Entanglement and Criticality in Quantum Impurity Systems

We investigate the entanglement between a spin and its environment in impurity systems which exhibit a second-order quantum phase transition separating a delocalized and a localized phase for the spin. As an application, we employ the spin-boson model, describing a two-level system (spin) coupled to a sub-Ohmic bosonic bath with power-law spectral density, $\mathcal{J}(\omega )\propto {\omega }^s$ and $0<s<1$. Combining Wilson’s numerical renormalization group method and hyperscaling relations, we demonstrate that the entanglement between the spin and its environment is always enhanced at the quantum phase transition resulting in a visible cusp (maximum) in the entropy of entanglement. We formulate a correspondence between criticality and impurity entanglement entropy, and the relevance of these ideas to nanosystems is outlined.

Figure 1

Unification of entanglement and criticality at a second-order impurity quantum phase transition.

Figure 2

NRG results for the entanglement at $T=ϵ=0$ in the sub-Ohmic case $s=0.5$, which can be realized through a qubit coupled to an $RLC$ transmission line; the arrow marks the position of the quantum phase transition. The relevant susceptibilies ${\overline{\chi }}_z$ and ${\chi }_{\perp }$ are presented in insets.

Figure 3

The entropy of entanglement $E$, obtained from NRG, displays a cusplike behavior for all $0<s<1$. For $s\to 1$, $E$ becomes rapidly suppressed at ${\Delta }_c^{-}$ which is a reminiscence of the ohmic case (KT transition).

Figure 4

$E(\Delta )$ at $s=0.5$ and $\alpha =0.1$ for several values of $ϵ$. Inset: universal scaling in the critical region for three different values of ${\Delta }_c$ (from NRG we obtain $\kappa \sim 0.6$ and the exact exponent is $\kappa =2/3=0.66\dots$).

Figure 5

Exponent $\kappa$ versus $s$ from NRG. For all $0<s<1$, $\kappa =2/\delta$ and for $0<s\le 1/2$, $\kappa =2/\delta =1/\overline{\delta }$.