Studies On Quantum Shannon Entropy In Quantum Multi-body System

Anderson localization is one of the most interesting phenomena in condensed matter physics.P.W.Anderson studied electrons in the crystal scattered by impurities and found that the coherence of the scattering path may lead to the complete disappearance of the conductance,i.e.,Anderson transition transition.Later,it can also be observed in other waves,such as light,microwave,ultrasonic wave,and matter wave.On the other hand,quantum phase transition is a very important phenomenon in condensed matter physics.Recently,scientists have begun to pay more attention to quantum phase transition in spin chains from the perspective of quantum informatics with the rapid development of quantum information science.The main works of this paper are follows:Firstly,an eigenstate of a Hamiltonian with periodic boundary condition is expaned as the superposition of all eigenstates of the same Hamiltonian but with antiperiodic boundary condition.In the view of quantum measurement,these eigenstates can constitute a set of measurement operator and they satisfy the completeness set.Thus,a boundary-condition-based quantum Shannon entropy is proposed,which simultaneously relates to periodic and antiperiodic boundary conditions and is more sensitive to system boundary conditions.Secondly,the boundary-condition-based quantum Shannon entropy is applied to several typical nonuniform potential energy system models,including one-dimensional Aubry-André-Harper model,Slowly varying potential model,Random-dimer potential model and three-dimensional Anderson model.Numerical results show that the averaged quantum Shannon entropy are relative larger for delocalized states and smaller for localized states.For critical states,they change rapadily.Therefore,it can be as a sensitivity measure to characterize Anderson transitions.Thirdly,quantum phase transitions in one-dimensional XY model in the transverse field are studied.A quantum Shannon entropy based on fidelity spectrum are used.Numerical results show that the entropy can well reflect the rich quantum phase transitions in the model.At the same time,at different kinds of phase transition points,the finite-scale scaling are different.Therefore,the quantum Shannon entropy is a very nice complement to existing quantities to reflect quantum phase transtions.