Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

This paper presents an exact, wave-based approach for determining Bloch waves in two-dimensional periodic lattices. This is in contrast to existing methods which employ approximate approaches (e.g., finite difference, Ritz, finite element, or plane wave expansion methods) to compute Bloch waves in general two-dimensional lattices. The analysis combines the recently introduced wave-based vibration analysis technique with specialized Bloch boundary conditions developed herein. Timoshenko beams with axial extension are used in modeling the lattice members. The Bloch boundary conditions incorporate a propagation constant capturing Bloch wave propagation in a single direction, but applied to all wave directions propagating in the lattice members. This results in a unique and properly posed Bloch analysis. Results are generated for the simple problem of a periodic bi-material beam, and then for the more complex examples of square, diamond, and hexagonal honeycomb lattices. The bi-material beam clearly introduces the concepts, but also allows the Bloch wave mode to be explored using insight from the technique. The square, diamond, and hexagonal honeycomb lattices illustrate application of the developed technique to two-dimensional periodic lattices, and allow comparison to a finite element approach. Differences are noted in the predicted dispersion curves, and therefore band gaps, which are attributed to the exact procedure more-faithfully modeling the finite nature of lattice connection points. The exact method also differs from approximate methods in that the same number of solution degrees of freedom is needed to resolve low frequency, and arbitrarily high frequency, dispersion branches. These advantageous features may make the method attractive to researchers studying dispersion characteristics, band gap behavior, and energy propagation in two-dimensional periodic lattices.     

Diffusion in Lorentz lattice gas cellular automata: The honeycomb and quasi-lattices compared with the square and triangular lattices

We study numerically the nature of the diffusion process on a honeycomb and a quasi-lattice, where a point particle, moving along the bonds of the lattice, scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules. For the honeycomb lattice fully occupied by fixed rotators two (symmetric) isolated critical points appear to be present, with the same hyperscaling relation as for the square and the triangular lattices. No such points appear to exist for the quasi-lattice. A comprehensive comparison is made with the behavior on the previously studied square and triangular lattices. A great variety of diffusive behavior is found, ranging from propagation, superdiffusion, normal, quasi-normal, and anomalous, to absence of diffusion. The influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far is summarized.     

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.     

Enlargement of absolute photonic band gap in modified 2D anisotropic annular photonic crystals

We analyze the absolute photonic band gap in two dimensional (2D) square, triangular and honeycomb lattices composed of air holes or rings with different geometrical shapes and orientations in anisotropic tellurium background. Using the numerical plane wave expansion method, we engineer the absolute photonic band gap in modified lattices, achieved by addition of circular, elliptical, rectangular, square and hexagonal air hole or ring into the center of each lattice unit cell. We discuss the maximization of absolute photonic band gap width as a function of main and additional air hole or ring parameters with different shapes and orientation.     

Ferromagnetism in the Hubbard model: instability of the Nagaoka state on the triangular,honeycomb and kagome lattices

In order to analyse the lattice dependence of ferromagnetism in the two-dimensional Hubbard model we investigate the instability of the fully polarised ferromagnetic ground state (Nagaoka state) on the triangular, honeycomb and kagome lattices. We mainly focus on the local instability, applying single spin flip variational wave functions which include majority spin correlation effects. The question of global instability and phase separation is addressed in the framework of Hartree-Fock theory. We find a strong tendency towards Nagaoka ferromagnetism on the non-bipartite lattices (triangular, kagome) for more than half filling. For the triangular lattice we find the Nagaoka state to be unstable above a critical density of n = 1.887 at U = ∞, thereby significantly improving former variational results. For the kagome lattice the region where ferromagnetism prevails in the phase diagram widely exceeds the flat band regime. Our results even allow the stability of the Nagaoka state in a small region below half filling. In the case of the bipartite honeycomb lattice several disconnected regions are left for a possible Nagaoka ground state.     

基于平面波展开法的二维光子晶体表面模式研究

采用平面波展开法研究了四种二维光子晶体结构(圆柱介质柱四方晶格、圆柱介质柱三角晶格、正方介质柱四方晶格、正方介质柱三角晶格)的带隙宽度随介质柱尺寸变化的关系.使用平面波展开法计算常规晶格和表面缺陷晶格的模式并进行结果叠加,研究了各结构的二维光子晶体在带隙宽度最大时的表面模式.结果表明,同种晶格的光子晶体带隙宽度随着介质柱的尺寸增大呈先增后减趋势,存在最大值.随着表面介质柱尺寸的增加,四种晶格表面模式曲线均呈下降趋势.四方晶格光子晶体与三角晶格相比,表面介质柱尺寸的变化范围更大,但能获取表面模式频率范围较小.     

