Entanglement, quantum phase transition, and scaling in the $\mathrm{XXZ}$ chain

Motivated by recent development in quantum entanglement, we study relations among concurrence C, ${\mathrm{SU}}_q\mathrm{}\left(2\right)\mathrm{}$ algebra, quantum phase transition and correlation length at the zero temperature for the $\mathrm{XXZ}$ chain. We find that at the SU(2) point, the ground state possesses the maximum concurrence. When the anisotropic parameter $\Delta$ is deformed, however, its value decreases. Its dependence on $\Delta$ scales as ${C=C}_0-C_1(\Delta -{1)}^2$ in the $\mathrm{XY}$ metallic phase and near the critical point (i.e., $1<\mathrm{}\Delta <1.3)$ of the Ising-like insulating phase. We also study the dependence of C on the correlation length $\xi ,$ and show that it satisfies ${C=C}_0-1/2\xi$ near the critical point. For different sizes of the system, we show that there exists a universal scaling function of C with respect to the correlation length $\xi .$