field theory one-point functions found for chiral primaries might be tested for non-protected operators as well
Summary
AdS/CFT set-ups relating probe brane systems with fluxes to defect conformal field theories allow for non-trivial one-point functions. Aiming only at tree-level one-point functions it suffices to know the conformal operators of the theory to one-loop order It is well-known that the conformal operators in the SU(2) subsector of N = 4 SYM at one-loop order can be identified with the zero-momentum Bethe eigenstates of the XXX1/2 Heisenberg spin chain upon mapping each Z-field to a spin up and each W -field to a spin down [16]. The second term in (3.2) does not contribute to the inner product between the matrix product state and the Bethe state In this way the result for the one-point function corresponding to a Bethe eigenstate with half filling followed from the overlap formula for the Neel state derived in [20], see [21, 22].
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