Nodal domain statistics for quantum maps, percolation, and stochastic Loewner evolution

We develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting statistical properties of the nodal domains is demonstrated for perturbed cat maps and supports the use of percolation theory to describe the wave functions of general Hamiltonian systems. We also demonstrate that the nodal domains of the perturbed cat maps obey the Cardy crossing formula and find evidence that the boundaries of the nodal domains are described by stochastic Loewner evolution with diffusion constant close to the expected value of 6, suggesting that quantum chaotic wave functions may exhibit conformal invariance in the semiclassical limit.     

XAuBi (X=Eu, Ba)拓扑nodal chain半金属的第一性原理研究

由于在磁性材料体系中缺失时间反演对称性,导致nodal chain被破坏,所以nodal chain通常存在于非磁材料中。但是,磁性材料EuAuBi是与常规磁性材料不同。本工作以第一性原理计算为研究方法,预言了在不考虑自旋轨道相互作用时,磁性材料EuAuBi体系为新型拓扑nodal chain半金属;当考虑自旋轨道耦合时,EuAuBi会退化为外尔半金属。对于非磁材料BaAuBi来说,在不考虑自旋轨道相互作用时,它同样是一种拓扑nodal chain半金属;当考虑自旋轨道相互作用时,由于C3旋转对称性的存在,BaAuBi会退化为狄拉克半金属。在XAuBi (X=Eu, Ba)中发现nodal chain半金属,会促进对六角材料的拓扑性质研究以及开拓其新实际应用领域。     

Arithmetic,Zeros, and Nodal Domains on the Sphere

We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelöf hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelöf.     

Topological Nodal Line Semimetal in Non-Centrosymmetric PbTaS_2

Topological semimetals are a new type of matter with one-dimensional Fermi lines or zero-dimensional Weyl or Dirac points in momentum space. Here using first-principles calculations, we find that the non-centrosymmetric PbTaS2 is a topological nodal line semimetal. In the absence of spin-orbit coupling(SOC), one band inversion happens around a high symmetrical H point, which leads to forming a nodal line. The nodal line is robust and protected against gap opening by mirror reflection symmetry even with the inclusion of strong SOC. In addition, it also hosts exotic drumhead surface states either inside or outside the projected nodal ring depending on surface termination. The robust bulk nodal lines and drumhead-like surface states with SOC in PbTaS_2 make it a potential candidate material for exploring the freakish properties of the topological nodal line fermions in condensed matter systems.     

Kinks, nodal bilayer splitting, and interband scattering in YBa2Cu3O(6+x)

We apply the new-generation angle-resolved photoemission spectroscopy methodology to the most widely studied cuprate superconductor YBa2Cu3O(6+x). Considering the nodal direction, we found noticeable renormalization effects known as kinks both in the quasiparticle dispersion and scattering rate, the bilayer splitting, and evidence for strong interband scattering--all the characteristic features of the nodal quasiparticles detected earlier in Bi2Sr2CaCu2O(8+delta). The typical energy scale and the doping dependence of the kinks clearly point to their intimate relation with the spin-1 resonance seen in the neutron scattering experiments. Our findings strongly suggest a universality of the electron dynamics in the bilayer superconducting cuprates and a dominating role of the spin fluctuations in the formation of the quasiparticles along the nodal direction.     

基于平面通量展开的三维Pin by Pin输运SP3计算

基于Pin by Pin输运SP3计算是下一代反应堆物理计算方法中一个重要的研究方向.节块法一般采用高阶通量和中子源展开以适应较大的节块尺寸,因此对于Pin by Pin计算来说效率偏低.由于反应堆堆芯不均匀性更多发生在径向,本文提出一种径向基于二维平面通量展开结合轴向常规节块法的综合方法进行三维Pin by Pin输...     

Nodal line optimization and its application to violin top plate design

In the literature, most problems of structural vibration have been formulated to adjust a specific natural frequency: for example, to maximize the first natural frequency. In musical instruments like a violin; however, mode shapes are equally important because they are related to sound quality in the way that natural frequencies are related to the octave. The shapes of nodal lines, which represent the natural mode shapes, are generally known to have a unique feature for good violins. Among the few studies on mode shape optimization, one typical study addresses the optimization of nodal point location for reducing vibration in a one-dimensional beam structure. However, nodal line optimization, which is required in violin plate design, has not yet been considered. In this paper, the central idea of controlling the shape of the nodal lines is proposed and then applied to violin top plate design. Finite element model for a violin top plate was constructed using shell elements. Then, optimization was performed to minimize the square sum of the displacement of selected nodes located along the target nodal lines by varying the thicknesses of the top plate. We conducted nodal line optimization for the second and the fifth modes together at the same time, and the results showed that the nodal lines obtained match well with the target nodal lines. The information on plate thickness distribution from nodal line optimization would be valuable for tailored trimming of a violin top plate for the given performances.     

