On factorized overlaps: Algebraic Bethe Ansatz, twists, and Separation of Variables

Abstract

We investigate the exact overlaps between eigenstates of integrable spin chains and a special class of states called "integrable initial/final states". These states satisfy a special integrability constraint, and they are closely related to integrable boundary conditions. We derive new algebraic relations for the integrable states, which lead to a set of recursion relations for the exact overlaps. We solve these recursion relations and thus we derive new overlap formulas, valid in the XXX Heisenberg chain and its integrable higher spin generalizations. Afterwards we generalize the integrability condition to twisted boundary conditions, and derive the corresponding exact overlaps. Finally, we embed the integrable states into the "Separation of Variables" framework, and derive an alternative representation for the exact overlaps of the XXX chain. Our derivations and proofs are rigorous, and they can form the basis of future investigations involving more complicated models such as nested or long-range deformed systems.

arXiv comment34 pages, 3 figures; v2: references added, minor modifications, v3: minor modification

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• Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT
• Integrable domain walls in ABJM theory
• Crosscap States in Integrable Field Theories and Spin Chains
• On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains
• Integrable quenches in the Hubbard model
• Duality Relations for Overlaps of Integrable Boundary States in AdS/dCFT
• Form-factors and complete basis of observables via separation of variables for higher rank spin chains
• Integrable Crosscap States: From Spin Chains to 1D Bose Gas
• Integrable crosscaps in classical sigma models
• Three-Point Functions in ABJM and Bethe Ansatz
• SO(n) Affleck-Kennedy-Lieb-Tasaki states as conformal boundary states of integrable SU(n) spin chains
• Integrability and Conformal Blocks for Surface Defects in $\mathcal{N}=4$ SYM
• Chiral Integrable Boundary States in the SU(4) Alternating Spin Chain