Parity Symmetry Protected Spin Entanglement in the Two Impurity Model

We consider the Two Impurity Kondo Model where two spin ½ impurity spins, separated by a distance R, are coupled via a Heisenberg interaction to the electron spin densities of a conduction band.
It has been recently shown that for a 1D inversion symmetric dispersion and the specific distances k_FR=nπ, with k_F being the Fermi momentum, conduction electron states with even (n={1,3,…}) or odd (n={0,2,…}) parity symmetry decouple from the impurities at low temperatures.
In this work, we study the non-equilibrium dynamics of the impurity spin correlation function 〈 S₁S₂ 〉(R,t) after different quenches. Starting from initially decoupled impurity spins, the build up of this spatial correlation function is investigated and it is shown that spatial and temporal correlations propagate along a light-cone determined by the Fermi-velocity. We demonstrate that for the specific distances k_FR=nπ the decoupling of conduction band states at low energy scales locks the impurity spin correlation function to a fixed value and, therefore, prevents it from decaying to its equilibrium value for long times.