Photonic Floquet topological insulators in atomic ensembles

The feasibility of realizing a photonic Floquet topological insulator (PFTI) in an atomic ensemble is demonstrated. The interference of three coupling fields will split energy levels periodically, to form a periodic refractive index structure with honeycomb profile that can be adjusted by different frequency detunings and intensities of the coupling fields. This in turn will affect the appearance of Dirac cones in momentum space. When the honeycomb lattice sites are helically ordered along the propagation direction, gaps open at Dirac points, and one obtains a PFTI in an atomic vapor. An obliquely incident beam will be able to move along the zigzag edge of the lattice without scattering energy into the PFTI, due to the confinement of edge states. The appearance of Dirac cones and the formation of a photonic Floquet topological insulator can be shut down by the third‐order nonlinear susceptibility and opened up by the fifth‐order one.

     

Modulation of the photonic band structure topology of a honeycomb lattice in an atomic vapor

In an atomic vapor, a honeycomb lattice can be constructed by utilizing the three-beam interference method. In the method, the interference of the three beams splits the dressed energy level periodically, forming a periodic refractive index modulation with the honeycomb profile. The energy band topology of the honeycomb lattice can be modulated by frequency detunings, thereby affecting the appearance (and disappearance) of Dirac points and cones in the momentum space. This effect can be usefully exploited for the generation and manipulation of topological insulators.     

Vector valley Hall edge solitons in the photonic lattice with type-II Dirac cones

Topological edge solitons represent a significant research topic in the nonlinear topological photonics. They maintain their profiles during propagation, due to the joint action of lattice potential and nonlinearity, and at the same time are immune to defects or disorders, thanks to the topological protection. In the past few years topological edge solitons were reported in systems composed of helical waveguide arrays, in which the time-reversal symmetry is effectively broken. Very recently, topological valley Hall edge solitons have been demonstrated in straight waveguide arrays with the time-reversal symmetry preserved. However, these were scalar solitary structures. Here, for the first time, we report vector valley Hall edge solitons in straight waveguide arrays arranged according to the photonic lattice with innate type-II Dirac cones, which is different from the traditional photonic lattices with type-I Dirac cones such as honeycomb lattice. This comes about because the valley Hall edge state can possess both negative and positive dispersions, which allows the mixing of two different edge states into a vector soliton. Our results not only provide a novel avenue for manipulating topological edge states in the nonlinear regime, but also enlighten relevant research based on the lattices with type-II Dirac cones.     

Aspects of Floquet bands and topological phase transitions in a continuously driven superlattice

The recent creation of novel topological states of matter via periodic driving fields has attracted much attention. To contribute to the growing knowledge on this subject, we study the well-known Harper-Aubry-André model modified by a continuous time-periodic modulation and report on its topological properties along with several other interesting features. The Floquet bands are found to have non-zero Chern numbers which are generally different from those in the original static model. Topological phase transitions (discontinuous change of Chern numbers) take place as we tune the amplitude or period of the driving field. We demonstrate that the non-trivial Floquet band topology manifests via the quantized transport of Wannier states in the lattice space. For certain parameter choices, very flat yet topologically non-trivial Floquet bands emerge, a feature potentially useful for simulating the physics of strongly correlated systems. In some cases with an even number of Floquet bands, the spectrum features linearly dispersing Dirac cones which hold potential for the simulation of high energy physics or Klein tunnelling. Taking open boundary conditions, we observe anomalous counter-propagating chiral edge modes and degenerate zero modes. We end by discussing how these theoretical predictions may be verified experimentally.     

空间盘绕型声学超材料的亚波长拓扑谷自旋态

声子晶体的Dirac线性色散关系,使其具有奇特的声拓扑特性,在声波控制领域具有良好的应用前景.目前,声子晶体的拓扑边缘态主要基于Bragg散射所产生的能带结构,难以实现低频声波的受拓扑保护单向边缘传输.本文引入空间盘绕结构,设计了具有C_(3v)对称性的空间盘绕型声学超材料,并研究其布里渊区高对称点(K/K'点)的亚波长Dirac锥形线性色散.接着,通过旋转打破空间盘绕型声学超材料的镜像对称性,使其Dirac简并锥裂开而产生亚波长拓扑相变和亚波长拓扑谷自旋态.最后,采用拓扑相位互逆的声学超材料构造拓扑界面,实现声拓扑谷自旋传输.空间盘绕型声学超材料的亚波长Dirac线性色散与亚波长拓扑谷自旋态突破了声子拓扑绝缘体的几何尺寸限制,为声拓扑稳健传输在低频段的应用提供理论基础.     

