Superfluid to Mott insulator transition in one, two, and three dimensions

No Heading We have created one-, two-, and three-dimensional quantum gases and study the superfluid to Mott insulator transition. Measurements of the transition using Bragg spectroscopy show that the excitation spectra of the low-dimensional superfluids differ significantly from the three-dimensional case.PACS numbers: 05.30.Jp, 03.75.Kk, 03.75.Lm, 73.43.Nq     

Electron-Doped Cuprates: A “Stripe-Free” Link between Hole-Doped Cuprates and Heavy-Fermion Physics?

Recent evidence suggests that in electron-doped cuprates, superconductivity arises near a quantum critical point associated with Mott gap collapse. A similar calculation for hole-doped cuprates is complicated by the presence of competing instabilities.     

Raman spectroscopy of polyconjugated molecules and materials: confinement effect in one and two dimensions

The effect of the confinement of pi electrons in one- and two-dimensional domains is illustrated with several examples ranging from linear polyene chains to planar molecules with honeycomb structure. Theoretical computations and specific Raman experiments on molecular materials demonstrate that a molecular approach provides a unified key to the interpretation of the Raman response both of linear polyconjugated polymers (polyacetylene) and of nanostructured graphitic materials.     

Correlation between Superconducting Gap and Pseudogap in High-T c Cuprates

We studied the low-temperature energy gap 2 ₀ on Bi ₂ Sr ₂ CaCu ₂ o ₈₊ (Bi2212) and La _2–x Sr _x CuCO ₄ (La214) systematically over a wide range of doping level p using STS, break junction tunneling spectroscopy, Raman scattering and low-T electronic specific heat data. We have also studied the electronic specific heat of La214 in the normal state at T > T _c , and confirmed that pseudogap behavior appears at around T*, below which the in-plane resistivity and magnetic susceptibility tend to be slightly suppressed. Similar suppression appears in and of Bi2212 below the onset temperature of pseudogap T*. It is pointed out in the present study that 2 ₀ is closely related to T* in both Bi2212 and La214 systems; T* 2 ₀ /4.3k _B . It is also pointed out that 2 ₀ is in almost linear proportion to k _B T _max ( T*), where T _max is the temperature exhibiting a broad peak in –T curves and k _B T _max can be considered to give a measure of the effective antiferromagnetic exchange energy J _eff. The factors in 2 ₀ k _B T _max J _eff are 1 for La214 and 2 for Bi2212, respectively. We also report that in both Bi22l2 and La214 systems T _c roughly scales with p ₀ except in highly doped samples, where T _c 2 ₀ .     

Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions

The spatial resolution of eigenfunctions of Sturm–Liouville equations in one-dimension is frequently measured by examining the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster like O(1/N ²) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In this work an alternative measure, ℓ, is proposed for estimating the characteristic length scale of eigenfunctions of Sturm–Liouville equations that does not rely on knowledge of the roots. It is first shown that ℓ is a reasonable measure for evaluating the eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing the minimum distance between Fekete points to ℓ shows that ℓ is a reasonably good measure of the characteristic length scale in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle proposed by Taylor MA, Wingate BA [(2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement.     

Micromagnetism in two dimensions

The two-dimensional approximation to the magnetostatic problem in solving the equilibrium equations of micromagnetism is reviewed. The ripple equation leads to the effective demagnetizing field, which can be determined by means of high sensitive susceptibility measurements. A method for characterizing the quality of ferromagnetic thin film samples by conventional susceptibility measurement is described.     

Unsteady-state radiative heat transfer between two opaque bodies of finite dimensions

The unsteady-state radiative interaction of two opaque gray bodies according to the Stefan-Boltzmann law is considered. The problem is reduced to a non-linear Volterra integral equation for the net radiation density. A solution is obtained for a linear approximation. The general case and the short-time behavior are discussed.     

Dispersion and pollution of the FEM solution for the Helmholtz equation in one,two and three dimensions

For high wave numbers, the Helmholtz equation suffers the so‐called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p‐method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual‐free bubble [3,4] is effective in one dimension but not in higher dimensions. The least‐square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss‐cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd.     

Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram

We propose to use a mask with a nonredundant array (NRA) of multiple apertures to measure spatial coherence in two dimensions. The spatial distribution of the apertures in the mask is made in such a way that we obtain a quasi-uniform sampling in the coherence domain. The spatial coherence is obtained by Fourier transform of the interferogram generated by the mask when it is illuminated by the light field under analysis.     

Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions

In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method (DGMPM) on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems, a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multidimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant number for a given space discretization. Although the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.     

