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Journal of Physics A Mathematical and General

Entanglement of periodic anisotropicXYchains

Lifa Zhang, Peiqing Tong

Published Aug 3, 2005• 12 citations
Publisher

Abstract

By using the concept of concurrence, the entanglement of periodic anisotropic XY chains in a transverse field is studied numerically. It is found that the derivatives ∂ λ C (1) of nearest-neighbour concurrence diverge at quantum critical points. By proper scaling, we found that all the derivatives ∂ λ C (1) for periodic XY chains in the vicinity of quantum critical points have the same behaviours as that of a uniform chain.

DOI
10.1088/0305-4470/38/33/011

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By using the concept of concurrence, the entanglement of periodic anisotropic XY chains in a transverse field is studied numerically. It is found that the derivatives ∂ λ C (1) of nearest-neighbour concurrence diverge at quantum critical points. By proper scaling, we found that all the derivatives ∂ λ C (1) for periodic XY chains in the vicinity of quantum critical points have the same behaviours as that of a uniform chain.
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The nearest-neighbor concurrence and block–block entanglement of a period-two XX chain with a transverse field h and a uniform long-range interaction I in the z -direction are studied numerically. It is found that when I > 0, the nearest-neighbor concurrence and block–block entanglement are both discontinuous at the critical points, where the system undergoes first-order quantum phase transitions. When the long-range interaction is weak , there are two critical points and . Between these two points there is a critical region, where the block–block entanglement . In one noncritical region is equal to a non-zero constant while in the other noncritical region is always zero. Furthermore, the effect of periodicity on the concurrence and block–block entanglement is discussed.
The well-known anisotropic XY spin chains in a transverse magnetic field are studied in the context of bipartite entanglement and quantum discord. By using Jordan–Wigner transformation, the model mapped onto a noninteracting spinless fermion model then, by accessing the energy spectrum we have found analytical formulas for concurrence and quantum discord based on reduced density matrix. We used the approach to go beyond the trivial nearest neighbours information, namely the second and third nearest neighbours. It is shown that far distance derivatives of quantum discord are able to capture XY-critical line and furthermore a signature of factorization points is observed.
In this paper the entanglement and quantum phase transition of the anisotropic spin-1/21/21/2 XY\mathit{XY}XY model are studied by using the quantum renormalization-group method. By solving the renormalization equations, we get the trivial and nontrivial fixed points, which correspond to the phase of the system and the critical point, respectively. The concurrence between two blocks are calculated and it is found that when the number of iterations of the renormalization tends to infinity, the concurrence develops two saturated values that are associated with two different phases, i.e., Ising-like and spin-fluid phases. We also investigate the first derivative of the concurrence and find that there exists nonanalytic behaviors at the quantum critical point, which are directly associated with the divergence of the correlation length. To gain further insight, the scaling behaviors of the system are analyzed and it is shown that the maximum value of the first derivative of the concurrence reaches infinity and the critical point is approached as the size of the system increases.
We have applied our recent approach [M. Kargarian et al., Phys. Rev. A 76, 060304(R) (2007)] to study the quantum-information properties of the anisotropic s=1/2s=1/2s=1/2 Heisenberg chain. We have investigated the underlying quantum-information properties such as the evolution of concurrence, entanglement entropy, nonanalytic behaviors, and the scaling close to the quantum critical point of the model. Both the concurrence and the entanglement entropy develop two saturated values after enough iterations of the renormalization of coupling constants. These values are associated with the two different phases, i.e., N\'eel and spin liquid phases. The nonanalytic behavior comes from the divergence of the first derivative of both measures of entanglement as the size of the system becomes large. The renormalization scheme demonstrates how the minimum value of the first derivative and its position scales with an exponent of the system size. It is shown that this exponent is directly related to the critical properties of the model, i.e., the exponent governing the divergence of the correlation length close to the quantum critical point. We also use a renormalization method based on the quantum group concept in order to get more insight about the critical properties of the model and the renormalization of entanglement.