Macroscopic Quantum Entanglement of a Kondo Cloud at Finite Temperature

We propose a variational approach for computing the macroscopic entanglement in a many-body mixed state, based on entanglement witness operators, and compute the entanglement of formation (EoF), a mixed-state generalization of the entanglement entropy, in single- and two-channel Kondo systems at finite temperature. The thermal suppression of the EoF obeys power-law scaling at low temperature. The scaling exponent is halved from the single- to the two-channel system, which is attributed, using a bosonization method, to the non-Fermi liquid behavior of a Majorana fermion, a “half” of a complex fermion, emerging in the two-channel system. Moreover, the EoF characterizes the size and power-law tail of the Kondo screening cloud of the single-channel system.

Figure 1

(a) Single-channel Kondo (1CK) and (b) two-channel Kondo (2CK) systems. In the 1CK (2CK) system, a spin-$1/2$ impurity is antiferromagnetically coupled with the spin(s) of one (two) channel(s) of a conduction electron bath at the impurity site [23, 37]; we consider a one-dimensional semi-infinite bath, without loss of generality. (c) In the 1CK system, the Kondo cloud, a macroscopic electronic object of size ${\xi }_{1\mathrm{CK}}$, forms to screen the impurity spin at zero temperature. The cloud spin ($\uparrow$, $\downarrow$) entangles with the impurity spin ($\Uparrow ,\Downarrow$), forming the Kondo singlet of the Bell-state type $|\Uparrow \downarrow ⟩-|\Downarrow \uparrow ⟩$. The entanglement of formation ${\mathcal{E}}_F$ between the impurity at $x=0$ and the bath electrons inside distance $L$ ($x\le L$) quantifies how the macroscopic entanglement of the cloud spatially extends. ${\mathcal{E}}_F$ is reduced from the value at $L\to \infty$, if the entanglement between electrons outside $L$ (which are traced out) and the rest exists.

Figure 2

Entanglement of formation ${\mathcal{E}}_F$ between the Kondo impurity and the bath electrons inside $L$ at temperature $T$ for single-channel Kondo (1CK) and two-channel Kondo (2CK) systems. (a),(b) ${\mathcal{E}}_F$ versus $T$ at $L\to \infty$ for (a) the 1CK system and (b) the 2CK system. (c) ${\mathcal{E}}_F$ versus $L$ at $T=0$ for the 1CK system. Blue circles (green triangles) denote the lower (upper) bounds of ${\mathcal{E}}_F$. These bounds are close to each other, especially at $T<T_{1\mathrm{CK},2\mathrm{CK}}$ and $L>{\xi }_{1\mathrm{CK}}$, enough to predict scaling behavior. Insets: the universal scaling behavior of ${\mathcal{E}}_F$ versus $T/T_{1\mathrm{CK},2\mathrm{CK}}\ll 1$ or $L/{\xi }_{1\mathrm{CK}}\gg 1$, well fitted by power laws (dashed lines), see Eqs. (4) and (5). The NRG parameters used for this plot and the expression of $T_{1\mathrm{CK},2\mathrm{CK}}$ are given in Ref. [37].