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arXiv:1804.07563 · Physical Review A

Ring-shaped fractional quantum Hall liquids with hard-wall potentials

E. Macaluso, Iacopo Carusotto

Published Jul 5, 2018• 9 citations
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Abstract

We study the physics of $\ensuremath{\nu}=1/2$ bosonic fractional quantum Hall droplets confined in a ring-shaped region delimited by two concentric cylindrically symmetric hard-wall potentials. Trial wave functions based on an extension of the Jack polynomial formalism including two different chiral edges are proposed and validated for a wide range of confinement potentials in terms of their excellent overlap with the eigenstates numerically found by exact diagonalization. In the presence of a single repulsive potential centered in the origin, a recursive structure in the many-body spectra and a massively degenerate ground-state manifold are found. The addition of a second hard-wall potential confining the fractional quantum Hall droplet from the outside leads to a nondegenerate ground state containing a well-defined number of quasiholes at the center and, for suitable potential parameters, to a clear organization of the excitations on the two edges. The utility of this ring-shaped configuration in view of theoretical and experimental studies of subtle aspects of fractional quantum Hall physics is outlined.

DOI
10.1103/physreva.98.013605

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We study the physics of \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 bosonic fractional quantum Hall droplets confined in a ring-shaped region delimited by two concentric cylindrically symmetric hard-wall potentials. Trial wave functions based on an extension of the Jack polynomial formalism including two different chiral edges are proposed and validated for a wide range of confinement potentials in terms of their excellent overlap with the eigenstates numerically found by exact diagonalization. In the presence of a single repulsive potential centered in the origin, a recursive structure in the many-body spectra and a massively degenerate ground-state manifold are found. The addition of a second hard-wall potential confining the fractional quantum Hall droplet from the outside leads to a nondegenerate ground state containing a well-defined number of quasiholes at the center and, for suitable potential parameters, to a clear organization of the excitations on the two edges. The utility of this ring-shaped configuration in view of theoretical and experimental studies of subtle aspects of fractional quantum Hall physics is outlined.
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.
In a Laughlin fractional quantum Hall state, one- and two-quasihole states can be obtained by diagonalizing the many-body Hamiltonian with a trapping potential or, for larger systems, from the linear combination of the edge Jack polynomials. The quasihole states live entirely in the subspace of the lowest-energy branch in the energy spectrum with a fixed number of orbits, or a hard-wall confinement. The reduction in the Hilbert space dimension facilitates the study of time evolution of the quasihole states after, say, the removal of the trapping potential. We explore the quench dynamics under a harmonic external potential, which rotates the quasiholes in the droplet, and discuss the effect of long-range interaction and more realistic confinement. Accurate evaluation of the mutual statistics phase of anyons for a wide range of anyon separation can be achieved from the Berry-phase calculation.
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 study the edge-mode excitations of a fractional quantum Hall droplet by expressing the edge-state wave functions as linear combinations of Jack polynomials with a negative parameter. We show that the exact diagonalization within a subspace of Jack polynomials can be used to generate the chiral edge-mode excitation spectrum in the Laughlin phase and the Moore-Read phase with realistic Coulomb interaction. The truncation technique for the edge excitations simplifies the procedure to reliably extract the edge-mode velocities, which avoids the otherwise complicated analysis of the full spectrum that contains both edge and bulk excitations. Generalization to the Read-Rezayi state is also discussed.
We study a quantum point contact in the fractional quantum Hall regime at Landau level filling factors $\ensuremath{\nu}=1/3$ and 5/25/25/2. By using exact diagonalizations in the cylinder geometry, we identify the edge modes in the presence of a parabolic confining potential. By changing the sign of the potential, we can access both the tunneling through the bulk of the fluid and the tunneling between spatially separated droplets. This geometry is realized in the quantum point contact geometry for two-dimensional electron gases. In the case of the model Moore-Read Pfaffian state at filling factor $\ensuremath{\nu}=5/2$, we identify the conformal towers of many-body eigenstates including the non-Abelian sector. By a Monte-Carlo technique, we compute the various scaling exponents that characterize the edge modes. In the case of hard-core interactions whose ground states are exact model wave functions, we find equality of neutral and charged velocities, both bosonic and fermionic, for the Pfaffian state.