Multipartite entanglement from matrix-product states and a quantum phase transition

We investigate multipartite entanglement and a quantum phase transition for spin chain systems by making use of the density-matrix renormalization-group method in the form of matrix-product states. It is found that multipartite entanglement changes dramatically near the critical region and the residual-to-global entanglement ratio reaches its maximum at the critical point of the quantum phase transition. These results indicate that multipartite entanglement as well as the residual-to-global entanglement ratio can serve as good indicators to detect a quantum phase transition. We also demonstrate a linear relation between the next-nearest-neighbor interaction strength and the critical magnetic field for an $XYZ$ spin chain by using multipartite entanglement and quantum coherence.

Figure 1

Quantum-critical behavior of $x$ (blue solid line) and $z$ (black dashed line) components of magnetization for the $XY$ spin chain with transverse magnetic field with the anisotropy parameter $\gamma =0.15$ and the chain length $N=100$. The $x$-direction magnetization component $M^x$ vanishes for $\lambda <{\lambda }_c$ and has finite value for $\lambda >h_c$ while $M^z$ is always different from zero.

Figure 2

The residual entanglement $\tau$ as a function of the exchange coupling $\lambda$ for the $XY$ spin chain with transverse magnetic field for three different values of the anisotropy parameter $\gamma$ ($\gamma =0.1$, orange solid line; $\gamma =0.15$, black dot-dashed line; $\gamma =0.2$, blue dashed line). The residual entanglement $\tau$ has a sharp drop near the quantum critical point ${\lambda }_c=1$ for all three curves.

Figure 3

The residual-to-global entanglement ratio $R=\tau /{\tau }_1$ as a function of the exchange coupling $\lambda$ ($\gamma =0.1$, orange solid line; $\gamma =0.15$, black dot-dashed line; $\gamma =0.2$, blue dashed line). The value of $R$ indicates the amount of entanglement stores in the residual entanglement for the $XY$ spin chain with transverse magnetic field. All three curves reach their maximum at $\lambda =1.00$, which is the quantum critical point of this system.

Figure 4

Quantum-critical behavior of $x$ (red solid line) and $z$ (black dashed line) components of magnetization for the $XYZ$ spin chain with next-nearest-neighbor interaction strength $J_2=0.2$ and the chain length $N=100$. The $x$ component of magnetization $M^x$ vanishes for $h>h_c$ and has finite value for $h<h_c$ while $M^z$ is always different from zero.

Figure 5

The residual entanglement $\tau$ as a function of the reduced magnetic field $h$ for the $XYZ$ spin chain with three different values of the next-nearest-neighbor interaction strength $J_2$ ($J_2=0.1$, green solid line; $J_2=0.2$, black dot-dashed line; $J_2=0.3$, red dashed line). The residual entanglement $\tau$ has a sharp drop and recovery near the critical field $h_c$ of each corresponding $J_2$.

Figure 6

The residual-to-global entanglement ratio $R=\tau /{\tau }_1$ as a function of the reduced magnetic field $h$ ($J_2=0.1$, green solid line; $J_2=0.2$, black dot-dashed line; $J_2=0.3$, red dashed line). The value of $R$ indicates the amount of entanglement stores in the residual entanglement for the $XYZ$ spin chain with next-nearest-neighbor interaction. All three curves emphasize that residual entanglement plays a more important role than bipartite entanglement at the critical region.

Figure 7

The residual-to-global entanglement ratio $R=\tau /{\tau }_1$ for the $XYZ$ spin chain with next-nearest-neighbor interaction plotted in the parameter space $(J_2,h)$. The peak of $R$ implies a linear relation between the next-nearest-neighbor interaction strength $J_2$ and the critical magnetic field $h_c$.

Figure 8

The linear relation between the next-nearest-neighbor interaction strength $J_2$ and the critical magnetic field $h_c$. Red rectangles are data points; the black solid line is $h_c=1.6031+1.6586J_2$.

Figure 9

The relative entropy of coherence $C_{RE}$ for the $XYZ$ spin chain with next-nearest-neighbor interaction plotted in the parameter space $(J_2,h)$. The value of $C_{RE}$ has two patterns: finite value for the Néel-ordered phase and approximately zero for the disordered phase. The black dashed line is $h_c=1.6031+1.6586J_2$.