Diffusion in lattice Lorentz gases : Nearest scatterers approximation

The expressions for diffusion coefficients and for velocity autocorrelation functions of lattice Lorentz gases are derived both in the nearest scatterers and Boltzmann approximations. The results are obtained for linear chain, square, triangular, simple cubic, body centred cubic, face centred cubic and face centred hyper cubic lattices. The diffusion coefficients are compared with those from the effective medium approximation for the square lattice and with computer simulation results for triangular, simple cubic and body centred cubic lattices.     

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.     

Ultracold fermions and the SU(N) Hubbard model

We investigate the fermionic SU(N) Hubbard model on the two-dimensional square lattice for weak to moderate interactions using renormalization group and mean-field methods. For the repulsive case U>0 at half filling and small N the dominant tendency is towards breaking of the SU(N) symmetry. For N>6 staggered flux order takes over as the dominant instability, in agreement with the large-N limit. Away from half filling for N=3 two flavors remain half filled by cannibalizing the third flavor. For U<0 and odd N a full Fermi surface coexists with a superconductor. These results may be relevant to future experiments with cold fermionic atoms in optical lattices.     

Anti-ferroelectric polarization transitions in quantum-dot-quantum-well arrays

With the improvement in fabrication techniques it is now possible to produce atom-like semiconductor structures with unique electronic properties. This makes possible periodic arrays of nanostructures in which the Coulomb interaction, polarizability and tunneling may all be varied. This theoretical study investigates the collective properties of 2D arrays and 3D face-centered cubic lattices of singly charged nanospherical shells, sometimes called 'quantum dot-quantum wells' or 'core-shell quantum dots'. We find that, for square arrays, the classical ground state is an Ising anti-ferroelectric (AFE), while the quantum ground state undergoes a transition from a uniform state to an AFE. The triangular lattice, in contrast, displays properties characteristic of frustration. Three-dimensional face-centered cubic lattices polarize in planes, with each layer alternating in direction. We discuss the possible experimental signals of these transitions.     

Metal-insulator transition in the Hubbard model with incommensurate magnetic structures

The metal-insulator transition for the square, simple cubic, and body centered cubic lattices has been studied within the Hubbard model at half-filling taking into account nearest- and next-nearest-neighbor electron hopping. Both staggered antiferromagnetic and incommensurate magnetic states (spin-spiral wave) have been considered. The inclusion of the latter states for the three-dimensional lattices does not change the general pattern of the metal-insulator transition, but opens the fundamentally new possibility of the metal-insulator transition of the first order between the magnetically ordered states for the square lattice.     

Vortex-lattice dynamics in rotating spinor Bose-Einstein condensates

We observe interlaced square vortex lattices in rotating dilute-gas spinor Bose-Einstein condensates (BEC). After preparing a hexagonal vortex lattice in a one-component BEC in an internal atomic state |1, we coherently transfer a fraction of the superfluid to a different state |2. The subsequent evolution of this pseudo-spin-1/2 superfluid towards a state of offset square lattices involves an intriguing interplay of phase-separation and -mixing dynamics, both macroscopically and on the length scale of the vortex cores, and a stage of vortex turbulence. The stability of the square structure is proved by its response to applied shear perturbations. An interference technique shows the spatial offset between the two vortex lattices. Vortex cores in either component are filled by fluid of the other component, such that the spin-1/2 order parameter forms a Skyrmion lattice.     

Renormalization group approach to the surface and defect critical behavior in the potts model

A simple real-space renormalization group method with two-terminal clusters is used to treat the critical behavior of Potts ferromagnet with free surface and defect plane on the same footing both for square and cubic lattices. For a square lattice, quite different critical behaviors are found for the cases of line defect and free surface. Whenq is larger than three, like the case ofa line type defect in ‘diamond’ hierarchical lattice, the order parameter on a defect line increases discontinuously at the bulk critical point if the defect interaction is sufficiently strong. This behavior, on the contrary, does not occur on the surface of a semi-infinite plane. For a cubic lattice, the phase diagram and renormalization group flow properties are obtained explicitly for bothq=1 (bond percolation) andq=2 (Ising model). In both cases, our calculations whow that the critical behavior on the surface of a semi-infinite system belongs to a different universality class from the critical behavior on the defect plane of a bulk system.     

