Scaling of Tripartite Entanglement at Impurity Quantum Phase Transitions

The emergence of a diverging length scale in many-body systems at a quantum phase transition implies that total entanglement has to reach its maximum there. In order to fully characterize this, one has to consider multipartite entanglement as, for instance, bipartite entanglement between individual particles fails to signal this effect. However, quantification of multipartite entanglement is very hard, and detecting it may not be possible due to the lack of accessibility to all individual particles. For these reasons it will be more sensible to partition the system into relevant subsystems, each containing a few to many spins, and study entanglement between those constituents as a coarse-grain picture of multipartite entanglement between individual particles. In impurity systems, famously exemplified by two-impurity and two-channel Kondo models, it is natural to divide the system into three parts, namely, impurities and the left and right bulks. By exploiting two tripartite entanglement measures, based on negativity, we show that at impurity quantum phase transitions the tripartite entanglement diverges and shows scaling behavior. While the critical exponents are different for each tripartite entanglement measure, they both provide very similar critical exponents for the two-impurity and the two-channel Kondo models, suggesting that they belong to the same universality class.

Figure 1

Schematics of the impurity systems. (a) The 2IKM in which the impurities are coupled to their bulks through impurity coupling $J^{\prime }$ and interact with each other via RKKY coupling $K$. (b) The 2CKM in which a single impurity is coupled to two bulks via couplings $J^{\prime }$ and $\mathrm{\Gamma }{J}^{\prime }$. In both (a) and (b) the system is partitioned into three blocks for studying tripartite entanglement.

Figure 2

Tripartite entanglement in 2IKM. (a) Tripartite entanglement $E_1^{2\mathrm{IKM}}$ versus $K$ in a chain with $J^{\prime }=0.4$. (b) Tripartite entanglement $E_2^{2\mathrm{IKM}}$ versus $K$ in a chain with $J^{\prime }=0.4$. (c) Scaling of $E_1^{2\mathrm{IKM}}(K_c)$ in terms of $N^{0.19}$. (d) Scaling of $E_2^{2\mathrm{IKM}}(K_c)$ in terms of $N^{0.46}$.

Figure 3

Tripartite entanglement in 2CKM. (a) Tripartite entanglement $E_1^{2\mathrm{CKM}}$ versus $\mathrm{\Gamma }$ in a chain with $J^{\prime }=0.4$. (b) Tripartite entanglement $E_2^{2\mathrm{CKM}}$ versus $\mathrm{\Gamma }$ in a chain with $J^{\prime }=0.4$. (c) Scaling of $E_1^{2\mathrm{CKM}}({\mathrm{\Gamma }}_c)$ in terms of $N^{0.19}$. (d) Scaling of $E_2^{2\mathrm{CKM}}({\mathrm{\Gamma }}_c)$ in terms of $N^{0.5}$.

Figure 4

Finite-size scaling in 2IKM. Data collapse of Eq. (7) for $E_1^{2\mathrm{IKM}}$ in a chain with (a) $J^{\prime }=0.4$ and (b) $J^{\prime }=0.5$. Similar data collapse for $E_2^{2\mathrm{IKM}}$ in a chain with (c) $J^{\prime }=0.4$ and (d) $J^{\prime }=0.5$.

Figure 5

Finite-size scaling in 2CKM. Data collapse of Eq. (7) for $E_1^{2\mathrm{CKM}}$ in a chain with (a) $J^{\prime }=0.4$ and (b) $J^{\prime }=0.5$. Similar data collapse for $E_2^{2\mathrm{CKM}}$ in a chain with (c) $J^{\prime }=0.4$ and (d) $J^{\prime }=0.5$.