Singular behavior of an entangled state for a one-dimensional quantum spin system

We studied the entangled state for a one-dimensional $S=1/2$ antiferromagnetic quantum spin chain in a transverse field. We calculate the ground state using the density matrix renormalization group and discuss how the entangled state changes around a quantum phase transition (QPT) point. By analyzing concurrence $C(\rho)$ for two-qubit density matrix $\rho$ after the Lewenstein-Sanpera decomposition, $\rho=\Lambda \rho_s + (1-\Lambda) \rho_e$, where $\rho_s$ is a separable density matrix and $\rho_e$ is a pure entangled state obtained by a linear combination of Bell states, we find singular behaviors both in $C(\rho_e)$ and $1-\Lambda$ at the QPT point.　 $C(\rho_e)$ includes the effects of quantum fluctuations, which manifest the competition between the antiferromagnetic spin fluctuation and the effect of transverse field in the transverse Ising model. The quantum fluctuation shows the singular maximum at the QPT point as expected from the general picture of phase transition. In contrast, $1-\Lambda$ reveals the singular minimum at QPT point.