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arXiv:2007.08870 · Physical review. B./Physical review. B

Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models

Bartholomew Andrews, Madhav Mohan, Titus Neupert

Published Feb 18, 2021• 27 citations
arXivHTMLPDF

Abstract

Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component $\ensuremath{\nu}=2/5$ and $3/7$ states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the $\ensuremath{\nu}=2/5$ and $3/7$ states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.

DOI
10.1103/physrevb.103.075132

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Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component \ensuremathν=2/5\ensuremath{\nu}=2/5\ensuremathν=2/5 and 3/73/73/7 states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the \ensuremathν=2/5\ensuremath{\nu}=2/5\ensuremathν=2/5 and 3/73/73/7 states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.
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.
The entanglement entropy of the ν=1/3 and ν=5/2 quantum Hall states in the presence of short-range random disorder has been calculated using direct diagonalization. A microscopic model of electron–electron interaction is used and spin-polarized electrons are confined to a single Landau level and interact with long-range Coulomb interaction. In the case of very weak disorder, the values of topological entanglement entropy are roughly consistent with the expected theoretical results. By considering a broad range of disorder strengths, entanglement entropy was studied in an effort to detect quantum phase transitions. In particular, there is a signature of the transition as the function of the disorder strength for the ν=5/2 state. Prospects of using the density matrix renormalization group to compute the entanglement entropy for larger system sizes are discussed.
We investigate the ground-state properties of a bosonic Harper-Hofstadter model with local interactions on a finite cylindrical lattice with filling fraction $\ensuremath{\nu}=1/2$. We find that our system supports topologically ordered states by calculating the topological entanglement entropy, and its value is in good agreement with the theoretical value for the $\ensuremath{\nu}=1/2$ Laughlin state. By exploring the behavior of the density profiles, edge currents, and single-particle correlation functions, we find that the ground state on the cylinder shows all signatures of a fractional quantum Hall state even for large values of the magnetic flux density. Furthermore, we determine the dependence of the correlation functions and edge currents on the interaction strength. We find that depending on the magnetic flux density, the transition toward Laughlin-like behavior can be either smooth or it can happen abruptly for some critical interaction strength.
We study the transition of $\ensuremath{\nu}=1/3$ and 2/52/52/5 fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moir\'e superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moir\'e systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moir\'e Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized $\ensuremath{\nu}=1/3$ and 2/52/52/5 states corresponding to the honeycomb Hofstadter model survive up to $\ensuremath{\approx}30%$ of typical moir\'e superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moir\'e superlattice Hamiltonians, which implies generally that fractional states for moir\'e superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.
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.
We investigate the ground-state properties of the fractional quantum Hall effect at the filling factor $\ensuremath{\nu}=2/3$ and 2+2/32+2/32+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+2/32+2/32+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 2/32/32/3 (2+2/3)(2+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 1/31/31/3 channel and the inner composite channel made of a charged mode and neutral modes.