Mean Field Decoupling of Single Impurity Anderson Model through Auxiliary Majorana Fermions

We present a method to study the time evolution of the single impurity Anderson model which exploits a mean field decoupling of the interacting impurity and the non-interacting bath (in form of a chain). This is achieved by the introduction of a pair of auxiliary Majorana fermions between the impurity and the chain. After decoupling, we obtain a self-consistent set of equations for the impurity and chain. First, we study the behavior of the system in equilibrium at zero temperature. We obtain a phase transition as a function of the interaction at the impurity and the coupling between the impurity and the chain between the Kondo regime, where the mean field parameters are zero and, hence, we have a well-defined spin at the impurity, to a phase where mean field parameters acquire finite values leading to a screening of the impurity spin by conduction bath electrons. In the latter case, we observe charge and spin fluctuations at the impurity site. Starting from this equilibrium ground state at zero temperature we quench in the interaction strength at the impurity and/or the hybridization strength between the impurity and the chain and study the time evolution of the system. We find that for quenches to weak to intermediate coupling the system converges to the equilibrium state defined by the final set of parameters after the quench. We analyze the oscillation frequency and as well as the thermalization rate during this quench. A quench to a strong interaction value results in persistent oscillations and a trapping of the system in a non-thermal state. We speculate that these two regimes of different long-time behavior are separated by a dynamical phase transition.