Research On Quantization Condition Of Scarring States In Complex Soft-wall Quantum Billiards

Semi-classical theory,as a commonly method to study classical quantum correspondence,has one of important research directions which is the correspondence between classical periodic orbits and quantum scars.Quantum scars,as an abnormal condensation phenomenon of wavefunctions around unstable periodic orbits,play an important role in the study of quantum chaos and quantum transport and can be observed by the Scanning Gate Microscopy in recent years,so it has received extensive attention.At the same time,due to the difficulty of obtaining high-purity samples,exploring the influence of impurities on the properties of samples has also become a popular research topic.This thesis focus on a smooth confinement potential which possesses complex classical dynamics and investigate the quantization rule of the quantum scars,these results may be exploited in understanding the measurements of density of states and transport properties in two dimensional electron systems with random impurities.In the first chapter,we introduce the background knowledge of this thesis and explain the basic concepts,including classical quantum correspondence,quantum scars,soft-wall quantum billiards,disordered system.In the second chapter,we firstly introduce the Lyapunov index which can be used to measure the chaotic characteristics of classical systems and the Poincarésection,a common tool for analyzing classical chaos.Then we introduce a quantum method to express classical chaos,level spacing statistics.In addition,we also introduce the semi-classical quantization which can achieve the correspondance between classical periodic orbit and quantum scars for integrable system and chaotic system.In the third chapter,we investigate the quantization conditions for Lissajous scars which appear in a typical soft-wall system,a two-dimensional harmonic oscillator under small disturbance.We firstly verified the semi-classical formula for scars corresponding to the bouncing ball orbits,and then verified the quantization conditions for various Lissajous scars caused by different frequency ratios in the direction of x and y,and obtained good corresponding results.In the fourth chapter,we add a gaussian potential field on the basis of a model in chapter 3 and explore the quantization conditions of scars.The system has complex dynamics due to the complex potential field distribution.We firstly do the Poincaré section and classify the stable periodic orbits for this system and then associate these stable periodic orbits with their corresponding quantum scars by semi-classical formula.In the end,we give the possible quantization conditions for these scars.In the fifth chapter,we focus on a disordered system,doped graphene.Doping refers to the addition of a randomly distributed gaussian potential on the basis of tight-binding model to simulate the real impurity distribution.We also analyzed the level spacing statistics,participation ratio and length spectrum properties of the system and also find some dominant wave functions which correspond to shortperiod orbits caused by impurity distribution.In the sixth chapter,we make a summary for this thesis and do some introductions for the later work.