Quantum Integrability of Hamiltonians with Time-Dependent Interaction Strengths and the Renormalization Group Flow

In this paper we consider quantum Hamiltonians with time-dependent interaction strengths, and following the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that constraints imposed by integrability take the same form as the renormalization group flow equations corresponding to the respective Hamiltonians with constant interaction strengths. As a concrete example, we consider the anisotropic time-dependent Kondo model characterized by the time-dependent interaction strengths $J_{\parallel}(t)$ and $J_{\perp}(t)$. We construct an exact solution to the time-dependent Schrodinger equation and by applying appropriate boundary conditions on the fermion fields we obtain a set of matrix difference equations called the quantum Knizhnik-Zamolodchikov (qKZ) equations corresponding to the XXZ R-matrix. The consistency of these equations imposes constraints on the time-dependent interaction strengths $J_{\parallel}(t)$ and $J_{\perp}(t)$, such that the system is integrable. Remarkably, the resulting temporal trajectories of the couplings are shown to coincide exactly with the RG flow trajectories of the static Kondo model, establishing a direct and universal correspondence between integrability and renormalization-group flow in time-dependent quantum systems.