Dissipation-induced topological phase transition and periodic-driving-induced photonic topological state transfer in a small optomechanical lattice

We propose a scheme to investigate the topological phase transition and the topological state transfer based on the small optomechanical lattice under the realistic parameters regime.We find that the optomechanical lattice can be equivalent to a topologically nontrivial Su-Schrieffer Heeger(SSH)model via designing the effective optomechanical coupling.Especially,the optomechanical lattice experiences the phase transition between topologically nontrivial SSH phase and topologically trivial SSH phase by controlling the decay of the cavity field and the opto mechanical coupling.We stress that the to pological phase transition is mainly induced by the decay of the cavity field,which is counter-intuitive since the dissipation is usually detrimental to the system.Also,we investigate the photonic state transfer between the two cavity fields via the topologically protected edge channel based on the small optomechanical lattice.We find that the quantum st ate transfer assisted by the topological zero energy mode can be achieved via implying the external lasers with the periodical driving amplitudes into the cavity fields.Our scheme provides the fundamental and the insightful explanations towards the mapping of the photonic topological insulator based on the micro-nano optomechanical quantum optical platform.     

类石墨烯复杂晶胞光子晶体中的确定性界面态

拓扑绝缘体是当前凝聚态物理领域研究的热点问题.利用石墨烯材料的特殊能带特性来实现拓扑输运特性在设计下一代电子和能谷电子器件方面具有较广泛的应用前景.基于光子与电子的类比,利用光子拓扑材料实现了确定性界面态;构建了具有C_(6v)。对称性的类似石墨烯结构的的光子晶体复杂晶格;通过多种方式降低晶格对称性来获得具有C_(3v),C_3,C_(2v)和C_2对称的晶体,从而打破能谷简并实现全光子带隙结构;将体拓扑性质不同的两种光子晶体摆放在一起,在此具有反转体能带性质的界面上,实现了具有单向传输特性的拓扑确定性界面态的传输.利用光子晶体结构的容易加工性,可以简便地调控拓扑界面态控制光的传播,可为未来光拓扑绝缘体的研究提供良好的平台.     

Conical Diffraction from Approximate Dirac Cone States in a Superhoneycomb Lattice

Superhoneycomb lattice is an edge‐centered honeycomb lattice that represents a hybrid fermionic and bosonic system. It contains pseudospin‐1/2 and pseudospin‐1 Dirac cones, as well as a flat band in its band structure. In this paper, we cut the superhoneycomb lattice along short‐bearded boundaries and obtain the corresponding band structure. The states very close to the Dirac points represent approximate Dirac cone states that can be used to observe conical diffraction during light propagation in the lattice. In comparison with the previous literature, this research is carried out using the continuous model, which brings new results and is simple, direct, accurate, and computationally efficient.     

Strain engineering of topological properties in lead‐salt semiconductors

Rock‐salt chalcogenide SnTe represents the simplest realization of a topological insulator where a crystal symmetry allows for the appearance of surface metallic states. Here, we theoretically predict that strain, as realized in thin films grown on (001) substrates, may induce a transition to a topological crystalline insulating phase in related lead‐salt chalcogenides. Furthermore, relevant topological properties of the surface states, such as the location of the Dirac cones on the surface Brillouin zone or the decay length of edge states, appear to be tunable with strain, with potential implications for technological devices benefiting from those additional degrees of freedom. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transport properties in the photonic super‐honeycomb lattice — a hybrid fermionic and bosonic system

We report on the transport properties of the super‐honeycomb lattice, the band structure of which possesses a flat band and Dirac cones, according to the tight‐binding approximation. The super‐honeycomb model combines the honeycomb lattice and the Lieb lattice and displays the properties of both. It also represents a hybrid fermionic and bosonic system, which is rarely seen in nature. By choosing the phases of input beams properly, the flat‐band mode of the super‐honeycomb lattice will be excited and the input beams will exhibit strong localization during propagation. On the other hand, if the modes of Dirac cones of the super‐honeycomb lattice are excited, one will observe conical diffraction. Furthermore, if the input beam is properly chosen to excite a sublattice of the super‐honeycomb lattice and the modes of Dirac cones with different pseudospins, e.g., by the three‐beam interference pattern, the pseudospin‐mediated vortices will be observed.

