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Physical Review B

Topological quantum phase transitions and topological flat bands on the star lattice

Wen-Chao Chen, Ru Liu, Yi-Fei Wang, Chang-De Gong

Published Aug 16, 2012• 52 citations
Publisher

Abstract

We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the $1/2$ fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.

DOI
10.1103/physrevb.86.085311

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We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the 1/21/21/2 fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.
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.
Models with topological flat bands (TFBs) have been proposed on which the fractional Chern insulators (FCIs) have been studied. To flatten the lowest energy band, some long-range hopping terms are considered in Chern insulator (CI) models. In this paper, we study a tight-binding model on a maple leaf lattice with staggered magnetic fluxes. When the nearest-neighbor hopping term with a suitable phase factor is considered, a TFB with a large flatness ratio emerges, suggesting that staggered magnetic fluxes can give rise to TFBs. By adding the next-nearest-neighbor (NNN) hopping term, there are two types of topological states at one-third filling, i.e., the CI and the higher-order topological insulator (HOTI) states. Simultaneously, we find the CI with a high Chern number can transition to the HOTI state by tuning the NNN hopping parameter. When we consider more distant-neighbor hopping terms, a TFB with Chern number two is found. We further investigate FCI states when hard-core bosons are taken into account in these TFB models.
Recent proposals of topological flat band models have provided a new route to realize the fractional quantum Hall effect without Landau levels. We study hard-core bosons with short-range interactions in two representative topological flat band models, one of which is the well-known Haldane model (but with different parameters). We demonstrate that fractional quantum Hall states emerge with signatures of an even number of quasidegenerate ground states on a torus and a robust spectrum gap separating these states from the higher energy spectrum. We also establish quantum phase diagrams for the filling factor 1/2 and illustrate quantum phase transitions to other competing symmetry-breaking phases.
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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.
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.