Negativity as the entanglement measure to probe the Kondo regime in the spin-chain Kondo model

We study the entanglement of an impurity at one end of a spin chain with a block of spins using negativity as a true measure of entanglement to characterize the unique features of the gapless Kondo regime in the spin-chain Kondo model. For this spin chain in the Kondo regime we determine—with a true entanglement measure—the spatial extent of the Kondo screening cloud, we propose an ansatz for its ground state and demonstrate that the impurity spin is indeed maximally entangled with the cloud. To better evidence the peculiarities of the Kondo regime, we carry a parallel analysis of the entanglement properties of the Kondo spin-chain model in the gapped dimerized regime. Our study shows how a genuine entanglement measure stemming from quantum information theory can fully characterize also nonperturbative regimes accessible to certain condensed matter systems.

Figure 1

(Color online) (a) Kondo Spin chain with next nearest neighbor Heisenberg interaction with one impurity at one end. (b) The chain is divided into three parts, an impurity, a block $A$ and a block $B$. Entanglement is computed between the impurity and block $B$.

Figure 2

(Color online) (a) DMRG representation of the state of the chain keeps two intermediate spins in ordinary computational basis and the left and right blocks in a truncated DMRG basis. (b) The intermediate spin next to the impurity is traced out from the density matrix of the chain. This tracing is equivalent to adding the traced out spin to block $A$. (c) The basis of the right block of DMRG representation is transformed so that a single spin in the left side of the right block is represented in the computational basis while the state of the new right block is given in a DMRG basis.

Figure 3

(Color online) (a) $L^{\ast }$ vs $1/\sqrt{J^{\prime }}$ for both Kondo $\left(J_2=0\right)$ and dimer regime $\left(J_2=0.42\right)$. (b) Entanglement vs $L/N$ for fixed $N/L^{\ast }=4$ when $J_2=0$. (c) Entanglement vs $L/N$ for fixed $N/L^{\ast }=4$ at the critical point $J_2=J_2^c$. (d) Entanglement vs $L/N$ for fixed $N/L^{\ast }=4$ in the dimer regime $\left(J_2=0.42\right)$.

Figure 4

(Color online) (a) Entanglement vs $L$ for the two fixed points $J^{\prime }\to 0$ and $J^{\prime }\to 1$ in the Kondo regime $J_2=0$. (b) Exponential decay of entanglement in terms of $L$ in a chain of length $N=250$ for $J^{\prime }=0.6$. (c) Nonexponential decay for small $J^{\prime }$ in the dimer regime $J_2=0.42$.