Edge Excitations of Quantum Hall States

Andrea Cappelli A, Carlo A. Trugenberger, Guillermo R. Zemba

建一 浅野

Xiao-Gang Wen

We propose a scheme to construct the edge states for the Fractional Quantum Hall (FQH) states obtained by the Jain’s scheme. The low energy effective theory of the edge excitations is obtained. We calculate the number of the branches and the electron propagators of the edge states for various FQH states. We demonstrate that two FQH states with the same filling fraction may have different topological orders and may support different edge excitations.

In this dissertation, we investigate the edge spin excitations and reconstructions of quantum Hall (QH) liquids. Edge spin reconstructions reflect the interplay of confining potential, electron-electron interaction, and Zeeman splitting at the edges of QH liquids. The main method used in this dissertation is the exact diagonalization calculation for finite size systems. The QH liquids studied in this dissertation are classified into three kinds, based on their topological properties and statistics of quasiparticles (quasiholes). The first kind of QH liquids is spin-polarized QH liquids. The QH liquids with filling factors ν = 1 and 1/3 are studied in this kind. We find the low-energy excitations of the ν = 1 and 1/3 are bosonic edge spin waves. Instabilities of these ferromagnetic state with altering confinement strength result from the softening of these spin waves and formation of edge spin textures. A microscopic trial wave function is used to study the edge instabilities of larger systems with particle number up to 40, and the obtained results are consistent to the ones calculated by exact diagonalization. The second kind of QH liquids is spin-unpolarized QH liquids. The liquids with ν = 2, 2/5 and 2/3 are studied in this kind. When ν = 2, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial ν = 2 instability is toward the compact state with total spin 1. At ν = 2/5, the edge instabilities is also triggered by softening of the edge spin waves with smoother confinement. The situation of ν = 2/3 unpolarized QH liquids is a little different, because it has one backward-moving edge spin wave mode and one forward-moving magnetoplasmon mode. There is no mystery about what kind of instability would occur with stronger or smoother confinement. The exact diagonalization calculation of finite systems can reproduce the instability induced by edge spin wave at the stronger confinement side and gives an estimate about the critical value of confining potential. But instability due to edge magnetoplasmon mode can not be realized in small size systems because of the high velocity of this mode. The last kind is the non-Abelian QH liquid (at ν = 5/2). Its edge instability triggered by softening of edge magnetoplasmon excitations with smoother confinement. Three different confinements are compared in the study of integer QH liquids in finite size systems. And some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit. The effect of Zeeman splitting to the edge instabilities is also considered in this dissertation.

Sergio De Filippo, Carmine Lubritto

Xiao-Gang Wen

We study topological order and edge excitations of the \ensuremath{\nu}=1/2 fractional quantum Hall state (FQH) of spin polarized electrons in a single-layer system. We find that the 1/2 FQH state obtained in the numerical study of Greiter et al. has a non-Abelian topological order suggested by Moore and Read. The edge excitations in such a non-Abelian FQH state are found to be described by the U(1) Kac-Moody algebra plus a chiral Majorana fermion theory. The electron and the quasiparticle propagators are calculated. Some experimental consequences are also discussed.