Compact RBF meshless methods for photonic crystal modelling

Meshless methods based on compact radial basis functions (RBFs) are proposed for modelling photonic crystals (PhCs). When modelling two-dimensional PhCs two generalised eigenvalue problems are formed, one for the transverse-electric (TE) mode and the other for the transverse-magnetic (TM) mode. Conventionally, the Band Diagrams for two-dimensional PhCs are calculated by either the plane wave expansion method (PWEM) or the finite element method (FEM). Here, the eigenvalue equations for the two-dimensional PhCs are solved using RBFs based meshless methods. For the TM mode a meshless local strong form method (RBF collocation) is used, while for the tricker TE mode a meshless local weak form method (RBF Galerkin) is used (so that the discontinuity of the dielectric function ?(x)$?(x)$ can naturally be modelled). The results obtained from the meshless methods are found to be in good agreement with the standard PWEM. Thus, the meshless methods are proved to be a promising scheme for predicting photonic band gaps.     

Three ways to solve the Poisson equation on a sphere with Gaussian forcing

Motivated by the needs of vortex methods, we describe three different exact or approximate solutions to the Poisson equation on the surface of a sphere when the forcing is a Gaussian of the three-dimensional distance, ∇²ψ=exp(-2?²(1-cos(θ))-C^Gauss(?)${\nabla }^2\psi =\exp (-2{?}^2(1-\cos (\theta ))-C^{\mathit{Gauss}}(?)$. (More precisely, the forcing is a Gaussian minus the “Gauss constraint constant”, C^Gauss$C^{\mathit{Gauss}}$; this subtraction is necessary because ψ$\psi$ is bounded, for any type of forcing, only if the integral of the forcing over the sphere is zero [Y. Kimura, H. Okamoto, Vortex on a sphere, J. Phys. Soc. Jpn. 56 (1987) 4203–4206; D.G. Dritschel, Contour dynamics/surgery on the sphere, J. Comput. Phys. 79 (1988) 477–483]. The Legendre polynomial series is simple and yields the exact value of the Gauss constraint constant, but converges slowly for large ?$?$. The analytic solution involves nothing more exotic than the exponential integral, but all four terms are singular at one or the other pole, cancelling in pairs so that ψ$\psi$ is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of ?$?$ that converge very rapidly for the large values of ?  (?>40)$(?>40)$ appropriate for geophysical vortex computations. The series converges to a nonzero O(exp(-4?²))$O(\exp (-4{?}^2))$ error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.     

Non-Abelian global strings at chiral phase transition

We construct non-Abelian global string solutions in the UL_(N)×UR_(N)$U{(N)}_{\mathrm{L}}\times U{(N)}_{\mathrm{R}}$ linear sigma model. These strings are the most fundamental objects which are expected to form during the chiral phase transitions, because the Abelian η^′${\eta }^{\prime }$ string is marginally decomposed into N   of them. We point out Nambu–Goldstone modes of CPN−1$\mathbf{C}{P}^{N-1}$ for breaking of SUV_(N)$\mathit{SU}{(N)}_{\mathrm{V}}$ arise around a non-Abelian vortex.     

Vortex properties of mesoscopic superconducting samples

In this work we investigated theoretically the vortex properties of mesoscopic samples of different geometries, submitted to an external magnetic field. We use both London and Ginzburg–Landau theories and also solve the non-linear Time Dependent Ginzburg–Landau equations to obtain vortex configurations, equilibrium states and the spatial distribution of the superconducting electron density in a mesoscopic superconducting triangle and long prisms with square cross-section. For a mesoscopic triangle with the magnetic field applied perpendicularly to sample plane the vortex configurations were obtained by using Langevin dynamics simulations. In most of the configurations the vortices sit close to the corners, presenting twofold or three-fold symmetry. A study of different meta-stable configurations with same number of vortices is also presented. Next, by taking into account de Gennes boundary conditions via the extrapolation length, b, we study the properties of a mesoscopic superconducting square surrounded by different metallic materials and in the presence of an external magnetic field applied perpendicularly to the square surface. It is determined the b  -limit for the occurrence of a single vortex in a mesoscopic square of area d²$d^2$, for 4ξ(0)?d?10ξ(0)$4\xi (0)?d?10\xi (0)$.     

