Bulk Excitation And Phase Transition Of The Fractional Quantum Hall Effect In Disk Geometry

The quantum Hall effect has generated a wealth of successful theories and new concepts in condensed matter physics,which won Nobel Prize three times.As a major theory to promote the progress of human society,the research of quantum Hall effect has attracted the interest of theoretical and experimental physicists,successfully.The integer quantum Hall effect was discovered in Silicon Metal-oxide-Semiconductor system by American physicist Klaus von Klitzing in 1980.Furtherly,with the improvement of the samples quality and technology of low temperature and high magnetic field,the fractional quantum Hall(FQH)effect was discovered in cleaner AlxGa1_xAs/GaAs samples by Daniel Chee Tsui et al.in 1982.FQHE has obtained extensive attention for its potential application in quantum computers.Since the kinetic energy of electrons has been frozen by a strong magnetic field,the FQHE system is typically strongly correlated which cannot be treated by perturbation approach.The most useful numerical tool for studying the FQHE is the exact diagonalization to a microscopic Hamiltonian for small number of electrons,or other advanced numerical methods,like DMRG or MPS.The numerical calculation can be applied in different geometries for different purposes,theoretically.For the case of compact geometries without edge,like putting electrons on the surface of a torus or sphere,one can consider the bulk topological properties of the FQH states,i.e.,the ground state topological degeneracy and magneto-roton excitation.This thesis focuses on the rapidly rotating dipole fermions,and the fractional quantum Hall state in different spaces is discussed.We assume that all the dipoles are polarized by an external orienting field.The dipolar-dipolar interaction will change if we tune the angle of the external orienting field.The dipolar-dipolar interaction can be divided into two parts,one is interaction along z direction another is along the x direction.The topological properties of ground state,such as energy spectrum,mean momentum and density et al.are discussed.The properties of dipolar-dipolar interaction in Lz space is discussed at first.Adding a suitable confine potential,v=1/3 Laughlin state is stable in the lowest Landau level(LLL);instead,the most stable Laughlin state in the first Landau level(1LL)is the v=2+1/5 Laughlin state.The phase transformation point is determined by mean total angular momentum,that is,the point where the mean angular momentum changes suddenly is the point where the topological phase transformation happened.The fractional quantum Hall effect state of the dipolar-dipolar interaction is determined by the confining potential and the angle of the external orienting field.The density of dipolar-dipolar interaction evolves gradually from the isolated density island to the fused elliptical density,which indicates the superposition and competition of different phases,leading to the emergence of complex phases.The entanglement spectrum cannot give the quantitative properties of the edge excitation of the FQH droplet,such as the edge velocities or edge reconstruction.In a realistic system,the existence of an edge is unavoidable.There are more parameters,such as the strength of the background confinement and edge potential to tune the system.With these knobs,the nature of the FQH edge are extremely explored.Since the edge excitation is gapless and that overwhelms the bulk excitation in the low-energy sector.As a result,the bulk excitation,such as the magneto-roton is rarely discussed in disk geometry and the topological phase transition accompanying gap closing in the bulk is hard to describe.In this thesis,we propose a method to dig out the magneto-roton excitation of the FQHE liquid in disk geometry and this method determined by the bulk energy gap.Taking the dipolar interaction neutral atoms in fast rotated trap as an example,we observe multiple branches in the magneto-roton by varying a phase transition from the FQHE region to molecular phase as buck gap closing or soften the magneto-roton excitation.The magneto-roton and the density of phase transition et al.about dipolar interaction in center of mass space is calculated in this thesis,and our result shows that the magneto-roton on disk looks very similar to that on torus and sphere.We find that the extrapolated energy gap in thermodynamic limit is consistent both for Coulomb interactiont and the model Hamiltonian.What's more,The gap in disk geometry has much fewer finite size effects for no curvature.The anisotropic Laughlin state flows to the most excited state when all the dipoles are parallel to plane.