On exact overlaps of integrable matrix product states: inhomogeneities, twists and dressing formulas

TL;DR

The paper develops a twisted Yangian-based framework to compute exact overlaps between integrable matrix product states and Bethe eigenstates in the SO(6) spin chain, incorporating inhomogeneities and twists. It introduces a quantum dressing procedure that, starting from a trivial boundary representation, generates K-matrices corresponding to evaluation representations of the twisted Yangian $Y^+(4)$, yielding universal formulas for overlaps in terms of fused transfer matrix eigenvalues. The authors derive two generalized dressing formulas, one for integer and one for half-integer spins, enabling closed expressions for the MPS overlaps relevant to the D3-D5 domain wall in ${ m N}=4$ SYM and filling a gap in the analytical understanding of the D3-D5 overlap in the SO(6) sector. The results are universal, extendable to any quantum space and twist, and provide a path toward analytic tree-level one-point functions in holographic defect setups.

Abstract

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 ${\cal N}=4$ 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.

Figures (4)

• Figure 1: A graphical representation of the monodromy matrices and the RTT-relations. The red and blue lines correspond to the auxiliary space ($i,j=1,\dots 4$). The black lines are the six-dimensional representations. The intersection of two red lines represents the R-matrix, $R(u)$, and the intersection of a red and a blue line the crossed R-matrix, $\bar{R}(u)$.
• Figure 2: The graphical presentation of the $K$-matrix and the boundary state. The red and blue lines correspond to the auxiliary space ($i,j=1,\dots 4$). The black lines are the six-dimensional representations ($a,b=1,\dots 6$). The boundary space is denoted by orange lines ($\alpha,\beta =1,\dots, d_B$).
• Figure 3: The graphical presentation of the KT-relation. The red and blue lines correspond to the auxiliary space. The black lines are the six-dimensional representations. The boundary space is denoted by orange lines.
• Figure 4: The graphical presentation fusion leading to the boundary state. The red lines correspond to the auxiliary space ($i,j=1,\dots 4$). The boundary space is denoted by orange lines. The intersection of two red lines denote the R-matrix $R(u)$. Here, we used the shorthand notations $u^\pm=u \pm 1/2$ for the spectral parameter dependence. The rounded rectangles are the two-site boundary operators $\psi^{(0)}_{i,j}(u)=\mathbf{K}_{5-j,i}(u)$ for the defining representation of $\mathfrak{gl}_4$ .