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arXiv:2109.00781 · Physical review. B./Physical review. B

Abelian origin of ν=2/3 and 2+2/3 fractional quantum Hall effect

Liangdong Hu, W. Zhu

Published Apr 22, 2022• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.165145

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Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for the exotic fundamental physics they would exhibit, and for universal quantum computing applications. The fractional quantum Hall (FQH) state at a filling factor of ν=12/5 is a promising candidate; however, its precise nature is still under debate and no consensus has been achieved so far. Here, we investigate the nature of the FQH ν=13/5 state and its particle-hole conjugate state at 12/5 with the Coulomb interaction, and we address the issue of possible competing states. Based on a large-scale density-matrix renormalization group calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at ν=13/5 and 12/5 is captured by the k=3 parafermion Read-Rezayi state (RR_{3}), including a robust excitation gap and the topological fingerprint from the entanglement spectrum and topological entanglement entropy. Furthermore, by considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a signature of the RR_{3} state at 13/5 and 12/5 filling numbers.
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We investigate the structure of gapless edge modes propagating at the boundary of some fractional quantum Hall states. We show how to deduce explicit trial wavefunctions from the knowledge of the effective theory governing the edge modes. In general, quantum Hall states have many edge states. Here, we discuss the case of fractions having only two such modes. The case of spin-polarized and spin-singlet states at filling fraction ν = 2/5 is considered. We give an explicit description of the decoupled charged and neutral modes. Then we discuss the situation involving negative flux acting on the composite fermions. This happens notably for the filling factor ν = 2/3 which supports two counterpropagating modes. Microscopic wavefunctions for spin-polarized and spin-singlet states at this filling factor are given. Finally, we present an analysis of the edge structure of a non-Abelian state involving also negative flux. Counterpropagating modes involve, in all cases, explicit derivative operators diminishing the angular momentum of the system.
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.
We study the many-body ground states of three-component quantum particles in two prototypical topological lattice models under strong intercomponent and intracomponent repulsions. At band filling $\ensuremath{\nu}=3/4$ for hard-core bosons, we demonstrate the emergence of the three-component fractional quantum Hall (FQH) effect characterized by the K\mathbf{K}K matrix, through an exact diagonalization study of fourfold quasidegenerate ground states with a robust spectrum gap and the combined density-matrix renormalization group calculation of fractional drag charge pumping. Further, we formulate the topological characterization of FQH states of three-component Bose-Fermi mixtures at various fillings by the K\mathbf{K}K matrix. Finally, we discuss the possible generalization of our approach to identify non-Abelian three-component spin-singlet FQH states.
While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $\ensuremath{\nu}=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by sixfold degenerate ground states and a fractionally quantized Chern-number matrix.