基于平面波展开法的二维光子晶体表面模式研究

采用平面波展开法研究了四种二维光子晶体结构(圆柱介质柱四方晶格、圆柱介质柱三角晶格、正方介质柱四方晶格、正方介质柱三角晶格)的带隙宽度随介质柱尺寸变化的关系.使用平面波展开法计算常规晶格和表面缺陷晶格的模式并进行结果叠加,研究了各结构的二维光子晶体在带隙宽度最大时的表面模式.结果表明,同种晶格的光子晶体带隙宽度随着介质柱的尺寸增大呈先增后减趋势,存在最大值.随着表面介质柱尺寸的增加,四种晶格表面模式曲线均呈下降趋势.四方晶格光子晶体与三角晶格相比,表面介质柱尺寸的变化范围更大,但能获取表面模式频率范围较小.     

蜂巢光子晶格中光波的无衍射和反常折射

空间频率模式的光子带隙反映了光波在周期性结构中的线性传输特性.以这种线性传输特性为基础,研究了蜂巢光子晶格中光波的无衍射和反常折射.通过详细分析带隙结构第一通带上的衍射与折射特性,得出了光波发生反常衍射和折射的入射条件.匹配不同的入射条件,数值模拟了光波的无衍射传输和反常折射现象.结果表明:将入射光束的波矢设置在蜂巢晶格布里渊区中正常、反常衍射区的交界处,可使高斯光束沿x轴、y轴方向的衍射得到有效抑制;以多光束干涉场作为入射光场,可对蜂巢晶格进行模式匹配,激发第二布里渊区的传输模式;进一步将模式匹配后入射光场的波矢设置在反常折射区,可实现光波的反常折射.     

Propagation of elastic waves through two-dimensional lattices of cylindrical empty or water-filled inclusions in an aluminum matrix

The layer-multiple-scattering method is developed to study wave propagation through two-dimensional lattices of cylindrical inclusions in an elastic medium. The lattices are a series of periodically spaced infinite one-dimensional periodic gratings (or rows) of inclusions. The layer-multiple-scattering method allows the analysis of the reflection and transmission properties of the two-dimensional lattice, provided those of each row are known. These are later determined by means of an exact multiple scattering formalism based on modal series developments. A new characteristic equation is obtained that describes the Bloch wave propagation into the infinite lattice. Lattices with empty and fluid-filled inclusions are compared. The comparison shows the existence of pass and stop bands due to the resonances of the fluid-filled inclusions. Resonant inclusions allow the opening of narrow pass bands inside phononic stop band, which is an interesting phenomenon for demultiplexing problems. It is worth noting that inclusion resonances have nothing to do with resonances due to defects, as they involve the whole lattice. In addition, it is shown that stop bands, at an oblique incidence, due to a strong coupling between longitudinal and transverse waves, are related to dispersive guided waves that propagate in the direction of the reticular planes of the lattices.     

Dimers Belonging to Three Orientations on Plane Honeycomb Lattices

It is well known that there are three types of dimers belonging to the three different orientations in a honeycomb lattice, and in each type all dimers are mutually parallel. Based on a previous result, we can compute the partition function of the dimer problem of the plane (free boundary) honeycomb lattices with three different activities by using the number of its pure dimer coverings (perfect matchings). The explicit expression of the partition function and free energy per dimer for many types of plane honeycomb lattices with fixed shape of boundaries is obtained in this way (for a shape of plane honeycomb lattices, the procedure that the size goes to infinite, corresponds to a way that the honeycomb lattice goes to infinite). From these results, an interesting phenomena is observed. In the case of the regions of the plane honeycomb lattice has zero entropy per dimer—when its size goes to infinite—though in the thermodynamic limit, there is no freedom in placing a dimer at all, but if we distinguish three types of dimers with nonzero activities, then its free energy per dimer is nonzero. Furthermore, a sufficient condition for the plane honeycomb lattice with zero entropy per dimer (when the three activities are equal to 1) is obtained. Finally, the difference between the plane honeycomb lattices and the plane quadratic lattices is discussed and a related problem is proposed.     

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.     

Orbital order in Mott insulators of spinless p-band fermions

A gas of strongly interacting single-species (spinless) p-orbital fermionic atoms in 2D optical lattices is proposed and studied. Several interesting new features are found. In the Mott limit on a square lattice, the gas is found to be described effectively by an orbital exchange Hamiltonian equivalent to a pseudospin-1/2 XXZ model. For a triangular, honeycomb, or kagome lattice, the orbital exchange is geometrically frustrated and described by a new quantum 120 degrees model. We determine the orbital ordering on the kagome lattice, and show how orbital wave fluctuations select ground states via the order by disorder mechanism for the honeycomb lattice. We discuss experimental signatures of various orbital ordering.     

Nonlinear dynamical stability of gap solitons in Bose-Einstein condensate loaded in a deformed honeycomb optical lattice

We investigate the existence and dynamical stability of multipole gap solitons in Bose-Einstein condensate loaded in a deformed honeycomb optical lattice. Honeycomb lattices possess a unique band structure, the first and second bands intersect at a set of so-called Dirac points. Deformation can result in the merging and disappearance of the Dirac points, and support the gap solitons. We find that the two-dimensional honeycomb optical lattices admit multipole gap solitons. These multipoles can have their bright solitary structures being in-phase or out-of-phase. We also investigate the linear stabilities and nonlinear stabilities of these gap solitons. These results have applications of the localized structures in nonlinear optics, and may helpful for exploiting topological properties of a deformed lattice.     

