【作者】 徐玉良；

【导师】 孔祥木；

【作者基本信息】 曲阜师范大学 ， 凝聚态物理， 2012， 硕士

【摘要】 量子纠缠具有奇特的相干性与非定域性，既是量子力学中的重要概念之一，也成为实现量子信息处理的关键。固态自旋系统展现出丰富的纠缠特性，引起了量子信息学与凝聚态物理领域的共同关注。由于精确求解的困难，对于自旋系统上热纠缠问题的研究大都集中在小尺度一维体系上。本文利用统计物理中的实空间重整化群方法，研究了分形晶格上多体（大尺度）自旋系统的热纠缠性质。研究了有限温度情况下分形维数d f=2的钻石型等级晶格上量子Heisenberg自旋系统的纠缠特性。采用格点消约重整化群方法并结合纠缠负值度（negativity）的定义计算了晶格上相距较远的外端点间的热纠缠。结果表明温度T、各向异性参数Δ和系统尺度L对端点热纠缠具有重要影响。各向异性参数确定后，在温度为零或极低时端点纠缠最大；随着温度的升高纠缠单调减小，而且存在临界温度TC，温度高于此值时纠缠消失；随着各向异性参数的减小，纠缠最大值降低而临界温度上升。温度一定时，端点纠缠在各向异性参数大于等于零时不存在；随着各向异性参数在小于零的范围内增大时纠缠先增大，在Δ=0附近纠缠迅速减小为零。当系统尺度增大（自旋数目增多）时，纠缠的最大值以及系统临界温度均降低，纠缠减小的越缓慢。当系统尺度很大时纠缠依然存在，这表明系统的热纠缠具有一定稳定性。研究了特殊的多分支分形维数d f=2.32以及d f=2.58的钻石型等级晶格上量子Heisenberg自旋系统的热纠缠。计算了等级晶格上两外端点自旋间的热纠缠。研究发现纠缠表现出一定的共性：当各向异性参数确定时，端点纠缠随温度升高单调减小，温度高于临界值时端点之间不存在纠缠。当温度一定时，随着各向异性参数增大时端点纠缠先增大后迅速降为零。研究还发现了这两种晶格与d f=2的晶格的端点纠缠的不同之处，随系统尺度增大时下降更缓慢，而且当各向异性参数取一定值时，随系统尺度的增大，纠缠出现“交叉”现象，系统的临界温度TC没有降低反而升高并趋于定值，这些结果表明，对于温度的引起的退相干，纠缠表现出更强的鲁棒性。系统的分形结构不同可以影响系统的能级结构，从而导致热纠缠的特性不同。更多还原

【Abstract】 Quantum entanglement, which shows amazing coherence and nonlocality, is not only one ofthe important basic concept in quantum mechanics but also the key resource to realize quantuminformation process. Solid spin system, which displays rich entanglement properties, hasreceived much attention both in quantum information and condensed matter physics recently.Due to the mathematical difficulty in exact solution, previous studies of the thermalentanglement on spin systems have mostly focused on the small size one-dimensional systems.In this thesis we study the thermal entanglement of many-body (large size) spin system on thediamond-type hierarchical lattices by using renormalization group (RG) methods.The thermal entanglement of quantum Heisenberg spin system on the diamond-typehierarchical lattice with fractal dimension d f=2is studied. By using the decimation RGmethod and the negativity, we calculate the entanglement of two distant terminal spins on thelattices, and it is found that temperature T the anisotropic parameter Δ and the size of systemL have important effect on the entanglement. When Δ is fixed and the temperatureapproaching zero or very low, the entanglement of terminal spins is the larges and with thetemperature increasing, the entanglement decreases monotonically. There exists the criticaltemperatureTC above which the entanglement vanishes. With the decrease of Δ, the maximumvalue of entanglement decrease andTC increases. At certain temperature, there exist noentanglement when the anisotropic parameter Δ≥0. In the range of Δ <0, the entanglementincreases with Δ increasing and sharply decreases to zero in the vicinity of Δ=0. The maximalvalue of thermal entanglement and the critical temperatureTC decrease as the size of the systembecomes large. The entanglement shows stability and exists when L is very large.By using the same method, we calculate the thermal entanglement of quantum Heisenbergspin system on the diamond-type hierarchical lattice with fractal dimension d f=2.32andd f=2.58. It is found that some similar properties of the entanglement on these two lattices andthe above lattice. The entanglement decreases with T increasing when Δ is certain and itincreases firstly and then decreases with Δ increasing at certain temperature. We also find thatthe entanglement of these two lattices compared with that of d f=2lattice exhibit differentfeatures that there occurs “entanglement crossing”. With the size of the system becoming large,the entanglement decrease much more slowly and the critical temperatureTC doesn’t decreasesbut increases and tends to the stable value. These results indicate that the entanglement is muchmore robust against the decoherence caused by temperature. The different fractal can affect different the energy level structure of system, which can lead to the different entanglementproperties.更多还原