Quantum Entanglement in the S=1/2 Spin Ladder with Ring Exchange
  In this paper we study the concurrence and the block-block entanglement in the $S=1/2$ spin ladder with four-spin ring exchange by the exact diagonalization method of finite cluster of spins. The relationship between the global phase diagram and the ground-state entanglement is investigated. It is shown that the block-block entanglement of different block size and geometry manifests richer information of the system. We find that the extremal point of the two-site block-block entanglement on the rung locates a transition point exactly due to SU(4) symmetry at this point. The scaling behavior of the block-block entanglement is discussed. Our results suggest that the block-block entanglement can be used as a convenient marker of quantum phase transition in some complex spin systems. 
Jun-Liang Song, Shi-Jian Gu, and Hai-Qing Lin