Non-abelian quantum Hall effect in topological flat bands

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.     

Multiple coherence resonances by time-periodic coupling strength in scale-free networks of bursting neurons

We investigate the algebraic structure of flat energy bands a partial filling of which may give rise to a fractional quantum anomalous Hall effect (or a fractional Chern insulator) and a fractional quantum spin Hall effect. Both effects arise in the case of a sufficiently flat energy band as well as a roughly flat and homogeneous Berry curvature, such that the global Chern number, which is a topological invariant, may be associated with a local non-commutative geometry. This geometry is similar to the more familiar situation of the fractional quantum Hall effect in two-dimensional electron systems in a strong magnetic field.     

The quantum anomalous Hall effect on a star lattice with spin-orbit coupling and an exchange field

We study a star lattice with Rashba spin-orbit coupling and an exchange field and find that there is a quantum anomalous Hall effect in this system, and that there are five energy gaps at Dirac points and quadratic band crossing points. We calculate the Berry curvature distribution and obtain the Hall conductivity (Chern number ν) quantized as integers, and find that ν?=-?1,2,1,1,2 when the Fermi level lies in these five gaps. Our model can be viewed as a general quantum anomalous Hall system and, in limit cases, can give what the honeycomb lattice and kagome lattice give. We also find that there is a nearly flat band with ν?=?1 which may provide an opportunity for realizing the fractional quantum anomalous Hall effect. Finally, the chiral edge states on a zigzag star lattice are given numerically, to confirm the topological property of this system.     

Rashba自旋轨道耦合下square-octagon晶格的拓扑相变

本文研究了各向同性square-octagon晶格在内禀自旋轨道耦合、Rashba自旋轨道耦合和交换场作用下的拓扑相变,同时引入陈数和自旋陈数对系统进行拓扑分类.系统在自旋轨道耦合和交换场的影响下会出现许多拓扑非平庸态,包括时间反演对称破缺的量子自旋霍尔态和量子反常霍尔态.特别的是,在时间反演对称破缺的量子自旋霍尔效应中,无能隙螺旋边缘态依然能够完好存在.调节交换场或者填充因子的大小会导致系统发生从时间反演对称破缺的量子自旋霍尔态到自旋过滤的量子反常霍尔态的拓扑相变.边缘态能谱和自旋谱的性质与陈数和自旋陈数的拓扑刻画完全一致.这些研究成果为自旋量子操控提供了一个有趣的途径.     

Current carrying states in the disordered quantum anomalous Hall effect

In a quantum Hall effect, flat Landau levels may be broadened by disorder. However, it has been found that in the thermodynamic limit, all extended (or current carrying) states shrink to one single energy value within each Landau level. On the other hand, a quantum anomalous Hall effect consists of dispersive bands with finite widths. We numerically investigate the picture of current carrying states in this case. With size scaling, the spectrum width of these states in each bulk band still shrinks to a single energy value in the thermodynamic limit, in a power law way. The magnitude of the scaling exponent at the intermediate disorder is close to that in the quantum Hall effects. The number of current carrying states obeys similar scaling rules, so that the density of states of current carrying states is finite. Other states in the bulk band are localized and may contribute to the formation of a topological Anderson insulator.     

The topological quantum phase transitions in Lieb lattice driven by the Rashba SOC and exchange field

The quantum spin Hall (QSH) effect and the quantum anomalous Hall (QAH) effect in Lieblattice are investigated in the presence of both Rashba spin-orbit coupling (SOC) anduniform exchange field. The Lieb lattice has a simple cubic symmetry, which ischaracterized by the single Dirac-cone per Brillouin zone and the middle flat band in theband structure. The intrinsic SOC is essentially needed to open the full energy gap in thebulk. The QSH effect could survive even in the presence of the exchange field. In terms ofthe first Chern number and the spin Chern number, we study the topological nature and thetopological phase transition from the time-reversal symmetry broken QSH effect to the QAHeffect. For Lieb lattice ribbons, the energy spectrum and the wave-function distributionsare obtained numerically, where the helical edge states and the chiral edge states revealthe non-trivial topological QSH and QAH properties, respectively.     

Topological phase transition in anisotropic square-octagon lattice with spin–orbit coupling and exchange field

We investigate the topological phase transitions in an anisotropic square-octagon lattice in the presence of spin–orbit coupling and exchange field. On the basis of the Chern number and spin Chern number, we find a number of topologically distinct phases with tuning the exchange field, including time-reversal-symmetry-broken quantum spin Hall phases, quantum anomalous Hall phases and a topologically trivial phase. Particularly, we observe a coexistent state of both the quantum spin Hall effect and quantum anomalous Hall effect. Besides, by adjusting the exchange filed, we find the phase transition from time-reversal-symmetry-broken quantum spin Hall phase to spin-imbalanced and spin-polarized quantum anomalous Hall phases, providing an opportunity for quantum spin manipulation. The bulk band gap closes when topological phase transitions occur between different topological phases. Furthermore, the energy and spin spectra of the edge states corresponding to different topological phases are consistent with the topological characterization based on the Chern and spin Chern numbers.     

Topological invariants for fractional quantum hall states

We calculate a topological invariant, whose value would coincide with the Chern number in the case of integer quantum Hall effect, for fractional quantum Hall states. In the case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In the case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.     

