Quasiparticle Mass Enhancement as a Measure of Entanglement in the Kondo Problem

We analyze the quantum entanglement between opposite spin projection electrons in the ground state of the Anderson impurity model. In this model, a single level impurity with intralevel repulsion $U$ is tunnel coupled to a free electron gas. The Anderson model presents a strongly correlated many body ground state with mass enhanced quasiparticle excitations. We find, using both analytical and numerical tools, that the quantum entanglement between opposite spin projection electrons is a monotonic universal function of the quasiparticle mass enhancement $Z$ in the Kondo regime. This indicates that the interaction induced mass enhancement, which is generally used to quantify correlations in quantum many body systems, could be used as a measure of entanglement in the Kondo problem.

Figure 1

Sketch of the Anderson model, for an impurity with Kramers degeneracy, including the two partitions of the total Hilbert space $\mathcal{H}$ considered to calculate the entanglement entropy: (a) $\mathcal{H}={\mathcal{H}}_i\otimes {\mathcal{H}}_{\mathrm{bath}}$. (b) $\mathcal{H}={\mathcal{H}}_{\uparrow }\otimes {\mathcal{H}}_{\downarrow }$. The Coulomb repulsion $U$ between opposite spin electrons at the impurity and the tunnel coupling $V$ between the impurity level and the conduction electron bath, are indicated in the figure.

Figure 2

Spin entanglement entropy $S_{\uparrow }$ for a variety of the model parameters. (a) $S_{\uparrow }$ vs $U$ for an electron-hole symmetric situation $ϵ=-U/2$. The inset shows the impurity level double occupancy probability. (b) $S_{\uparrow }$ vs $ϵ$ for fixed local interaction as indicated in the figure. The entropy is symmetric under the transformation $ϵ\to -U-ϵ$ due to the electron-hole symmetry of the electron bath. The inset shows the level occupancy.

Figure 3

Spin entanglement $S_{\uparrow }$ as a function of $Z$ for a variety of model parameters. In the low $Z$ regime, the data fall into a universal curve.

Figure 4

Spin entanglement entropy $S_{\uparrow }$ as a function of the exchange coupling $J$ in the Kondo model calculated using DMRG. The solid line is a fit using the functional form expected from perturbation theory in $D/\mathcal{J}$.