5/2 Filling Factor Fractional Quantum Hall State On Disk

Since the quantum Hall effect has been discovered in experiment,it has become an important research field in condensed matter physics after several decades of development.The integer quantum Hall effect and odd denominator filling factor in lowest Landau level has been a good explanation of the theory.The even denominator filling factor in first Landau level such as 5/2,its ground state is still puzzles us.The 5/2 filling quantum Hall state has non-Abelian properties,which may have a great application in the quantum computation of topology protection.Exploring the ground state of 5/2 filling factor has been an import direction in the field of quantum Hall effect in these years.In this thesis we focus on the state of 5/2 filling factor.We consider the layer-thickness,Landau level mixing and background potential in disk geometry.We compute the fidelity between the ground state and Pfaffian state and anti-Pfaffian state,the phase diagram,entanglement entropy and entanglement spectrum.The content of this thesis are arranged as follows:Chapter 1 The introduction of Hall effect,including the classic Hall effect and quantum Hall effect.The research background and the concepts used in this thesis.Chapter 2 The derivation and expression of the matrix element and Haldane pseudopotentials of the fractional quantum Hall effect in plane and spherical geometry.The mapping between the sphere geometry and disk geometry.Chapter 3 The perturbation calculation of Landau level mixing including two-body and three-body correction consider with the layer thickness.The numerical results show in tables.Chapter 4 The projection of the three-body interaction of fermions in plane geometry.The CG coefficients of the relative angular momentum of 3,5,6,7,8,9 are calculated respectively.Especially at relative momentum m=9,the pseudopotential is a 2 × 2 matrix,the CG coefficients of the two states are discussed respectively.The expressions are given.Chapter 5 The fidelity between the 5/2 filling factor state and the Pfaffian state and anti-Pfaffian state in disk geometry consider the layer thickness,background potential and Landau level mixing.The phase diagram of 5/2 filling factor state,the entanglement entropy and entanglement spectrum of the ground state.We get the conclusion: when the Landau level mixing strength is relative small,the ground state is more favor Pfaffian state,when the Landau level mixing strength is relative large the ground state is more favor the anti-Pfaffian state.