Integrability of the Kondo model with time-dependent interaction strength

In this Letter we consider the time-dependent Kondo model where a magnetic impurity interacts with the electrons through a time-dependent interaction strength $J(t)$. We develop an alternative framework based on the Bethe ansatz and construct an exact solution to the time-dependent Schr"odinger equation. We show that when periodic boundary conditions are applied, the consistency of the solution results in a constraint equation which relates the amplitudes corresponding to a certain ordering of the particles in the configuration space. This constraint equation takes the form of a matrix difference equation, and the associated consistency conditions restrict the interaction strength $J(t)$ for the system to be integrable. For a given $J(t)$ satisfying these constraints, the solution to the matrix difference equations provides the exact many-body wave function that satisfies the time-dependent Schr"odinger equation. We provide a concrete example of $J(t)$ which satisfies these constraint equations. We show that in this case, the matrix difference equations turn into quantum Knizhnik-Zamolodchikov (qKZ) equations, which are well studied in the literature. The framework developed in this Letter allows one to probe the nonequilibrium physics of the Kondo model, and being general, it also allows one to solve a new class of Hamiltonians with time-dependent interaction strengths which are based on quantum Yang-Baxter algebra.