Matrix regularization of embedded 4-manifolds

We consider products of two 2-manifolds such as S²×S²$S^2\times S^2$, embedded in Euclidean space and show that the corresponding 4-volume preserving diffeomorphism algebra can be approximated by a tensor product SU(N)⊗SU(N)$\mathit{SU}(N)\otimes \mathit{SU}(N)$ i.e. functions on a manifold are approximated by the Kronecker product of two SU(N)$\mathit{SU}(N)$ matrices.     

Atomic resolution and vortex lattice studies of magnetic superconductors: A first approach in the nickel borocarbide TmNi2B2C

We present new scanning tunneling microscopy measurements in the superconductor TmNi₂B₂C. The topography shows in some areas flat surfaces, where atomic size modulations can be identified. We find a hexagonal vortex lattice between 0.15 T and 1.4 T, when the magnetic field is applied along the basal plane of the tetragonal crystal structure (B⊥c)$(B\perp c)$, and a hexagonal to square transition around 0.15 T when the field is applied along the c  -axis (B‖c)$(B\Vert c)$. Measured intervortex distance are smaller than expected at high field, due to the internal field being larger than the applied field.     

Determination of the Landau–Lifshitz damping parameter by means of complex susceptibility measurements

A new experimental method for the determination of the Landau–Lifshitz damping parameter, α$\alpha$, based on measurements of the frequency and field dependence of the complex magnetic susceptibility, χ(ω,H)=χ^′(ω,H)-iχ^″(ω,H)$\chi (\omega ,H)={\chi }^{\prime }(\omega ,H)-\mathrm{i}{\chi }^{″}(\omega ,H)$, is proposed. The method centres on evaluating the ratio of f_max/f_res, where f_res is the resonance frequency and f_max is the maximum absorption frequency at resonance, of the sample susceptibility spectra, measured in strong polarizing fields. We have investigated three magnetic fluid samples, namely sample 1, sample 2 and sample 3. Sample 1 consisted of particles of Mn_0.6Fe_0.4Fe₂O₄ dispersed in kerosene, sample 2 consisted of magnetite particles dispersed in Isopar M and sample 3 was composed of particles of Mn_0.66Zn_0.34Fe₂O₄ dispersed in Isopar M  . The results obtained for the mean damping parameter of particles within the magnetic fluid samples are as follows: 〈α(_{Mn0.6Fe0.4Fe2O4})〉=0.057$〈\alpha ({\mathrm{Mn}}_{0.6}{\mathrm{Fe}}_{0.4}{\mathrm{Fe}}_2{\mathrm{O}}_4)〉=0.057$ with the corresponding standard deviation SD=0.0104$\mathrm{SD}=0.0104$; 〈α(_Fe3O4)〉=0.1105$〈\alpha ({\mathrm{Fe}}_3{\mathrm{O}}_4)〉=0.1105$ with the corresponding standard deviation, SD=0.034$\mathrm{SD}=0.034$ and 〈α(_{Mn0.66Zn0.34Fe2O4})〉=0.096$〈\alpha ({\mathrm{Mn}}_{0.66}{\mathrm{Zn}}_{0.34}{\mathrm{Fe}}_2{\mathrm{O}}_4)〉=0.096$ with the corresponding standard deviation, SD=0.037$\mathrm{SD}=0.037$.     

Random ballistic growth and diffusion in symmetric spaces

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a N  -column box is viewed as a time-ordered product of (N×N)$(N\times N)$-matrices consisting of a single _sl2${\mathit{sl}}_2$-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space _HN=SL(N,R)/SO(N)$H_N=\mathit{SL}(N,R)/\mathit{SO}(N)$. In particular, the distribution of the maximal height of a growing heap is connected with the distribution of the maximal distance for the diffusion process in _HN$H_N$. The coordinates of _HN$H_N$ are interpreted as the coordinates of particles of the one-dimensional Toda chain. The group-theoretic structure of the system and links to some random matrix models are also discussed.     

