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arXiv:1103.0772 · Physical Review B

Fractional quantum Hall states at 1 3 and 5 2 filling: Density-matrix renormalization group calculations

Jize Zhao, D. N. Sheng, F. D. M. Haldane

Published May 31, 2011• 48 citations
arXivHTMLPDF

Abstract

In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions $\ensuremath{\nu}=1/3$ and $5/2$. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at $\ensuremath{\nu}=5/2$ for large systems with electron numbers up to ${N}_{e}=34$, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.

DOI
10.1103/physrevb.83.195135

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In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 and 5/25/25/2. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=5/2 for large systems with electron numbers up to Ne=34{N}_{e}=34Ne​=34, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.
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 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.
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We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.
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 it is generally believed that a crystal state is favored at filling factors ν<1/6\nu< 1/6ν<1/6 in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences ν=n/(6n±1)\nu=n/(6n\pm 1)ν=n/(6n±1) and ν=n/(8n±1)\nu=n/(8n\pm 1)ν=n/(8n±1). To address this paradox, we investigate the nature of the state at filling factors ν=1/7\nu=1/7ν=1/7, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.