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 a new framework based on Bethe ansatz and construct an exact solution to the time-dependent Schrodinger 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 wavefunction that satisfies the time-dependent Schrodinger 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 work allows one to probe the non-equilibrium physics of the Kondo model, and being general, it also allows one to solve new class of Hamiltonians with time-dependent interaction strength which are based on quantum Yang-Baxter algebra.