The quantum Hall effect is due to discontinuities in the chemical potential at certain bulk electron densities. It has been conjectured that the resulting edge states in some fractional Hall gaps can have an intricate composite structure which reflects the correlations in the incompressible bulk fluid. We present direct numerical evidence obtained from exact-diagonalization studies in favor of this conjecture. We show that the states on the \ensuremath{\nu}=2/3 fractional quantum Hall edge constitute a two-component system in which a \ensuremath{\nu}=1/3 droplet of holes is contained within a \ensuremath{\nu}=1 droplet of electrons.
The quantum Hall effect is due to discontinuities in the chemical potential at certain bulk electron densities. It has been conjectured that the resulting edge states in some fractional Hall gaps can have an intricate composite structure which reflects the correlations in the incompressible bulk fluid. We present direct numerical evidence obtained from exact-diagonalization studies in favor of this conjecture. We show that the states on the \ensuremath{\nu}=2/3 fractional quantum Hall edge constitute a two-component system in which a \ensuremath{\nu}=1/3 droplet of holes is contained within a \ensuremath{\nu}=1 droplet of electrons.
The edge states in the fractional quantum Hall systems at filling factor $\ensuremath{\nu}=1/3$ are studied by the density matrix renormalization group method. It is shown that the density oscillation induced by the local boundary condition at the edge is characterized by the wave number of the minimum magnetoroton excitation, and this structure is partially reconstructed with the change in the confinement potential shape. In particular, the $\ensuremath{\nu}=1$ counterpropagating edge channel appears with the change in the chemical potential, which is consistent with recent experiments on heat transport. The stability of the bulk states against the change in the number of electrons confirms that the bulk part of the fractional quantum Hall state is incompressible, while the edge state is compressible.
There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\ensuremath{\nu}=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\ensuremath{\nu}=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.
We present numerical evidence in support of the view that the physical origin of the fractional quantum Hall effect at \ensuremath{\nu}=n/(2n+1) is associated with the fact that this is the largest filling factor at which an n-component system of electrons can avoid having any pair of electrons occur in a state of relative angular momentum 1. Our calculations, for n=2, are based on a model in which the Hilbert space is truncated to the two lowest Landau levels and the Landau-level separation, \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is a parameter of the model. For \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0 and short-range repulsive interactions, a zero-energy incompressible ground state occurs at \ensuremath{\nu}=2/5. Our numerical results suggest that, in agreement with arguments advanced by J. K. Jain, this incompressibility survives to the strong-field limit \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=\ensuremath{\infty}.
We investigate the ground-state properties of the fractional quantum Hall effect at the filling factor $\ensuremath{\nu}=2/3$ and 2+2/3, with a special focus on their typical edge physics. Via a topological characterization scheme in the framework of the density matrix renormalization group, the nature of the $\ensuremath{\nu}=2/3$ and 2+2/3 states is identified as an Abelian hole-type Laughlin state, as evidenced by the fingerprint of entanglement spectra, central charge, and topological spin. Crucially, by constructing an interface between the 2/3(2+2/3) state and different integer quantum Hall states, we study the structures of the interfaces from many aspects, including charge density and dipole moment. In particular, we demonstrate the edge reconstruction by visualizing edge channels comprised of two groups: the outermost 1/3 channel and the inner composite channel made of a charged mode and neutral modes.
The energy spectra and wave functions of up to 14 interacting quasielectrons (QE's) in the Laughlin $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors ${\ensuremath{\nu}}_{\mathrm{QE}}=\frac{1}{2},$ 31,32, corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$ 114,135. The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$ 51,72, corresponding to $\ensuremath{\nu}=\frac{3}{10},$ 134,175. All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.
We compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 52, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1) and paramagnetic (P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=21), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.