Entanglement structure of the two-channel Kondo model

Two electronic channels competing to screen a single impurity spin, as in the two-channel Kondo model, are expected to generate a ground state with a nontrivial entanglement structure. We exploit a spin-chain representation of the two-channel Kondo model to probe the ground-state block entropy, negativity, tangle, and Schmidt gap, using a density matrix renormalization group approach. In the presence of symmetric coupling to the two channels, we confirm field-theory predictions for the boundary entropy difference $ln(g_{\text{UV}}/g_{\text{IR}})=ln\left(2\right)/2$ between the ultraviolet and infrared limits and the leading $ln\left(x\right)/x$ impurity correction to the block entropy. The impurity entanglement $S_{\text{imp}}$ is shown to scale with the characteristic length ${\xi }_{2\text{CK}}$. We show that both the Schmidt gap and the entanglement of the impurity with one of the channels—as measured by the negativity—faithfully serve as order parameters for the impurity quantum phase transition appearing as a function of channel asymmetry, allowing for explicit determination of critical exponents, $\nu \approx 2$ and $\beta \approx 0.2$. Remarkably, we find the emergence of tripartite entanglement only in the vicinity of the critical channel-symmetric point.

Figure 1

(a) Kondo spin chain with a spin-1/2 impurity coupled to its left and right channels by $\mathrm{\Gamma }{J}^{\prime }$ and $J^{\prime }$, respectively. For $\mathrm{\Gamma }=1$, the impurity is screened by both channels representing the 2CK model, while for $\mathrm{\Gamma }\ne 1$, 1CK physics emerges. (b) The impurity entropy $S_{\text{imp}}$ is computed as the difference between the entropy of region $A$ with and without the impurity. (c) Partitioning of the system for computing the Schmidt gap.

Figure 2

(a) Scaling of $S_{\mathrm{imp}}^{\mathrm{sub}}[x/{\xi }_{\text{2CK}}\left(J^{\prime }\right)]$ for fixed $x/N=1/10$ ($N$ even). At $J^{\prime }=0.9, {\xi }_{\text{2CK}}\left(J^{\prime }\right)$ is arbitrarily fixed at $0.07747$ to coincide with the estimate from (b). Inset: ${\xi }_{\text{2CK}}\left(J^{\prime }\right)$ as a function of $1/J^{\prime }$. (b) DMRG results for $S_{\mathrm{imp}}^{\mathrm{sub}}(x;J^{\prime }=0.9,N=800)$. For ${\xi }_{2\text{CK}}\left(J^{\prime }\right)\ll x\ll N/2, S_{\mathrm{imp}}^{\mathrm{sub}}$ can be fit to the form $Aln(x/{\xi }_{2CK})/(x/{\xi }_{2\text{CK}})+B$ (red line) with ${\xi }_{2\text{CK}}(J^{\prime }=0.9)=0.07747, A=0.69$, and $B=0.34\sim ln\left(2\right)/2$ significantly better than to $\sim 1/x$ (green line). Inset: Convergence to the limiting form at $x\ll N/2$ with $N$.

Figure 3

(a) Negativity between the impurity and the right channel (i.e., $E_{0,R}$) vs $\mathrm{\Gamma }$ for $N=403$ and $J^{\prime }=0.4$. (b) Derivative of $E_{0,R}$ with respect to $\mathrm{\Gamma }$ for different system sizes. (c) Finite-size scaling of $E_{0,R}$. (d) Finite-size scaling of the Schmidt gap.

Figure 4

(a) The tripartite entanglement indicator $\tau$ vs $\mathrm{\Gamma }$. (b) Entanglement between the two channels vs $\mathrm{\Gamma }$. In both panels, $J^{\prime }=0.4$.