Abstract
We construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.
Highlights
Integrable spin chains are prominent examples of quantum integrable models
An obstacle that immediately arises is that the construction of the transfer matrix as presented in [4] assumes certain symmetries of the R-matrix and most of them are absent in the Lax operators used for the Qoperators construction [13]
A priori it is not clear which boundary Yang–Baxter equations and unitarity relations have to be satisfied in order to build members of the commuting family of operators from the two bulk Lax matrices used in [13]
Summary
Integrable spin chains are prominent examples of quantum integrable models. They bear a close relation to two-dimensional integrable quantum field theories as well as lattice models and rest on surprisingly rich mathematical structures. Formulated for closed systems, the quantum inverse scattering method was further extended to integrable models with open boundaries by Sklyanin [4]. This allowed the transfer matrix to be constructed for the open XXX Heisenberg spin chain. Baxter Q-operators are of distinguished importance in the theory of integrable systems They contain the information about the eigenfunctions as well as the Bethe roots. The results obtained for the spectral problem of planar N = 4 super Yang–Mills theory, see [10] and references therein, suggest that Q-operators play a central role in the study of the integrable structures appearing in the context of the AdS/CFT correspondence, see e.g.
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