     

人工带隙材料的拓扑性质

近年来,人工带隙材料(如声子晶体和光子晶体)由于其优异的性能,已成为新一代智能材料的研究焦点.另一方面,材料拓扑学由凝聚态物理领域逐渐延伸到其他粒子或准粒子系统,而研究人工带隙材料的拓扑性质更是受到人们的广泛关注,其特有的鲁棒边界态,具有缺陷免疫、背散射抑制和自旋轨道锁定的传输等特性,潜在应用前景巨大.本文简要介绍拓扑材料特有的鲁棒边界态的物理图像及其物理意义,并列举诸如光/声量子霍尔效应、量子自旋霍尔效应、Floquet拓扑绝缘体等相关工作;利用Dirac方程,从原理上分析光/声拓扑性质的由来;最后对相关领域的发展方向和应用前景进行了相应的讨论.     

Half-filled Kondo lattice on the honeycomb lattice

The unique linear density of state around the Dirac points for the honeycomb lattice brings much novel features in strongly correlated models. Here we study the ground-state phase diagram of the Kondo lattice model on the honeycomb lattice at half-filling by using an extended mean-field theory. By treating magnetic interaction and Kondo screening on an equal footing, it is found that besides a trivial discontinuous first-order quantum phase transition between well-defined Kondo insulator and antiferromagnetic insulating state, there can exist a wide coexistence region with both Kondo screening and antiferromagnetic orders in the intermediate coupling regime. In addition, the stability of Kondo insulator requires a minimum strength of the Kondo coupling. These features are attributed to the linear density of state, which are absent in the square lattice. Furthermore, fluctuation effect beyond the mean-field decoupling is analyzed and the corresponding antiferromagnetic spin-density-wave transition falls into the O(3) universal class. Comparatively, we also discuss the Kondo necklace and the Kane-Mele-Kondo (KMK) lattice models on the same lattice. Interestingly, it is found that the topological insulating state is unstable to the usual antiferromagnetic ordered states at half-filling for the KMK model. The present work may be helpful for further study on the interplay between conduction electrons and the densely localized spins on the honeycomb lattice.     

Optically engineering the topological properties of a spin Hall insulator

Time-periodic perturbations can be used to engineer topological properties of matter by altering the Floquet band structure. This is demonstrated for the helical edge state of a spin Hall insulator in the presence of monochromatic circularly polarized light. The inherent spin structure of the edge state is influenced by the Zeeman coupling and not by the orbital effect. The photocurrent (and the magnetization along the edge) develops a finite, helicity-dependent expectation value and turns from dissipationless to dissipative with increasing radiation frequency, signalling a change in the topological properties. The connection with Thouless' charge pumping and nonequilibrium zitterbewegung is discussed, together with possible experiments.     

Zitterbewegung in the honeycomb photonic lattice

The propagation of a wave packet in a honeycomb photonic lattice has been studied using the time-dependent wave packet dynamics. It is found that the wave packet, superposed from the positive and negative energy modes at the vicinity of the two inequivalent Dirac points, can transform into a double-ring structure, which is caused by the interference between the two positive and negative energy modes around the Dirac points and is closely related to the Zitterbewegung (ZB). Also, a possible way to detect the ZB effect is proposed in the honeycomb photonic lattice.     

Unveiling the Link Between Fractional Schrödinger Equation and Light Propagation in Honeycomb Lattice

We suggest a real physical system — the honeycomb lattice — as a possible realization of the fractional Schrödinger equation (FSE) system, through utilization of the Dirac‐Weyl equation (DWE). The fractional Laplacian in FSE causes modulation of the dispersion relation of the system, which becomes linear in the limiting case. In the honeycomb lattice, the dispersion relation is already linear around the Dirac point, suggesting a possible connection with the FSE, since both models can be reduced to the one described by the DWE. Thus, we propagate Gaussian beams in three ways: according to FSE, honeycomb lattice around the Dirac point, and DWE, to discover universal behavior — the conical diffraction. However, if an additional potential is brought into the system, the similarity in behavior is broken, because the added potential serves as a perturbation that breaks the translational periodicity of honeycomb lattice and destroys Dirac cones in the dispersion relation.     

