Anisotropic Phase Transition And Quasiparticle Tunneling Amplitude In The Fractional Quantum Hall Effects

The quantum Hall effect is one of the remarkable discoveries in the history of condensed matter physics.In?2+1?dimensional space-time,it reveals many exciting physical concepts including fractional charge excitation of quasiparticles,fractional and non-Abelian statistics,new classification of matter with topological order and chiral edge states.Current research shows that non-Abelian statistics exists in fractional quantum Hall?FQH?states with filling factor=5/2,which could be the first candidate state for topological quantum computation.In recent years,the study of FQH states is mostly carried out on the isotropic systems with rotational symmetry,however,the real FQH systems may be anisotropic.The anisotropy in effective mass tensor or dielectric tensor and the uniaxial strain on sample all break the rotational symmetry.This thesis focuses on the dipole-dipole interaction model on torus with the motion of a neutral particle in a rapidly rotating trap which is equivalent to that of a charged particle in a uniform magnetic field.It becomes possible to realize the FQH states in ultracold fermionic gases.We assume that the dipole momentum of all the particles are polarized by an external orienting field and the dipole-dipole interaction can be tuned to be anisotropic on the plane of the motion by tilting the direction of the dipole moments with respect to the rotation axis.We characterize the effective 2D dipole-dipole interaction systematically by the generalized pseudopotential especially when rotational symmetry is broken and find that while=1/3 Laughlin state is stable in the lowest Landau level?LLL?,it is not stable in the first Landau level?1LL?;instead,the most stable Laughlin state in 1LL is the=2+1/5Laughlin state.The FQH states keep robust against moderate anisotropy but large anisotropy induces a transition into a compressible phase in which all the particles are attracted and form a bound state.The anisotropy and the phase transition are systematically studied with the generalized pseudopotentials and characterized by the intrinsic metric,the wave function overlap,and the nematic order parameter.Quasiparticles tunneling between two counter-propagating edges through two-point contacts could provide information of the topological properties of the quasiparticles.Previous study on a disk found that in the limit of zero tunneling distance,the Abelian quasiparticle tunneling obeys a scaling law in which the scaling exponent is related to its charge and conformal dimension,while the non-Abelian quasiparticle exhibits some non-trivial behaviors.This thesis consider the scaling behavior of the Abelian and non-Abelian quasiholes in the_*Read-Rezayi states on a cylinder which has a full tunable tunneling distance by varying the aspect ratio.Via a finite size scaling analysis,we find that the scaling parameters of non-Abelian quasiholes should be considered in two regions.The scaling parameters are modified by the edge-edge interaction in region A when tunneling distance is close to the CFT limit and the correction term comes from the neutral fermionic mode of non-Abelian quasiparticles.In region B the scaling parameters are consistent with the effective field theory analysis for large tunneling distance.Besides,we think the two regions are separated by the radius of the quasihole which is a universal value around ₄₅?2.5l_-for model Hamiltonian.Comparing to the disk,the quasiparticle tunneling amplitudes of Laughlin state shows a better scaling behavior due to the advantages of the cylinder geometry including the smoothly tuned tunneling distance and no curvature difference between Landau orbitals.In particular,there are some crossover behaviors at two length scales when the two open edges are close to each other.These lengths are also reflected in the bipartite entanglement and the electron Green's function as either a singularity or a crosser.These two critical length scales of edge-edge distance,L_/⁰¹ and L_/⁰²,are found to be related to the dimension reduction and back scattering point respectively.It could be the benchmark of the sample size in designing an experimental setup of quasiparticle tunneling and interference.