Dispersion analysis of the gradient weighted finite element method for acoustic problems in one,two, and three dimensions

This paper reports a detailed analysis on the numerical dispersion error in solving one-, two-, and three-dimensional acoustic problems governed by the Helmholtz equation using the gradient weighted finite element method (GW-FEM) in comparison with the standard FEM and the modified methods presented in the literatures. The discretized system equations derived based on the gradient weighted operation corresponding to the considered method are first briefed. The discrete dispersion relationships relating the exact and numerical wave numbers defined in different dimensions are then formulated, which will be further used to investigate the dispersion effect mainly caused by the approximation of field variables. The influence of nondimensional wave number and wave propagation angle on the dispersion error is detailedly studied. Comparisons are made with the classical FEM and high-performance algorithms. Results of both theoretical and numerical experiments show that the present method can effectively reduce the pollution effect in computational acoustics owning to its crucial effectiveness in handing the dispersion error in the discrete numerical model.     

Hybrid numerical method for solution of the radiative transfer equation in one, two, or three dimensions

A hybrid method is presented by which Monte Carlo (MC) techniques are combined with an iterative relaxation algorithm to solve the radiative transfer equation in arbitrary one-, two-, or three-dimensional optical environments. The optical environments are first divided into contiguous subregions, or elements. MC techniques are employed to determine the optical response function of each type of element. The elements are combined, and relaxation techniques are used to determine simultaneously the radiance field on the boundary and throughout the interior of the modeled environment. One-dimensional results compare well with a standard radiative transfer model. The light field beneath and adjacent to a long barge is modeled in two dimensions and displayed. Ramifications for underwater video imaging are discussed. The hybrid model is currently capable of providing estimates of the underwater light field needed to expedite inspection of ship hulls and port facilities.     

Unsteady Stokes' flow in two dimensions

Some simple problems are considered which indicate how transient effects lead to the development of slow viscous flows. Explicit solutions are obtained for situations when a source, rotlet or stokeslet is impulsively introduced at a particular time, and the manner by which such flows, which are initially harmonic, are transformed into those satisfying the biharmonic equation is clearly displayed. Conclusions regarding the formation of separated regions are presented.     

Fractional Fourier transforms in two dimensions

We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle S1) in the one-dimensional case to U(2) (the four-parameter group of 2 x 2 unitary matrices) in the two-dimensional case [rather than simply U(1) x U(1)]. Our treatment clarifies the situation in the N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2 = S1 x S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 coordinates. FrFT's corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite-Gaussian beams and the Laguerre-Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, R) first-order system is essentially a FrFT.     

Superfluid flow dissipation in two dimensions

A model is presented describing the onset of nonlinear dissipation in two-dimensional helium films for allT _KT, whereT _KT is the Kosterlitz-Thouless transition temperature. The superfluid velocity is included in the renormalization equations of Kosterlitz and Thousless, and it is shown that the model predicts a characteristic velocity for a given temperature, film thickness, and frequency where onset occurs. No critical velocity is predicted except atT=0. The model neglects the effects of prinning, and preliminary measurements seem to substantiate this assumption.     

Electron-Lattice Coupling in Manganites and Cuprates

The lattice effects of charge localization in manganites and cuprates arediscussed from the local structural point of view. Pulsed neutron atomicpair-density function (PDF) analysis indicates that in La_1–x Sr _x MnO₃ the realstructure deviates from the average crystal structure, and the localJahn–Teller (JT) distortion of Mn³⁺O₆ octahedra remains even in themetallic phase. This leads to a concept of the critical ionic size factorfor the formation of JT polarons. The colossal magnetoresistance (CMR)phenomena are observed in the crossover region from localized to delocalizedcharge states. This concept also explains the asymmetry of the phase diagramwhich is difficult to understand in the current band picture. Similarconsideration may help understand the lattice effects in superconductingcuprates.     

Polaron Dynamics and Bipolarons in Cuprates

The Fröhlich electron–phonon interaction in cuprates, manganites, and other charge-transfer ionic Mott insulators is much stronger than any magnetic interaction. The polaron shift due to the Fröhlich interaction, which is about 1 eV, suggests that carriers in those systems are small (bi)polarons at all temperatures and doping levels. We show both analytically and numerically that (bi)polarons exist in the itinerant Bloch states at temperatures below the characteristic phonon frequency no matter which values the parameters of the system take. The small Fröhlich polaron has spectral features compatible with the single-particle tunneling and photoemission in cuprates. Whereas the band energy dispersion of intersite bipolarons is responsible for the d-wave symmetry of the condensate wave-function, the single-particle excitation spectrum is s-like in agreement with the tunneling data. Two different energy scales in Giaver tunneling and Andreev reflection experiments in cuprates can be understood in the framework of the bipolaron theory as well.     

Low Temperature Anharmonicity and Superconductivity in Cuprates

Temperature-dependent X-ray absorption spectra of the hole-doped La_2?x Sr _x CuO₄ and electron-doped Nd_2?x Ce _x CuO_4?δ high-temperature superconductors were investigated above the Cu K absorption edge. We observed strong anharmonicity in the superconductive CuO₂ plane. For x=0.15 it was shown that part of oxygen ions oscillate in a double-well potential and their vibrations are correlated with the local electron (hole) pair transfer. We suppose that at a low temperature the phase coherence of the local pair movement is determined by the peculiarities of the perovskite-like structure which includes the stiff CuO _n (n=4,6) complexes combined by collective rotational and breathing modes.