Study On Localization Properties Of Generalized Two-dimensional Lieb Lattice

Since P.W.Anderson proposed the concept of electronic Anderson localization in 1958,much attention has been paid to the study of localized electronic properties in disordered systems,and the disorder induced localization has gradually become a very important subject in condensed matter physics.In recent years,flat band(that is,an energy band with no dispersion in the entire k-space)has aroused a great interest in research,some lattices,such as Kagome lattice and Lieb lattice,have such flat band characteristics.This paper chooses the Lieb lattice as the research object.It has now been shown to be of experimental relevance in Wigner crystals,high-temperature superconductivity,photonic wave guide arrays,Bose-Einstein condensates,and ultracold atoms in optical lattices.In this paper,we study the electronic localization properties in Lieb lattice and extended Lieb lattices.Our focus is slightly different: instead of concentrating on the localization properties of the flat band states,In particular,we are interested in whether the presence of the flat band changes the localization properties and the finite-size scaling behaviour in disordered twodimensional(2D)Lieb model and its extensions.We study the localization properties of the 2D Lieb lattice and its extensions in the presence of disorder using transfer matrix method and finite-size scaling.We find that all states in the Lieb lattice and its extensions are localized for the disorder W?1,which is in accordance with the prediction of scaling theory.Clear differences in the localization properties between disordered flat band and disordered dispersive bands are identified.Our results complement previous experimental studies of ordered photonic Lieb lattices and provide information about their stability with respect to disorder.The thesis is divided into five chapters to study the localization properties of the disordered 2D Lieb lattice and its extended lattices.First,we give a brief introduction to the research background and review the basic properties of flat bands to understand our research purpose.Secondly,we use the tight-binding model to get the dispersion relationships between the ordered 2D Lieb lattice and its extended lattices,and make a comparative analysis with the disordered systems.At the same time,we pay attention to some special localization properties of the Lieb lattice and its extended lattices,mainly through direct diagonalization of the systems Hamiltonians,and other numerical methods to discuss the behaviors of density of states.Finally,we apply the standard Anderson tight-binding Hamiltonian to derive the transfer matrix for calculating the localized length.With the help of Mathematica scientific calculation software and the program of finite size scaling theory,we study the localization properties of Lieb lattice and its extensions.