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arXiv:1810.03458 · Physical review. B./Physical review. B

Characterization of quasiholes in two-component fractional quantum Hall states and fractional Chern insulators in |C| =2 flat bands

Błażej Jaworowski, Nicolas Regnault, Zhao Liu

Published Jan 23, 2019• 24 citations
arXivHTMLPDF

Abstract

We perform an exact-diagonalization study of quasihole excitations for the two-component Halperin (221) state in the lowest Landau level and for several $\ensuremath{\nu}=1/3$ bosonic fractional Chern insulators in topological flat bands with Chern number $|C|$=2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of $|C|$=2 fractional Chern insulators, although the latter possess a ``color-entangled'' nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having $|C|$=2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.

DOI
10.1103/physrevb.99.045136

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We perform an exact-diagonalization study of quasihole excitations for the two-component Halperin (221) state in the lowest Landau level and for several \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 bosonic fractional Chern insulators in topological flat bands with Chern number ∣C∣|C|∣C∣=2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of ∣C∣|C|∣C∣=2 fractional Chern insulators, although the latter possess a ``color-entangled'' nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having ∣C∣|C|∣C∣=2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.
Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translational symmetry. We conclusively show that a partially filled Chern insulator at 1/3 filling exhibits a fractional quantum Hall effect and rule out charge-density wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: The system is incompressible and has a 3-fold degenerate ground state whose momenta can be computed by postulating an generalized Pauli principle with no more than 1 particle in 3 consecutive orbitals. The ground state density is constant, and equal to 1/3 in momentum space. Excitations of the system are fractional statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit counting identical to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator - both the many-body energy gap and the entanglement gap close at the phase transition when the single-particle Hamiltonian goes from topologically nontrivial to topologically trivial. These facts clearly show that fractional many-body states are possible in topological insulators.
Recent theoretical works have demonstrated various robust Abelian and non-Abelian fractional topological phases in lattice models with topological flat bands carrying Chern number C=1C=1C=1. Here we study hard-core bosons and interacting fermions in a three-band triangular-lattice model with the lowest topological flat band of Chern number C=2C=2C=2. We find convincing numerical evidence of bosonic fractional quantum Hall effect at the $\ensuremath{\nu}=1/3$ filling characterized by threefold quasidegeneracy of ground states on a torus, a fractional Chern number for each ground state, a robust spectrum gap, and a gap in quasihole excitation spectrum. We also observe numerical evidence of a robust fermionic fractional quantum Hall effect for spinless fermions at the $\ensuremath{\nu}=1/5$ filling with short-range interactions.
The integer quantum Hall state occurs when Landau levels are fully occupied by fermions, while the fractional quantum Hall state usually emerges when the Landau level is partially filled by strongly correlated fermions or bosons. Here, we report two fractional Chern insulating states of hard-core bosons in a multiband lattice model hosting topological flat bands with a high Chern number. The previously proposed $\ensuremath{\nu}=1/3$ fractional Chern insulating state inherited from the high Chern number C=2C=2C=2 of the lowest topological flat band is revisited by the infinite density matrix renormalization group algorithm. In particular, we numerically identify a bosonic 1/21/21/2-Laughlin-like fractional Chern insulating state at the integer fillings. We show two lower topological flat bands jointly generate an effective C=1C=1C=1 Chern band with half filling. Furthermore, we find a strictly particle-hole-like symmetry between the $\ensuremath{\nu}$ and $3\ensuremath{-}\ensuremath{\nu}$ filling in our model. These findings extend our understanding of quantum Hall states and offer a route to realize fractional states in the system with multibands and high Chern numbers.
We study NNN-component interacting particles (hardcore bosons and fermions) loaded in topological lattice models with SU(N)\mathrm{SU}(N)SU(N)-invariant interactions based on exact diagonalization and density matrix renormalization group method. By tuning the interplay of interspecies and intraspecies interactions, we demonstrate that a class of SU(N)\mathrm{SU}(N)SU(N) fractional quantum Hall states can emerge at fractional filling factors $\ensuremath{\nu}=N/(N+1)$ for bosons [$\ensuremath{\nu}=N/(2N+1)$ for fermions] in the lowest Chern band, characterized by the nontrivial fractional Hall responses and the fractional charge pumping. Moreover, we establish a topological characterization based on the K\mathbf{K}K matrix and discuss the close relationship to the fractional quantum Hall physics in topological flat bands with Chern number NNN.
We numerically investigate the properties of the quasihole excitations above the bosonic fractional Chern insulator state at filling nu = 1/2, in the specific case of the Harper-Hofstadter Hamiltonian with hard-core interactions. For this purpose, we employ a tree tensor network technique, which allows us to study systems with up to N = 18 particles on a 16 x 16 lattice and experiencing an additional harmonic confinement. First, we observe the quantization of the quasihole charge at fractional values and its robustness against the shape and strength of the impurity potentials used to create and localize such excitations. Then, we numerically characterize quasihole anyonic statistics by applying a discretized version of the relation connecting the statistics of quasiholes in the lowest Landau level to the depletions they create in the density profile [E. Macaluso et al., Phys. Rev. Lett. 123, 266801 (2019)]. Our results give a direct proof of the anyonic statistics for quasiholes of fractional Chern insulators, starting from a realistic Hamiltonian. Moreover, they provide strong indications that this property can be experimentally probed through local density measurements, making our scheme readily applicable in state-of-the-art experiments with ultracold atoms and superconducting qubits.
The Halperin $({m}^{\ensuremath{'}},m,n)$ fractional quantum Hall effects of two-component quantum particles are studied in topological checkerboard lattice models. Here for $m\ensuremath{\ne}{m}^{\ensuremath{'}}$, we demonstrate the emergence of fractional quantum hall effects with the associated K=(m+211m)\mathbf{K}=\left(\begin{array}{cc}m+2& 1\\ 1& m\end{array}\right)K=(m+21​1m​) matrix (even m=2m=2m=2 for boson and odd m=3m=3m=3 for fermion) in the presence of both strong intercomponent and intracomponent repulsions. Through exact diagonalization and density-matrix renormalization group calculations, we elucidate their topological fractionalizations, including (i) the $det|\mathbf{K}|=({m}^{2}+2m\ensuremath{-}1)$-fold ground-state degeneracies and (ii) fractionally quantized topological Chern number matrix $\mathbf{C}={\mathbf{K}}^{\ensuremath{-}1}$. Our flat band model provides a paradigmatic example of a microscopic Hamiltonian featuring fractional quantum Hall effect with partial spinpolarization.