Orbital ordering and frustration of p-band Mott insulators

We investigate the general structure of orbital exchange physics in Mott-insulating states of p-orbital systems in optical lattices. Orbital orders occur in both the triangular and kagome lattices. In contrast, orbital exchange in the honeycomb lattice is frustrated as described by a novel quantum 120 degrees model. Its classical ground states are mapped into configurations of the fully packed loop model with an extra U(1) rotation degree of freedom. Quantum orbital fluctuations select a six-site plaquette ground state ordering pattern in the semiclassical limit from the "order from disorder" mechanism. This effect arises from the appearance of a zero energy flat band of orbital excitations.     

Numerical Study on the Waveguiding Properties of Photonic Crystal Fibers

The photonic band structure, mode field distribution, and leakage loss of different photonic crystal fibers are studied by the full vectorial plane wave expansion method and an extended beam propagation method. A comparison on the waveguiding mechanism is conducted among solid-core triangular-lattice PCF, air-core honeycomb PCF, and solid-core honeycomb PCF. For the latter two structures, which conduct light by the photonic bandgap effect, the advantages of down-doped solid-core honeycomb structure are demonstrated for its Gaussian-like mode field distribution and low leakage loss. A combined study on the leakage loss and band structure of solid-core honeycomb PCF has revealed the distinctive properties of photonic bandgap effect.     

BCx layers with honeycomb lattices on an NbB2(0001) surface

At elevated temperatures of 1000-1500 K, carbon (C) atoms that segregated to a surface and mixed with the boron (B) honeycomb lattice resulted in the formation of three different BC(x) layers as the topmost layers of NbB(2)(0001). Two of the layers were commensurate lattices: √7 × √7 and √3 × √3 structures; the third was incommensurate. The characteristic features of the √3 × √3 lattice with a honeycomb structure are discussed on the basis of the experimental data.     

Absolute band gap properties in two-dimensional photonic crystals composed of air rings in anisotropic tellurium background

We analyze the band gap spectra of two-dimensional photonic crystals created by square, triangular and honeycomb lattices of air rings with different geometrical shapes and orientations in anisotropic tellurium background. Specifically, five different shapes of rings (circular, hexagonal, elliptical, rectangular and square) are considered. Using the numerical plane wave method, we discuss the maximization of the absolute photonic band gap width as a function of air ring parameters with different shapes and orientations in three types of lattices.     

二维棋盘格子复式晶格的完全光子带隙研究

设计了一种棋盘格子复式晶格的二维光子晶体：在二维正方形格子中，把截面为正方形的柱子旋转45°，同时在每个原胞中心引入一个圆形截面的柱子构成的光子晶体结构. 用平面波展开计算棋盘格子复式晶格的完全光子带隙，结果表明：棋盘格子复式晶格的完全光子带隙的Δω/ω比值几乎是普通棋盘格子的5倍，完全光子带隙的个数也增加. 与其他复式结构相比较，发现其最佳的Δω/ω比值是一类粗锐复合结构光子晶体的2.1倍. 关键词： 二维光子晶体 复式晶格 完全光子带隙     

Band gap engineering in simultaneous phononic and photonic crystal slabs

We discuss the simultaneous existence of phononic and photonic band gaps in two types of phononic crystals slabs, namely periodic arrays of nanoholes in a Si membrane and of Si nanodots on a SiO₂ membrane. In the former geometry, we investigate in detail both the boron nitride lattice and the square lattice with two atoms per unit cell (these include the square, triangular and honeycomb lattices as particular cases). In the latter geometry, some preliminary results are reported for a square lattice.     

Quantum phase transition of cold atoms trapped in optical lattices

We review our recent theoretical advances in phase transition of cold atoms in optical lattices, such as triangular lattice, honeycomb lattice, and Kagomé lattice. By employing the new developed numerical methods called dynamical cluster approximation and cellular dynamical mean-field theory, the properties in different phases of cold atoms in optical lattices are studied, such as density of states, Fermi surface and double occupancy. On triangular lattice, a reentrant behavior of phase translation line between Fermi liquid state and pseudogap state is found due to the Kondo effect. We find the system undergoes a second order Mott transition from a metallic state into a Mott insulator state on honeycomb lattice and triangular Kagomé lattice. The stability of quantum spin Hall phase towards interaction on honeycomb lattice with spin-orbital coupling is systematically discussed. And we investigate the transition from quantum spin Hall insulator to normal insulator in Kagomé lattice which includes a nearest-neighbor intrinsic spin-orbit coupling and a trimerized Hamiltonian. In addition, we propose the experimental protocols to observe these phase transition of cold atoms in optical lattices.