Generalized diffusion and quasi-elastic scattering widths in two-dimensional systems

The paper presents approximate calculations of dimensionality and lattice symmetry effects in the wave vector and frequency dependent diffusional response of disordered lattices. Numerical results are given and comparatively discussed for the tracer diffusion correlation function and the quasi-elastic incoherent scattering width in triangular, simple square and simple cubic lattices, only the restriction of double occupancy avoidance being taken into account. The quasi-elastic coherent scattering width is estimated for a triangular system possessing liquid-like structural disorder by means of the simple inclusion of a spread in jump lengths around a preferred set of jumps. Qualitative contact is made with recent neutron scattering experiments on alkali-metal graphite intercalation compounds.     

Stochastic flows,reaction — Diffusion processes,and morphogenesis

Recently, anexact procedure has been introduced [C. A. Walsh and J. J. Kozak,Phys. Rev. Lett. 47:1500 (1981)] for calculating the expected walk length 〈n〉 for a walker undergoing random displacements on a finite or infinite (periodic)d-dimensional lattice with traps (reactive sites). The method (which is based on a classification of the symmetry of the sites surrounding the central deep trap and a coding of the fate of the random walker as it encounters a site of given symmetry) is applied here to several problems in lattice statistics for each of whichexact results are presented. First, we assess the importance of lattice geometry in influencing the efficiency of reaction-diffusion processes in simple and multiple trap systems by reporting values of 〈n〉 for square (cubic) versus hexagonal lattices ind=2, 3. We then show how the method may be applied to variable-step (distance-dependent) walks for a single walker on a given lattice and also demonstrate the calculation of the expected walk length 〈n〉 for the case of multiple walkers. Finally, we make contact with recent discussions of “mixing” by showing that the degree of chaos associated with flows in certain lattice systems can be calibrated by monitoring the lattice walks induced by the Poincaré map of a certain parabolic function.     

Arbitrary structuring of two-dimensional photonic crystals by use of phase-only Fourier gratings

The computer-generated holography technique is applied to the structuring of two-dimensional (2D) photonic crystals with inherently embedded arbitrary defects. The technique uses phase-only Fourier gratings as a generator of spot arrays in the focal plane, such that a single exposure produces a 2D array of focused spots with desired defects or modifications in the lattice structure. We demonstrate several types of large-area 2D lattice structures with square, hexagonal, or hybrid lattices embedded with point and (or) line defects. Scanning the Fourier plane in the depth direction throughout multiphoton polymerization media allows 3D lattices with stacked defect layers to be formed.     

A scaling theory of clusters in the lattice gas

It is proposed that the number ofp-particle,k-bond lattice gas cluster configurations is of the form exp{σpf(k/p)} in the limitp→t8. A simple modification permits application to finite clusters, with the consequence that asymptotically the cluster partition function is of the droplet form, i.e., Z _p =exp[kp{itμp^1?1/d }+O(ln p)]. The scaling function for two-dimensional lattices is determined numerically and is found to be qualitatively universal. The scaling theory is used to investigate the size dependence of the surface free energy. The surface tension of small clusters can significantly exceed its limiting value. For intermediate cluster sizes (～100 particles) there is a modest reduction in surface tension, in accord with Tolman's prediction and the results of recent Monte Carlo studies.     

Preparation,structural, and calorimetric characterization of bicomponent metallic photonic crystals

We report preparation and characterization of novel bicomponent metal-based photonic crystals having submicron three-dimensional (3D) periodicity. Fabricated photonic crystals include SiO₂ sphere lattices infiltrated interstitially with metals, carbon inverse lattices filled with metal or metal alloy spheres, Sb inverse lattices, and Sb inverse lattices filled with Bi spheres. Starting from a face centered SiO₂ lattice template, these materials were obtained by sequences of either templating and template extraction or templating, template extraction, and retemplating. Surprising high fidelity was obtained for all templating and template extraction steps. Scanning electron microscopy (SEM), small angle X-ray scattering (SAXS) and differential scanning calorimetry (DSC) were used to characterize the structure and the effects of the structure on calorimetric properties. To the best of our knowledge, SAXS data on metallic photonic crystals were collected for first time. PACS 42.70.Qs; 68.37.Hk; 61.10.Eq; 81.70.Pg     

Lattice animals: Supplementation of perimeter polynomial data by graph-theoretic methods

Application of graph-theoretic methods to new perimeter polynomials for connected clusters on a lattice yields extra data on the total number of clusters and for the coefficients in the series expansion for the mean size of clusters at low densities. The lattices studied are the square, the square with next nearest neighbors, the triangular, and the simple cubic.     