Diffusion on curved, periodic surfaces

We present a simulation algorithm for a diffusion on a curved surface given by the equation phi(r)=0. The algorithm is tested against analytical results known for diffusion on a cylinder and a sphere, and applied to the diffusion on the P, D, and G periodic nodal surfaces. It should find application in an interpretation of two-dimensional exchange NMR spectroscopy data of diffusion on biological membranes.     

Nodal deficiency,spectral flow,and the Dirichlet-to-Neumann map

Letters in Mathematical Physics - It has been recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the...     

基于非线性迭代的压水堆瞬态计算程序开发

采用目前工业上成熟的非线性迭代计算策略,基于两群粗网有限差分方法和多群UNM节块方法,开发了针对压水堆工况的三维瞬态扩散计算程序。UNM方法采用解析基函数作为基函数,通过方程变换解决了解析节块法在临界节块计算不稳定的问题,提高了计算精度。热工计算采用单通道模型和燃料棒一维导热模型,相比目前堆芯在用的经验关系式方法,该模型可以更加准确地计算燃料棒温度分布。采用基于横向积分方程的三节块方法,可以有效减轻控制棒尖齿效应对瞬态计算的影响。为测试程序性能,采用NEACRP等基准算例对程序进行了校验。数值结果表明,开发的程序计算结果正确,适用于压水堆堆芯瞬态过程的模拟。     

Nodal Topological Phases in s-wave Superfluid of Ultracold Fermionic Gases

The gapless Weyl superfluid has been widely studied in the three-dimensional ultracold fermionic superfluid.In contrast to Weyl superfluid, there exists another kind of gapless superfluid with topologically protected nodal lines,which can be regarded as the superfluid counterpart of nodal line semimetal in the condensed matter physics, just as Weyl superfluid with Weyl semimetal. In this paper we study the ground states of the cold fermionic gases in cubic optical lattices with one-dimensional spin-orbit coupling and transverse Zeeman field and map out the topological phase diagram of the system. We demonstrate that in addition to a fully gapped topologically trivial phase, some different nodal line superfluid phases appear when the Zeeman field is adjusted. The presence of topologically stable nodal lines implies the dispersionless zero-energy flat band in a finite region of the surface Brillouin zone. Experimentally these nodal line superfluid states can be detected via the momentum-resolved radio-frequency spectroscopy. The nodal line topological superfluid provide fertile grounds for exploring exotic quantum matters in the context of ultracold atoms.     

Can one count the shape of a drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study. This sequence is expressed as a trace formula, which consists of a smooth (Weyl-like) part which depends on global geometrical parameters, and a fluctuating part, which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical content of the nodal sequence is thus explicitly revealed.     

On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach

In finite element methods that are based on position and slope coordinates, a representation of axial and bending deformation by means of an elastic line approach has become popular. Such beam and plate formulations based on the so-called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories. Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect. A correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived. The strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the St. Venant-Kirchhoff material: therefore, the strain energy is derived for the latter case and compared to classical nonlinear rod theories. The error in the original absolute nodal coordinate formulation is documented by numerical examples. The numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach, while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications. The numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre-curved static and dynamic problems, including buckling and eigenvalue analysis. The resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy-momentum conservation.     

Weyl and Nodal Ring Magnons in Three-Dimensional Honeycomb Lattices

We study the topological properties of magnon excitations in a wide class of three-dimensional(3 D) honeycomb lattices with ferromagnetic ground states. It is found that they host nodal ring magnon excitations. These rings locate on the same plane in the momentum space. The nodal ring degeneracy can be lifted by the DzyaloshinskiiMoriya interactions to form two Weyl points with opposite charges. We explicitly discuss these physics in the simplest 3 D honeycomb lattice and the hyerhoneycomb lattice, and show drumhead and arc surface states in the nodal ring and Weyl phases, respectively, due to the bulk-boundary correspondence.     

