Quantum Renormalization-group In Two-dimensional Spin Ladder

The quantum information science is a cross discipline developed very fast in recentyears. The quantum entanglement is one of the important resources in quantuminformation theory. It plays a major role in quantum information processing, such asquantum cryptography, quantum communication, quantum computing, etc. Theinvestigation of the generation and manipulation of quantum entanglement in solid spinsystems is one of the hot topics and has attracted much attention due to the advantageousof small volume and easy integrated of a solid state system. In this paper, theentanglement dynamics of a spin ladder with four-spin interactions and Heisenberg spinladder with triangular interactions is theoretically studied.Firstly, the time-evolution of quantum entanglement and fidelity of the spin ladderwith Dzyaloshinsky–Moriya interaction (DM) are discussed. The effects of variations ofthe coupling between spins and the DM interaction on the entanglement and fidelity ofthe spin ladder are analyzed. If there is no influence of environment, the entanglementand fidelity are periodic functions of the time and oscillate between the maximum andminimum values. The oscillation period decreases with the increases of either coupling inthe spin ladder or the Dzyaloshinsky–Moriya interaction. If there is coupling between thesystem and the environment, both entanglement and fidelity oscillate with a damping rate.The entanglement and fidelity decrease and even disappear when the intrinsicdecoherence rate increases. It is obvious that the entanglement and fidelity will beinfluenced by the intrinsic decoherence rate.Secondly, the two-dimensional Heisenberg spin model is investigated usingquantum renormalization group theory. The effective Hamiltonian of sub-system consisting of a triangular-spin is calculated. Then the two sub-systems are combined toform a new sub-system. In this way, the calculation of a long spin ladder can be reduced.The analytical result of a long spin ladder can be obtained. According to the Kadanoff’sblock theory, the forms of the effective Hamiltonian are different if the divisions of theblocks are different. While the changes of the corresponding lowest state energy are quitesmall. If the system is coupled to the environment, the time evolution of quantumentanglement, entanglement of formation and quantum discord is analyzed when the DMinteraction parameter and spin-spin coupling interaction are changed. It is shown thatthey are affected by the intrinsic decoherence rate with various degrees of dampness. Theclassical and quantum correlations of the system are quite different in certain situation.