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arXiv:1407.1563 · Physical Review B

Fractional quantum spin Hall effect in flat-band checkerboard lattice model

Wei Li, D. N. Sheng, C. S. Ting, Yan Chen

Published Aug 4, 2014• 24 citations
arXivHTMLPDF

Abstract

By means of finite-size exact diagonalization we theoretically study the electronic many-body effects on the nearly flat-band structure with time-reversal symmetry in a checkerboard lattice model and identify the topological nature of two quantum phases, with ninefold and threefold degeneracy, that appear, respectively, at small and large values $\ensuremath{\lambda}$ of a nearest-neighbor spin-dependent interaction. Numerical evidence from the evolution of low-lying energy spectra and Berry phases with both spin-independent and spin-dependent twisted boundary conditions reveal that these two different ground states share the same topological spin Chern number. Quantum phase transition between these two states by tuning $\ensuremath{\lambda}$ is confirmed by evaluating the closing of energy and quasispin excitation spectra. Finally, the counting rules of spin excitation spectra are demonstrated as the fingerprints of the fractionalized quantum spin Hall states.

DOI
10.1103/physrevb.90.081102

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By means of finite-size exact diagonalization we theoretically study the electronic many-body effects on the nearly flat-band structure with time-reversal symmetry in a checkerboard lattice model and identify the topological nature of two quantum phases, with ninefold and threefold degeneracy, that appear, respectively, at small and large values \ensuremathλ\ensuremath{\lambda}\ensuremathλ of a nearest-neighbor spin-dependent interaction. Numerical evidence from the evolution of low-lying energy spectra and Berry phases with both spin-independent and spin-dependent twisted boundary conditions reveal that these two different ground states share the same topological spin Chern number. Quantum phase transition between these two states by tuning \ensuremathλ\ensuremath{\lambda}\ensuremathλ is confirmed by evaluating the closing of energy and quasispin excitation spectra. Finally, the counting rules of spin excitation spectra are demonstrated as the fingerprints of the fractionalized quantum spin Hall states.
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 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.
Inspired by the recent theoretical discovery of robust fractional topological phases without a magnetic field, we search for the non-abelian quantum Hall effect in lattice models with topological flat bands. Through extensive numerical studies on the Haldane model with three-body hard-core bosons loaded into a topological flat band, we find convincing numerical evidence of a stable ν=1 bosonic non-abelian quantum Hall effect, with the characteristic threefold quasidegeneracy of ground states on a torus, a quantized Chern number, and a robust spectrum gap. Moreover, the spectrum for two-quasihole states also shows a finite energy gap, with the number of states in the lower-energy sector satisfying the same counting rule as the Moore-Read pfaffian state.
We develop a method to characterize topological phase transitions for strongly correlated Hamiltonians defined on two-dimensional lattices based on the many-body Berry curvature. Our goal is to identify a class of quantum critical points between topologically nontrivial phases with fractionally quantized Hall (FQH) conductivity and topologically trivial gapped phases through the discontinuities of the many-body Berry curvature in the so-called flux Brillouin zone (fBZ), the latter being defined by imposing all possible twisted boundary conditions. For this purpose, we study the finite-size signatures of several quantum phase transitions between fractional Chern insulators and charge-ordered phases for two-dimensional lattices by evaluating the many-body Berry curvature numerically using exact diagonalization. We observe degeneracy points (nodes) of many-body energy levels at high-symmetry points in the fBZ, accompanied by diverging Berry curvature. We find a correspondence between the number and order of these nodal points, and the change of the topological invariants of the many-body ground states across the transition, in close analogy with Weyl nodes in noninteracting band structures. This motivates us to apply a scaling procedure, originally developed for noninteracting systems, for the Berry curvature at the nodal points. This procedure offers a useful tool for the classification of topological phase transitions in interacting systems harboring FQH like topological order.
We study the toplogical checkerboard lattice model around the $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall phase using numerical exact diagonalization without Landau level projections. We add local perturbations, modified hoppings, and on-site potentials, and observe phase transitions from the fractional quantum Hall phase to metallic and insulating phases when the strength and number of impurities is increased. In addition to evaluating the energy spectrum, we identify the phase diagrams by computing the topological Chern number of the many-body ground state manifold, and we show how the ground states lose their correlations due to the impurities by evaluating the spectrum of the one-body reduced density matrix. Our results show that the phase transition from the fractional quantum Hall phase to the metallic phase occurs for both impurity hoppings and potentials. Strong impurity hoppings cause a further transition into the insulating state, regardless of the sign of the hopping, when their density is high enough. In contrast, the same happens only for attractive potentials. Furthermore, the mixing to the higher band in a two-band model, generally denoted as Landau level mixing, is measured concluding that the lowest Landau level projection works well even with remarkably strong interactions and in the presence of impurities.
We consider a topologically nontrivial flat-band structure in one spatial dimension in the presence of nearest- and next-nearest-neighbor Hubbard interaction. The noninteracting band structure is characterized by a symmetry-protected topologically quantized Berry phase. At certain fractional fillings, a gapped phase with a filling-dependent ground-state degeneracy and fractionally charged quasiparticles emerges. At filling 13\frac{1}{3}31​, the ground states carry a fractional Berry phase in the momentum basis. These features at first glance suggest a certain analogy to the fractional quantum Hall scenario in two dimensions. We solve the interacting model analytically in the physically relevant limit of a large band gap in the underlying band structure, the analog of a lowest Landau level projection. Our solution affords a simple physical understanding of the properties of the gapped interacting phase. We pinpoint crucial differences to the fractional quantum Hall case by studying the Berry phase and the entanglement entropy associated with the degenerate ground states. In particular, we conclude that the ``fractional topological phase in one-dimensional flat bands'' is not a one-dimensional analog of the two-dimensional fractional quantum Hall states, but rather a charge density wave with a nontrivial Berry phase. Finally, the symmetry-protected nature of the Berry phase of the interacting phase is demonstrated by explicitly constructing a gapped interpolation to a state with a trivial Berry phase.