Constructing integrable spin-boson model with open boundary conditions

Abstract We investigate the integrability of the Rabi model, which is traditionally viewed as not Yang-Baxter integrable despite its solvability. Building on efforts by Bogoliubov, Amico, Batchelor, and Zhou, who explored special limiting cases of the model, we develop a spin-boson interaction Hamiltonian under more general boundary conditions, particularly focusing on open boundary conditions with off-diagonal terms. Our approach maintains the direction of the spin in the $z$ direction, and also preserves the boson particle number operator $a^\dagger a$, marking a progression beyond previous efforts that have primarily explored reduced forms of the Rabi model from Yang-Baxter algebra.
We also address the presence of ``unwanted" quadratic boson terms $a^2$ and $a^\dagger{}^2$, which share coefficients with the boson particle number operator. Interestingly, these terms vanish when spectral parameter $u = \pm \theta_s$, simplifying the model to a limiting case of operator-valued twists, a scenario previously discussed by Batchelor and Zhou.