Thermodynamic and magnetic properties in two artificial frustrated lattices

With the Monte Carlo simulation, we investigate the thermodynamics and magnetic properties of the artificial frustrated square and honeycomb lattices. The results from the Ising-like dipolar model show that there occurs one magnetic order transition for the square lattice while the honeycomb lattice exhibits two magnetic order phase transitions. When the magnetic field is applied perpendicular to one of sublattices, a sharp field-independent peak in the specific heat curves appears at a very low temperature for both frustrated lattices due to the occurrence of a long-range ordered state induced by the magnetic field. For the square lattice, the coercive field slightly increases with the angle of field relative to the vertical axis. For both frustrated lattices, the magnetic reversal is achieved mostly via flipping a chain of the nearest neighbor spins.     

Regular packings on periodic lattices

We investigate the problem of packing identical hard objects on regular lattices in d dimensions. Restricting configuration space to parallel alignment of the objects, we study the densest packing at a given aspect ratio X. For rectangles and ellipses on the square lattice as well as for biaxial ellipsoids on a simple cubic lattice, we calculate the maximum packing fraction φ(d)(X). It is proved to be continuous with an infinite number of singular points X(ν)(min), X(ν)(max), ν = 0, ±1, ±2,…. In two dimensions, all maxima have the same height, whereas there is a unique global maximum for the case of ellipsoids. The form of φ(d)(X) is discussed in the context of geometrical frustration effects, transitions in the contact numbers, and number-theoretical properties. Implications and generalizations for more general packing problems are outlined.     

Optical properties and photonic bands of GaAs photonic crystal waveguides with tilted square lattice

Reflectance measurements at variable angle of incidence are performed on GaAs photonic crystal waveguides with unconventional square lattices. The technique yields the dispersion of photonic bands for the investigated lattices, as first shown by Astratov et al. [Phys. Rev. B 60, R16225 (1999)]. A sample with a square lattice of air rings and small air fraction yields narrow resonant structures and a dispersion of photonic modes close to that of free photons. Another sample with a square lattice of dielectric squares and large air fraction leads to broader structures and to a dispersion of photonic modes which differs strongly for the two polarizations of light: this sample has a pseudo-gap around 1 micron wavelength. The experimental results are in good agreement with theoretical calculations of the reflectance and of the photonic mode dispersion in the photonic crystal slabs. Received 16 January 2002     

Spectral degeneracy of the lattice dirac equation as a function of lattice shape

We investigate the free Dirac equation in 2 + 1 dimensions on square, triangular, and hexagonal lattices. For each lattice the spectrum exhibits a degeneracy not present in the continuum limit. In the square and hexagonal cases there is a 4-fold degeneracy corresponding to 2 independent symmetries of the Hamiltonian; the degeneracy is eliminated by diagonalizing these symmetries and projecting onto the subspace characterized by a particular pair of eigenvalues. For the triangular case the degeneracy is 6-fold, but the naive Hamiltonian does not possess enough symmetry to eliminate the degeneracy. Certain ambiguities in the lattice Hamiltonian are pointed out and by the addition of terms which vanish in the continuum limit, it is cast in a form with sufficient symmetry to remove the degeneracy entirely, just as was done for the hexagonal and square lattices. It is found that after elimination of the spectral degeneracy the Hamiltonians for the hexagonal and triangular lattices are identical. The solutions to these theories are shown to have the correct continuum limit.     

Coulomb and liquid dimer models in three dimensions

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.     

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.     

The percolation perimeter for two-dimensional directed animals

The percolation perimeter to-size ratio of directed animals is investigated. For the square lattice it is found to be very close to 0.75 and the first correction is found on both the triangular and square lattices to be Bethe-like as for normal lattice animals.