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Physical review. B, Condensed matter

Elementary excitations in a finite fractional quantum Hall droplet

X. C. Xie, S. Das Sarma, Song He

Published Jun 15, 1993• 8 citations
Publisher

Abstract

We have carried out a numerical study of elementary excitations in a finite droplet of an incompressible quantum Hall liquid by using the few-particle exact diagonalization technique on disks and spheres in the presence of an external confining potential. We study both weak and strong confinements in a spherical droplet, finding in the latter case a possible destruction of the incompressibility gap for high-enough confinement strength. In the disk geometry, we find evidence for a linear gap mode and for a second edge mode at a fractional filling of 2/5.

DOI
10.1103/physrevb.47.15942

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We have carried out a numerical study of elementary excitations in a finite droplet of an incompressible quantum Hall liquid by using the few-particle exact diagonalization technique on disks and spheres in the presence of an external confining potential. We study both weak and strong confinements in a spherical droplet, finding in the latter case a possible destruction of the incompressibility gap for high-enough confinement strength. In the disk geometry, we find evidence for a linear gap mode and for a second edge mode at a fractional filling of 2/5.
We make use of numerical exact diagonalization calculations to explore the physics of $\ensuremath{\nu}=1/2$ bosonic fractional quantum Hall droplets in the presence of experimentally realistic cylindrically symmetric hard-wall potentials. This kind of confinement is found to produce very different many-body spectra compared to a harmonic trap or the so-called extremely steep limit. For a relatively weak confinement, the degeneracies are lifted and the low-lying excited states organize themselves in energy branches that can be explained in terms of their Jack polynomial representation. For a strong confinement, a strong spatial deformation of the droplet is found, with an unexpected depletion of its central density.
For the numerical simulation of the fractional quantum Hall (FQH) effects on a finite disk, the rotational symmetry is the only symmetry that is used in diagonalizing the Hamiltonian. In this work, we propose a method of using the weak translational symmetry for the center of mass of the many-body system. With this approach, the bulk properties, such as the energy gap and the magneto-roton excitation are consistent with those in the closed manifolds like the sphere and torus. As an application, we consider the FQH phase and its phase transition in the fast rotated dipolar fermions. We thus demonstrate the disk geometry having versatility in analyzing the bulk properties beside the usual edge physics.
The single-particle spectral function for an incompressible fractional quantum Hall state in the presence of a scalar short-ranged attractive impurity potential is calculated via exact diagonalization within the spherical geometry. In contrast to the noninteracting case, where only a single bound state below the lowest Landau level forms, electron–electron interactions strongly renormalize the impurity potential, effectively giving it a finite range, which can support many quasi-bound states (long-lived resonances). Averaging the spectral weights of the quasi-bound states and extrapolating to the thermodynamic limit, for filling factor ν = 1/3 we find evidence consistent with localized fractionally charged e /3 quasi-particles. For ν = 2/5, the results are slightly more ambiguous, due to finite size effects and possible bunching of Laughlin-quasi-particles.
We develop the density-matrix renormalization group (DMRG) technique for numerically studying incompressible fractional quantum Hall (FQH) states on the sphere. We calculate accurate estimates for ground-state energies and excitation gaps at FQH filling fractions nu=1/3 and nu=5/2 for systems that are considerably larger than the largest ever studied by exact diagonalization. We establish, by carefully comparing with existing numerical results on smaller systems, that DMRG is a highly effective numerical tool for studying incompressible FQH states.
The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.
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.