Discrete gauge symmetries in discrete MSSM-like orientifolds

Motivated by the necessity of discrete _ZN${\mathbb{Z}}_N$ symmetries in the MSSM to insure baryon stability, we study the origin of discrete gauge symmetries from open string sector U(1)$U(1)$?s in orientifolds based on rational conformal field theory. By means of an explicit construction, we find an integral basis for the couplings of axions and U(1)$U(1)$ factors for all simple current MIPFs and orientifolds of all 168 Gepner models, a total of 32 990 distinct cases. We discuss how the presence of discrete symmetries surviving as a subgroup of broken U(1)$U(1)$?s can be derived using this basis. We apply this procedure to models with MSSM chiral spectrum, concretely to all known U(3)×U(2)×U(1)×U(1)$U(3)\times U(2)\times U(1)\times U(1)$ and U(3)×Sp(2)×U(1)×U(1)$U(3)\times \mathit{Sp}(2)\times U(1)\times U(1)$ configurations with chiral bi-fundamentals, but no chiral tensors, as well as some SU(5)$\mathit{SU}(5)$ GUT models. We find examples of models with _Z2${\mathbb{Z}}_2$ (R-parity) and _Z3${\mathbb{Z}}_3$ symmetries that forbid certain B and/or L violating MSSM couplings. Their presence is however relatively rare, at the level of a few percent of all cases.     

Effect of external magnetic field on superconducting and spin density wave gaps of high-Tc superconductors

A theoretical model is addressed here to study the interplay of the superconductivity (SC) and the spin density wave (SDW) long range orders in underdoped region in the vicinity of on-set of superconductivity in presence of an external magnetic field. The order parameters are calculated by using Zubarev’s technique of Green’s functions and determined numerically self-consistently. The gap parameters are found to be strongly coupled to each other through their coupling constants. The interplay displays BCS type two gaps in the quasi-particle density of states (DOS) which resemble the tunneling conductance of STM experiments. The gap edges in the DOS appear at ±(z+z₁)$\pm (z+z_1)$ and ±(z-z₁)$\pm (z-z_1)$. The applied magnetic field further induces Zeeman splitting which is explained on the basis of spin-filter effect of tunneling experiment.     

Tensor decomposition in electronic structure calculations on 3D Cartesian grids

In this paper, we investigate a novel approach based on the combination of Tucker-type and canonical tensor decomposition techniques for the efficient numerical approximation of functions and operators in electronic structure calculations. In particular, we study applicability of tensor approximations for the numerical solution of Hartree–Fock and Kohn–Sham equations on 3D Cartesian grids. We show that the orthogonal Tucker-type tensor approximation of electron density and Hartree potential of simple molecules leads to low tensor rank representations. This enables an efficient tensor-product convolution scheme for the computation of the Hartree potential using a collocation-type approximation via piecewise constant basis functions on a uniform n×n×n$n\times n\times n$ grid. Combined with the Richardson extrapolation, our approach exhibits O(h³)$O(h^3)$ convergence in the grid-size h=O(n^-1)$h=O(n^{-1})$. Moreover, this requires O(3rn+r³)$O(3\mathit{rn}+r^3)$ storage, where r   denotes the Tucker rank of the electron density with r=O(logn)$r=O(\log n)$, almost uniformly in n  . For example, calculations of the Coulomb matrix and the Hartree–Fock energy for the CH₄${\text{CH}}_4$ molecule, with a pseudopotential on the C atom, achieved accuracies of the order of 10^-6${10}^{-6}$ hartree with a grid-size n of several hundreds. Since the tensor-product convolution in 3D is performed via 1D convolution transforms, our scheme markedly outperforms the 3D-FFT in both the computing time and storage requirements.