Resonance effects in nonlinear lattices

We study a class of one-dimensional nonlinear lattices with nearest-neighbour interactions described by a potential of the binomial type. This potential contains a free parameter which can be chosen to reproduce a variety of models, such as the Toda, the Fermi-Pasta-Ulam and the Coulomb-like lattices. Carrying out essentially numerical experiments, the effects of soliton propagation on a lattice with defects are investigated. In particular, the properties of the localized mode, generated by the propagation of the soliton through the defect, are discussed with respect to the defect mass and the potential parameter, in the light of a simple theoretical model. Furthermore, an interesting phenomenon is observed: the amplitude of the speed of the mass defect shows a sequel of resonance peaks in terms of the mass defect. The positions of these peaks appear to be independent of the potential parameter. Received 16 August 1999 and Received in final form 3 February 2000     

Efficient omnidirectional couplers achieved by anisotropic photonic crystal waveguides

Photonic crystals (PCs) have many potential applications because of their ability to control light-wave propagation. We have investigated omnidirectional couplers in two-dimensional anisotropic PC structures. The anisotropic PC coupler composed of an anisotropic-dielectric cylinder in air is studied by solving Maxwell's equations using the plane wave expansion method and finite-difference time-domain method. The photonic band structures are found to exhibit absolute bandgaps for the square lattices. Numerical simulations show that the incident light-waves at both transverse electric and transverse magnetic modes have efficient coupling in anisotropic PC coupler with square lattices. The guided modes and coupling length are analyzed by considering various line defect anisotropic PC waveguides and interaction regions of couplers. Such a mechanism of omnidirectional coupling should open up a new application for designing components in photonic integrated circuits.     

Vortices in quantum spin systems

We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices. Received 2 October 1998     

Critical behavior in a model of correlated percolation

We study the critical behavior of certain two-parameter families of correlated percolation models related to the Ising model on the triangular and square lattices, respectively. These percolation models can be considered as interpolating between the percolation model given by the + and – clusters and the Fortuin-Kasteleyn correlated percolation model associated to the Ising model. We find numerically on both lattices a two-dimensional critical region in which the expected cluster size diverges, yet there is no percolation.     

Periodic overlayers and moiré patterns: theoretical studies of geometric properties

Single crystal surfaces with periodic overlayers, such as graphene on hexagonal metal substrates, are found to exhibit, apart from their intrinsic periodicity, additional long-range order expressed by approximate surface lattices with large lattice constants. This phenomenon can be described as geometrically analogous to lateral interference effects resulting in periodic moiré patterns which are characterized by two-dimensional moiré lattices. Here we discuss in detail the mathematical formalism determining such moiré patterns based on concepts of two-dimensional Fourier transformation including coincidence lattices. The formalism provides simple relations that allow one to calculate possible moiré lattice vectors in their dependence on rotation angles α and scaling factors p1,p2 for periodic (p1 × p2)Rα overlayers on substrate surfaces described by general Bravais lattices. Specific emphasis will be given to hexagonal lattices where experimental data are available.     

Cluster synchronization and spatio-temporal dynamics in networks of oscillatory and excitable Luo-Rudy cells

We study collective phenomena in nonhomogeneous cardiac cell culture models, including one- and two-dimensional lattices of oscillatory cells and mixtures of oscillatory and excitable cells. Individual cell dynamics is described by a modified Luo-Rudy model with depolarizing current. We focus on the transition from incoherent behavior to global synchronization via cluster synchronization regimes as coupling strength is increased. These regimes are characterized qualitatively by space-time plots and quantitatively by profiles of local frequencies and distributions of cluster sizes in dependence upon coupling strength. We describe spatio-temporal patterns arising during this transition, including pacemakers, spiral waves, and complicated irregular activity.     

Coherent and Incoherent Interactions between Discrete-Soliton Trains in Two-Dimensional Light-Induced Photonic Lattices

The coherent and incoherent interactions between discrete-soliton trains are numerically investigated in lightinduced two-dimensional photonic lattices. The solutions of discrete-soliton trains for diamond and square lattices are obtained by Petviashvili iteration method. It is found that for both the kinds of lattices, two in-phase （out- of-phase） discrete-soliton trains attract （repel） each other, and the intermediates are always accompanied with energy transfer. While the interaction forces between two incoherent discrete-soliton trains are always attractive.     

From the Dirac operator to Wess-Zumino models on spatial lattices

We investigate two-dimensional Wess-Zumino models in the continuum and on spatial lattices in detail. We show that a non-antisymmetric lattice derivative not only excludes chiral fermions but in addition introduces supersymmetry breaking lattice artifacts. We study the non-local and antisymmetric SLAC derivative which allows for chiral fermions without doublers and minimizes those artifacts. The supercharges of the lattice Wess-Zumino models are obtained by dimensional reduction of Dirac operators in high-dimensional spaces. The normalizable zero modes of the models with N=1 and N=2 supersymmetry are counted and constructed in the weak- and strong-coupling limits. Together with known methods from operator theory this gives us complete control of the zero mode sector of these theories for arbitrary coupling.     

