One-point functions in AdS/dCFT from matrix product states

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 pre-factor. 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.

Circular Wilson loops in defect $$ \mathcal{N} $$ = 4 SYM: phase transitions, double-scaling limits and OPE expansions

“…(4.20). We see that our result non-trivially matches with the one-point functions derived in [10][11][12] if…”

“…There are a number of different directions that can be explored in order to improve the present investigations. First of all one should compute independently the Wilson coefficient for the Konishi in the circular BPS Wilson loop OPE: this would represent a non-trivial check of the result obtained for one-point functions in [10][11][12] or would enlighten potential subtleties in our OPE description.…”

Spin chain overlaps and the twisted Yangian

“…[1][2][3][4][5][6][7][8]. In holography the typical state of interest is a so-called matrix product state which encodes information about the vacuum of the holographic system and the overlaps give the expectation value of the theory's operators in the vacuum state [9][10][11][12][13][14].…”

“…One class of such quantities are the overlaps with the Bethe eigenstates of the system. Based on the experience from a number of concrete studies of overlaps [9][10][11][12][22][23][24][25] a proposal for an integrability criterion was put forward in [5]. An initial state or matrix product state was said to be integrable if it was annihilated by all the conserved charges of the integrable system, which were odd under space-time parity.…”

“…The dCFT versions of N = 4 SYM theory with non-vanishing vevs and their dual string theory configurations. The discussion of the integrability properties of the corresponding matrix product states can be found in [13,14] and the closed expression for the overlap formula for the D3-D5 case appears in [9][10][11]13] for tree level and in [26] for one-loop, see also [27,28].…”