Magnetic-Flux-Induced Tunable Completely Flat Band and Topological Nearly Flat Band in Square Kagome Lattice

The flat bands in a square kagome lattice containing square and triangle plaquettes, are respectively investigated, and the origin of the doubly degenerated completely flat bands and the corresponding compact localized states are elucidated. It is found that the introduction of external magnetic flux enriches the modulation parameters, making the system present many interesting results. In the case that the magnetic flux penetrates through each square plaquette, the tunable completely flat band has one more tunable parameter. And when the external magnetic flux penetrates through each triangle plaquette, excepting a completely flat band, the band dispersions of the system present the topological nearly flat band, which is very useful to realize the interesting fractional quantum Hall physics. The average density of states is also calculated to corroborate the completely FB generated by the highly localized eigenstates. Furthermore, the implementation of the square kagome lattice system based on the current photonic waveguide network technology is demonstrated. The scheme opens up a way to generate the tunable completely flat band and topological nearly flat band in square kagome lattice under multi-parameter variable conditions.     

声子晶体中的多重拓扑相

研究了圆环型波导依照蜂窝结构排列的声子晶体系统中的拓扑相变.利用晶格结构的点群对称性实现赝自旋,并在圆环中引入旋转气流来打破时间反演对称性.通过紧束缚近似模型计算的解析结果表明,没有引入气流时,调节几何参数,系统存在普通绝缘体和量子自旋霍尔效应绝缘体两个相;引入气流后,可以实现新的时间反演对称性破缺的量子自旋霍尔效应相,而增大气流强度,则可以实现量子反常霍尔效应相.这三个拓扑相可以通过自旋陈数来分类.通过有限元软件模拟了多个系统中边界态的传播,发现不同于量子自旋霍尔效应相,量子反常霍尔相系统的表面只支持一种自旋的边界态,并且它无需时间反演对称性保护.     

Z2 topological order and the quantum spin Hall effect

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2 classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multiband and interacting systems.     

Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions

An important characteristic of topological band insulators is the necessary presence of in-gap edge states on the sample boundary. We utilize this fact to show that when the boundary is reconnected with a twist, there are always zero-energy defect states. This provides a natural connection among novel defects in the two-dimensional p{x}+ip{y} superconductor, the Kitaev model, the fractional quantum Hall effect, and the one-dimensional domain wall of polyacetylene.     

Topological Mott insulators

We consider extended Hubbard models with repulsive interactions on a honeycomb lattice, and the transitions from the semimetal to Mott insulating phases at half-filling. Because of the frustrated nature of the second-neighbor interactions, topological Mott phases displaying the quantum Hall and the quantum spin Hall effects are found for spinless and spin fermion models, respectively. The mean-field phase diagram is presented and the fluctuations are treated within the random phase approximation. Renormalization group analysis shows that these states can be favored over the topologically trivial Mott insulating states.     

Entanglement spectrum of non-Abelian anyons

Non-Abelian anyons can emerge as fractionalized excitations in two-dimensional systems with topological order. One important example is the Moore—Read fractional quantum Hall state. Its quasihole states are zero-energy eigenstates of a parent Hamiltonian, but its quasiparticle states are not. Both of them can be modeled on an equal footing using the bipartite composite fermion method. We study the entanglement spectrum of the cases with two or four non-Abelian anyons. The counting of levels in the entanglement spectrum can be understood using the edge theory of the Moore—Read state, which reflects the topological order of the system. It is shown that the fusion results of two non-Abelian anyons is determined by their distributions in the bipartite construction.     

Interacting dirac fermions on a topological insulator in a magnetic field

We study the fractional quantum Hall states on the surface of a topological insulator thin film in an external magnetic field, where the Dirac fermion nature of the charge carriers have been experimentally established only recently. Our studies indicate that the fractional quantum Hall states should indeed be observable in the surface Landau levels of a topological insulator. The strength of the effect will however be different, compared to that in graphene, due to the finite thickness of the topological insulator film and due to the admixture of Landau levels of the two surfaces of the film. At a small film thickness, that mixture results in a strongly nonmonotonic dependence of the excitation gap on the film thickness. At a large enough thickness of the film, the excitation gap in the lowest two Landau levels are comparable in strength.     

石墨烯中新奇量子物态的研究

石墨烯特殊的晶格结构和能带结构赋予了它独特的电学性质. 近年来, 分数量子霍尔态、 魔角石墨烯中的 关联绝缘体态和超导态等现象的发现不断证明着石墨烯是一种理想的二维模型体系, 可用于实现一系列新奇的量子物态, 对石墨烯中新奇量子物态的探测和调控也一直是凝聚态物理领域的前沿研究热点之一. 本文将系统地介绍近年来石墨烯中对称性破缺量子物态的研究进展, 包括平带中强关联量子物态的研究以及谷赝自旋调控的研究, 并介绍一种在纳米尺度、 单电子精度上探测二维材料体系简并度及对称性破缺态的普适方法, 希望为相关领域的研究人员提供参考和借鉴.     

Higher-energy composite fermion levels in the fractional quantum Hall effect

Even though composite fermions in the fractional quantum Hall liquid are well established, it is not yet known up to what energies they remain intact. We probe the high-energy spectrum of the 1/3 liquid directly by resonant inelastic light scattering, and report the observation of a large number of new collective modes. Supported by our theoretical calculations, we associate these with transitions across two or more composite fermions levels. The formation of quasiparticle levels up to high energies is direct evidence for the robustness of topological order in the fractional quantum Hall effect.