Optically-induced defect states in photonic lattices: formation of?defect channels, directional couplers, and disordered lattices leading to Anderson-like light localization

Formation of defect states by optical induction in one-dimensional photonic lattices fabricated in photorefractive lithium niobate is investigated experimentally. First, by using a moving narrow laser beam for defect recording, we investigate light propagation in samples containing single line defects and adjacent channel defects forming directional couplers. Then, these results are used to create lattices with randomly distributed defects, resembling a disordered optical potential. In such lattices, wave propagation is found to change from ballistic transport to transverse Anderson-like light localization as a function of induced disorder.     

Monte Carlo simulation of Ising model on decagonal covering structure

We study the Ising model on a two-dimensional quasilattice developed from the decagonal covering structure. The periodic boundary conditions are applied to a patch of rhombus-like covering pattern. By means of the Monte Carlo simulation and the finite-size scaling analysis the critical temperature is estimated as 2.317±0.002. Two critical exponents are obtained being 1/v=0.992±0.003 and η=0.247±0.002, which are close to the values of the two-dimensional regular lattices as well as the Penrose tilings.     

Eight-parameter renormalization group for Penrose lattices

The Ising model and the bond percolation model are set up with eight parameters on two-dimensional Penrose lattices. The behavior of their phase transition is studied by the use of a real-space renormalization group method. The resulting critical indices suggest that they belong to the universality class of two-dimensional periodic lattices.     

Polymorphism and chevron layer structure of an antiferroelectric liquid crystal studied by X-ray diffraction

We report X-ray diffraction experiments performed on an antiferroelectric compound exhibiting a very rich polymorphism (). The structural study of the unknown phases only allows us to exclude some phenomenological models. The use of oriented planar samples prepared between solid glass plates generate by cooling from the phase a chevron structure of tilted layers already well characterized for the phase. The extensive analysis of the evolution of the chevron structure through the numerous smectic-smectic phase transitions provides some original information in three distinct areas: fundamental data on the important physical parameters in the chevron structure formation, detection of the smectic-smectic phases transition by small change of the chevron structure, and information on the local molecular order induced by the alignment layer (interaction with a rubbed polymer). Received: 13 November 1996 / Received in final form: 19 January 1997 / Accepted: 30 January 1998     

Graphene antidot lattices: designed defects and spin qubits

Antidot lattices, defined on a two-dimensional electron gas at a semiconductor heterostructure, are a well-studied class of man-made structures with intriguing physical properties. We point out that a closely related system, graphene sheets with regularly spaced holes ("antidots"), should display similar phenomenology, but within a much more favorable energy scale, a consequence of the Dirac fermion nature of the states around the Fermi level. Further, by leaving out some of the holes one can create defect states, or pairs of coupled defect states, which can function as hosts for electron spin qubits. We present a detailed study of the energetics of periodic graphene antidot lattices, analyze the level structure of a single defect, calculate the exchange coupling between a pair of spin qubits, and identify possible avenues for further developments.     

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.     

Partition function zeros for the two-dimensional Ising model

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated. For anisotropic square and triangular lattices the distribution function is two-dimensional and satisfies a partial differential equation derived from a generalized scaling theory. Corresponding results for the isotropic square, triangular and honeycomb lattices are also presented.     

Spontaneous plaquette formation in the SU(4) spin-orbital ladder

The low-energy properties of the SU(4) spin-orbital model on a two-leg ladder are studied by a variety of analytical and numerical techniques. As in the case of SU(2) models, there is a singlet-multiplet gap in the spectrum, but the ground state is twofold degenerate. An interpretation in terms of SU(4)-singlet plaquettes is proposed. The implications for general two-dimensional lattices are outlined.     

Isolated structures in two-dimensional optical superlattice

Overlaying commensurate optical lattices with various configurations called superlattices can lead to exotic lattice topologies and, in turn, a discovery of novel physics. In this study, by overlapping the maxima of lattices, a new isolated structure is created, while the interference of minima can generate various “sublattice” patterns. Three different kinds of primitive lattices are used to demonstrate isolated square, triangular, and hexagonal “sublattice” structures in a two-dimensional optical superlattice, the patterns of which can be manipulated dynamically by tuning the polarization, frequency, and intensity of laser beams. In addition, we propose the method of altering the relative phase to adjust the tunneling amplitudes in “sublattices”. Our configurations provide unique opportunities to study particle entanglement in “lattices” formed by intersecting wells and to implement special quantum logic gates in exotic lattice geometries.     

Left-Handed Properties in Two-Dimensional Photonic Crystals Formed by Holographic Lithography

We give an analysis of the frequency distribution trends in the four lowest bands of two-dimensional square lattices formed by holographic lithography （HL） and in the lattices of the same kind but with regular dielectric columns with increasing filling ratios, and then present a comparative study on the left-handed properties in these two kinds of structures using plane wave expansion method and finite-difference time-domain （FDTD） simulations. The results show that the left-handed properties are more likely to exist in structures with large high-epsilon filling ratios or in a connected lattice.     

Improved phenomenological renormalization schemes

An analysis is made of various methods of phenomenological renormalization based on finite-dimensional scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-dimensional scaling amplitude of the internal energy in the three-dimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-dimensional corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-dimensional equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical index.