Quantum Entanglement And Phase Transition On Single Ion Anisotropic Heisenberg Chains

Entanglement is a pure quantum correlation in the quantum many-body systems, which is different from classical physics. In recent years, quantum entanglement has attracted much attentions due to its particular properties and potential applications in quantum computers and other quantum information devices. Quantum phase transition(QPT) is induced by the change of an external parameter or coupling constant at absolute zero, which is a significant role in condensed matter physics such as high superconductivity and quantum Hall effect. Some interesting QPT phenomenon of spin-1/2 chain has been found by forerunners, which combined the quantum information theory with the quantum renormalization group(QRG) method. This paper study the quantum entanglement and QPT of the spin-1 Heisenberg chain with single ion anisotropy by the QRG method.We obtain the ground state phase diagram by using the scheme of Kadanoff block-spin remormalization group, which consists of Néel phase, Haldane phase and large D phase( for the single ion anisotropy parameter D is very large) for the anisotropy parameter ??0. Consider negativity as a measurement of the quantum entanglement, which can be defined by the partial transport reduced density matrix of the three-site block model. The negativity of the spins in a block has been checked, which can be affected by the anisotropy parameter and the single ion anisotropy parameter of the Hamiltonian. The negativity decreases as the parameter of single ion anisotropy increases. We check the tendency of the negativity as the scale of the lattice increases, it is found that the negativity tends to different saturated values despondence to different phase separated by the quantum critical points for a given value of the anisotropy parameter as the scale of the lattice tends to infinity, which shows the system has occured quantum phase transition. The negativity has a maximum saturated value in the Néel phase and has less one in the Haldane phase, while it disappears in the large D phase. The absolute value of the first partial derivative of the entanglement with respect to the single ion anisotropy parameter exhibits a nonanalytic behavior near the critical points,and the maximum tends to the critical point as the scale of the lattice increases. As the scale of the lattice tends to infinity, there is a growing peak at the Néel-Haldane critical point as well as the Néel-large D phase one, however, there is a maximum at the Haldane-large D phase critical point, all of them associate with the divergence of the correlation length directly. It is also found a similar phenomenon of the negativity and the first partial derivative of the negativity with respect to the anisotropy parameter for a given value of single ion anisotropy parameter as the scale of the lattice tends to infinity. Moreover, the critical exponents and the correlation length exponents are obtained, which are not reciprocal relationship.