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Physical review. B./Physical review. B

Topological states and flat bands on the maple leaf lattice

Ai‐Lei He, Xu‐Hui Yan, Lu Qi, Yongjun Liu, Ying Jie Han

Published Feb 9, 2024• 15 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.109.075118

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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.
We construct an islamic lattice by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-NN hoppings. This model supports abundant quantum phases for various values of filling fractions. At1/4filling, Chern insulator (CI) phases with Chern numbersC=±1, -2and a zero-Chern-number topological insulator (ZCNTI) phase exist. At3/8filling, several CI phases with Chern numbersC=±1, 3and the ZCNTI phase are obtained. For the filling fraction 3/4, CI phases with Chern numbersC=±1, 2and two ZCNTI phase areas appear. Interestingly, these ZCNTI phases host both robust corner states and gapless edge states which can be characterized by the quantized polarization and quadrupole moment. We further find that staggered magnetic fluxes can give rise to the ZCNTI state at1/4and3/4fillings. Phase diagrams for filling fractions1/8,1/2,5/8and7/8are presented as well. In addition, flat bands are obtained for various filling fractions by tuning the hopping parameters. At 1/8 filling, a best topological flat band (TFB) with flatness ratio about 12 appears. Several trivial flat bands but with total Chern number|C|=1emerge in this model and exactly flat band is found at 3/8 filling. We further investigateν=1/2fractional Chern insulate state when hard-core bosons fill into this TFB model.
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.
We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from C=1C=1C=1 to C=−1C=-1C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with 1/31/31/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy α(nA)α(n_A)α(nA​) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.
We study fermions on a triangular lattice model that exhibits topological flatbands characterized by nonzero Chern numbers. Our scheme stems from the well-known Hofstadter model but the next-nearest-neighbor hopping is introduced, which is crucial for tuning the lowest band to be nearly flat. Differing from previous proposals with the necessity of multiparticle interactions, we consider the more realistic long-range dipolar interaction combined with two-body short-range attractions between fermions. We show the realization of the non-Abelian $\ensuremath{\nu}=1/2$ Moore-Read fractional Chern insulators, and strong evidence for the existence of the more exotic $\ensuremath{\nu}=3/5$ Read-Rezayi fractional Chern insulators. Our results provide insights for the experimental realization of these exotic states by realistic two-body interactions and thus facilitates the implementation of the universal topological quantum computation.
The stability of the $\ensuremath{\nu}=1/3$ fractional Chern insulator (FCI) phase is analyzed on the example of a checkerboard lattice undergoing a transition into a Lieb lattice. The transition is performed by the addition of a second sublattice, whose coupling to the checkerboard sites is controlled by sublattice staggered potential. We investigate the influence of these sites on the many-body energy gap between three lowest energy states and the fourth state. We consider cases with different complex phases acquired in hopping and a model with a flattened topologically nontrivial band. We find that an interaction with the additional sites either open the single-particle gap or enlarge the existing one, which translates into similar effect on the many-particle gap. By looking at Berry curvature flatness we notice its strong correlation with the magnitude of the many-body energy gap, suggesting that the main mechanism of the FCI stabilization by additional atoms is via their influence on the Berry curvature. Evidence of the FCI phase for a region in a parameter space with larger energy gap is shown by looking at momenta of the threefold degenerate ground state, spectral flow, and quasihole excitation spectrum.
The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest-neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.