Study On Thermal Entanglement Of Quantum Spin System On Diamond-type Hierarchical Lattice

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.