Edge Modes And Disorder In The Fractional Quantum Hall Effects

The quantum Hall effect has generated a wealth of successful theories and new concepts in condensed matter physics.The integer quantum Hall effect was discovered by Klaus von Klitzing in 1980.Furtherly,with the improvement of the samples quality,high magnetic field and low temperature technology,the fractional quantum Hall(FQH)effect was discovered by Tsui et al.in 1982.FQH states support the fractional charged Abelian and non-Abelian quasiparticle excitations.Based on the non-trivial topological properties,the properties of the non-Abelian FQH have been explored extremely in recent decades.Moreover,the braiding operation based on exchanging two non-Abelian quasiparticles leads to the unitary transformation of the ground state in its topological degenerated Hilbert space and which can be used to realize the topological quantum computation,such as that in the Microsoft.This thesis focuses on the states at filling factors ?(28)1/ 3,5 / 2 and 12 / 5.The topological properties,such as the quasiparticle excitations,edge excitations and the entanglement et al.are discussed.The non-Abelian quasiparticles in the 5 / 2 and 12 / 5 states are studied by entanglement spectrum and topological entanglement entropy.For studying the robustness of the topological quantum states,the effect of the impurities are studied.Through studying topological entanglement entropy and energy level statistics,the ground state quantum phase transition and many body localization are found.In addition,simulating the generation of individual quasihole or quasiparticle by a STM/AFM tip potential in experiment,a Gaussian potential at arbitrary position is applied in the disk geometry.The quasiparticle excitation and the ground state entanglement are studied where the rotational symmetry is broken.The two point-contact interference experiment was proposed to verify the statistics of the quasiparticles.The edge velocities for different branches are important for estimating the coherence length and temperature in experiments.The complex structure of the edge excitations for non-Abelian FQH state has been studied clearly for large momentum.This thesis focuses on the edge excitations in the Moore-Read state and Read-Rezayi state.The accurate coherence length and coherence temperature in the thermodynamic limit are obtained numerically by Jack polynomial.Our results provide important reference data for the related experiments.In recent years,beside in the GaAs-based semiconductors,the quantum Hall effect was also been observed in some new two-dimensional materials,such as graphene,ZnO,MoS2,silicene,germanene,phosphorene and other group-VI transition-metal dichalcogenides.The electron-electron interaction is different from that in the traditional semiconductor due to the structure of these materials.We preliminarily study the possibility for the realization of the FQH in phosphorene.The main content of this thesis are arranged as follows:(1)The energy spectrum and properties of the edge excitations are obtained by using the Jack polynomial method.In ?(28)5 / 2,a neutral fermionic mode and a chiral bosonic mode appear in the edge excitation.From the analysis of the density profiles,the two branches in the edge spectrum are distinguishable.This method can also be used for edge reconstruction.It demonstrates that the first reconstructed mode is fermionic branch.In addition,the similar analysis for ?(28)12 / 5,the velocities of two branches in thermodynamic limit are obtained from large-scale numerical calculation.The coherence length and coherence temperature for the two point-contact interference experiment are also calculated.(2)The wavefunction of quasiparticles in Laughlin state,Moore-Read state and Read-Rezayi state are obtained on the cylinder geometry by Jack polynomial and the quantum dimensions of these quasiparticles are calculated from topological entanglement entropy.Furthmore,the topological properties of two types of quasiparticles can be understood from the counting in the entanglement spectrum.(3)Impurities are essential for realizing the Hall plateaus.The influence to the topological properties of the ground state are studied.We explored the energy level statistic and entanglement entropy as varying the disorder strength and found a many-body localization transition in the ground state.(4)We focus on the properties of quantum Hall states in phosphorene.As the method used in monolayer and bilayer graphene,the Landau level and pseudopotential in the monolayer phosphorene are obtained.The interaction in phosphorene is anisotropic from the structure and dispersion relation which is different from the GaAs system.Therefore,the Landau level mixing and anisotropic quantum Hall effect can be considered in phosphorenne for the future study.