Universal Spin Screening Clouds in Local Moment Phases

When a magnetic impurity is surrounded by conduction electrons, the impurity spin is screened via the Kondo effect. However, if the density of states (DOS) of electrons exhibits a gap or divergence at the Fermi level, the screening can become imperfect, leading to the emergence of a local moment (LM) phase. Conventionally, the LM phase is understood as lacking screening. In contrast to this common belief, we show that the impurity spin is still screened by the conduction electrons. Also, the electrons screening the spin are spatially distributed, which forms a spin cloud. To show that, we calculate quantum entanglement between the impurity spin and conduction electrons, by using numerical and analytical methods. For pseudogap and diverging DOS, we employ the numerical renormalization group method and show that the decay of this spin cloud follows an algebraic behavior. For DOS with a hard gap, we use the density matrix renormalization group method, and the spin cloud decays exponentially. In each case, we find that the spatial distribution of the spin cloud follows a single universal function of a rescaled distance by a characteristic length of the system.