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.
In this paper, we study the entanglement between two-neighboring sites and the rest of the system in a simple quantum phase transition of 1D transverse field Ising model. We find that the entanglement shows interesting scaling and singular behavior around the critical point, and then can be use as a convenient marker for the transition point.PACS numbers: 03.67. Mn, 03.65.Ud, 05.70.Jk, 75.10.Jm Entanglement is a one of the most fascinating features [1] of the quantum theory. It roots in the superposition principle of quantum mechanics and the tensor product structure of different Hilbert spaces, and implies the existence nonlocal correlation in quantum world. Since this correlation is absent in the classical world, it is a kind of pure quantum correlation and regarded as crucial resource in many quantum information processing tasks [2,3]. Actually, similar fascinating property also exists in the classic optics. By recording the interference pattern of two laser beams reflected from one object, the whole image can be recovered from even a small piece of holograph, although the resolution may be reduced. In other words, the superposition of two coherent electromagnetic waves enables us to learn some global information from a localized spatial area. Though it is not a precise analogy, people expect that the entanglement, which roots in the same superposition principle, can enable us to learn some global properties from a small part of the system. This observation may be one of the main motivations in the recent studies [4,5,6,7,8,9,10,11,12,13] on the role of entanglement between a small part, e.g. a block consisting of one or more sites, and the rest of the system in the quantum phase transition [14].Conceptually, the single-site and the two-site entanglement, and the block-block entanglement [11,12,15] are different notations of the same kind of measurement. They are defined as the von Neumann entropy of the reduced density matrix of a local part in the system. Studies of the block-block entanglement in the onedimensional spin models have established a close relationship between the conformal field theory and quantum information theory [12]. The block-block entanglement is also generalized in systems other than spin models. Many works are devoted to understand the local entanglement, a limiting case of the block-block entanglement, in the ground state of itinerant fermion systems. For example, in the extended Hubbard model, the global phase diagram can be sketched out by the contour map of the single-site local entanglement in the parameter space * Email: sjgu@phy.cuhk.edu.hk URL: http://www.phystar.net/ [4]. And the entanglement's scaling behavior is investigated in the one-dimensional Hubbard model at criticality point [6]. In the Hirsch model, the singularity in the first derivative of the single-site entanglement is used to locate the superconductor-insulator transition point [7]. However, in some spin systems with Z2 symmetry, the single-site entanglement usually does not contain enough information to des...
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