Graphene-like Be_3X_2(X=C,Si, Ge,Sn):A new family of two-dimensional topological insulators

Using first-principle calculations, we predict a new family of stable two-dimensional(2 D) topological insulators(TI),monolayer Be_3 X_2(X = C,Si, Ge, Sn) with honeycomb Kagome lattice. Based on the configuration of Be_3 C_2, which has been reported to be a 2 D Dirac material, we construct the other three 2 D materials and confirm their stability according to their chemical bonding properties and phonon-dispersion relationships. Because of their tiny spin-orbit coupling(SOC)gaps, Be_3 C_2 and Be_3 Si_2 are 2 D Dirac materials with high Fermi velocity at the same order of magnitude as that of graphene.For Be3 Ge2 and Be_3 Sn_2,the SOC gaps are 1.5 meV and 11.7 meV, and their topological nontrivial properties are also confirmed by their semi-infinite Dirac edge states. Our findings not only extend the family of 2 D Dirac materials, but also open an avenue to track new 2 DTI.     

A computational investigation of topological insulator Bi2Se3 film

Topological insulators have a bulk band gap like an ordinary insulator and conducting states on their edge or surface which are formed by spin–orbit coupling and protected by time-reversal symmetry. We report theoretical analyses of the electronic properties of three-dimensional topological insulator Bi₂Se₃ film on different energies. We choose five different energies (–123, –75, 0, 180, 350 meV) around the Dirac cone (–113 meV). When energy is close to the Dirac cone, the properties of wave function match the topological insulator’s hallmark perfectly. When energy is far way from the Dirac cone, the hallmark of topological insulator is broken and the helical states disappear. The electronic properties of helical states are dug out from the calculation results. The spin-momentum locking of the helical states are confirmed. A 3-fold symmetry of the helical states in Brillouin zone is also revealed. The penetration depth of the helical states is two quintuple layers which can be identified from layer projection. The charge contribution on each quintuple layer depends on the energy, and has completely different behavior along K and M direction in Brillouin zone. From orbital projection, we can find that the maximum charge contribution of the helical states is p_z orbit and the charge contribution on p_yand p_x orbits have 2-fold symmetry.     

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.     

Introduction to Topological Phases and Electronic Interactions in (2+1) Dimensions

A brief introduction to topological phases is provided, considering several two-band Hamiltonians in one and two dimensions. Relevant concepts of the topological insulator theory, such as: Berry phase, Chern number, and the quantum adiabatic theorem, are reviewed in a basic framework, which is meant to be accessible to non-specialists. We discuss the Kitaev chain, SSH, and BHZ models. The role of the electromagnetic interaction in the topological insulator theory is addressed in the light of the pseudo-quantum electrodynamics (PQED). The well-known parity anomaly for massless Dirac particle is reviewed in terms of the Chern number. Within the continuum limit, a half-quantized Hall conductivity is obtained. Thereafter, by considering the lattice regularization of the Dirac theory, we show how one may obtain the well-known quantum Hall conductivity for a single Dirac cone. The renormalization of the electron energy spectrum, for both small and large coupling regime, is derived. In particular, it is shown that massless Dirac particles may, only in the strong correlated limit, break either chiral or parity symmetries. For graphene, this implies the generation of Landau-like energy levels and the quantum valley Hall effect.     

Floquet engineering in topological crystalline insulator

There is a peculiar type of insulator, which is protected by the crystal symmetry known as topological crystalline insulator (TCI). In off-resonant cases, Floquet theory is an another way to study conventional Rabi oscillations. By using Floquet theory, the various type of Dirac fermionic systems and phases can be distinguished. In this article, it is shown that Floquet frequency can be used as a tool to distinguish different phases of TCI. The study of Bloch-Siegert shift has been performed and shown its variation in different phases of TCI. The collapse-revival spectra have also been studied in the perspective of Floquet theory and shown how quantum and classical Floquet oscillations are related to each other. The verification of the Floquet theory is justified by using numerical simulation.