Fermion nodes

The knowledge of the nodes of the many-fermion wave function would enable exact calculation of the properties of fermion systems by Monte Carlo methods. It is proved that fermion nodal regions have a tiling property, there is only one distinct kind of nodal region. All others are related to it by permutational symmetry. For some free particle systems, it is shown that there are only two nodal regions. An explicit form for the nodes of the many-fermion density matrix would enable exact simulations to be carried out at finite temperature. In the high-temperature limit, its nodes are related to Voronoi polyhedra. Twodimensional cross sections of nodes are depicted. General computable families of fermion wave functions and density matrices are discussed.     

广义振子强度矩阵元节线的规律

本文分析了从初态nl到末态n＇l＇(包括连续态)的广义振子强度矩阵元节线的规律(个数和位置)。利用我们提出的分区标度的方法,从氢原子广义振子强度节线的规律出发,通过简单的标度得到了多电子原子广义振子强度节线在高动量转移区的规津。在低动量转移区,还讨论了由于初态和末态量子数亏损的差Δμ对广义振子强度节线所产生的影响;Δμ>0的节线几何结构与Δμ<0的节线几何结构完全不相同。 关键词：     

Free vibration of two identical circular plates coupled with bounded fluid

This paper examines the hydroelastic vibration of two identical circular plates coupled with a bounded fluid. An analytical method based on the finite Fourier-Bessel series expansion and the Rayleigh-Ritz method is suggested. In the theory, it is assumed that a rigid cylindrical container is filled with the ideal fluid and the two plates are clamped along the container edges. The proposed method is verified by finite element analysis using commercial software with excellent accuracy. Two transverse vibration modes, in-phase and out-of-phase, are observed alternately in the fluid-coupled system when the number of nodal circles increases for the fixed nodal diameter. It is found that the normalized natural frequency of the system monotonically increases with an increase in the number of nodal diameters and circles by virtue of a decrease in relative hydrodynamic mass.     

Signatures of quantum chaos in the nodal points and streamlines in electron transport through billiards

Streamlines and the distributions of nodal points are used as signatures of chaos in coherent electron transport through three types of billiards: Sinai, Bunimovich, and rectangular. Numerical averaged distribution functions of the nearest distances between nodal points are presented. We find the same form for the Sinai and Bunimovich billiards and suggest that there is a universal form that can be used as a signature of quantum chaos for electron transport in open billiards. The universal distribution function is found to be insensitive to the way the averaging is performed (over the positions of the leads, over an energy interval with a few conductance fluctuations, or both). The integrable rectangu-lar billiard, on the other hand, displays a nonuniversal distribution with a central peak related to partial order of nodal points for the case of symmetric attachment of the leads. However, cases with asymmetric leads tend to the universal form. Also, it is shown how nodal points in the rectangular billiard can lead to “channeling of quantum flows,” while disorder in the nodal points in the Sinai billiard gives rise to unstable irregular behavior of the flow. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 6, 398–404 (25 September 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Accurate vibration analysis of thick, cracked rectangular plates

This work applies the Ritz method to accurately determine the frequencies and nodal patterns of thick, cracked rectangular plates analyzed using Mindlin plate theory. Two types of cracked configuration are considered, namely, side crack and internal crack. To enhance the capabilities of the Ritz method in dealing with cracked plates, new sets of admissible functions are proposed to represent the behaviors of true solutions along the crack. The proposed admissible functions appropriately describe the stress singularity behaviors around a crack tip and the discontinuities of transverse displacement and bending rotations across the crack. The present solutions monotonically converge to the exact frequencies as upper bounds when the number of admissible functions increases. The validity and accuracy of the present solutions are confirmed through comprehensive convergence studies and comparison with the published results based on the classical thin plate theory. The proposed approach is further employed to investigate the effects of the length, location, and orientation of crack on frequencies and nodal patterns of simply supported and cantilevered cracked rectangular plates. The results shown are the first ones available in the published literature.     

Polymorphic nodal elements and their application in discontinuous Galerkin methods

In this work, we discuss two different but related aspects of the development of efficient discontinuous Galerkin methods on hybrid element grids for the computational modeling of gas dynamics in complex geometries or with adapted grids. In the first part, a recursive construction of different nodal sets for hp finite elements is presented. They share the property that the nodes along the sides of the two-dimensional elements and along the edges of the three-dimensional elements are the Legendre–Gauss–Lobatto points. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. In the second part, these nodal elements are applied within the modal discontinuous Galerkin framework. We still use a modal based formulation, but introduce a nodal based integration technique to reduce computational cost in the spirit of pseudospectral methods. We illustrate the performance of the scheme on several large scale applications and discuss its use in a recently developed space-time expansion discontinuous Galerkin scheme.