Momentum-space Properties Research Articles

Observation of accurately designed bound states in the continuum in momentum space

Bound states in the continuum (BICs) in artificial photonic structures have received considerable attention since they offer unique methods for the extreme field localization and enhancement of light-matter interactions. Usually, the symmetry-protected BICs are located at high symmetric points, while the positions of accidental BICs achieved by tuning the parameters will appear at some points in momentum space. Up to now, to accurately design the position of the accidental BIC in momentum space is still a challenge. Here, we theoretically and experimentally demonstrate an accurately designed accidental BIC in a two-coupled-oscillator system consisting of bilayer gratings, where the optical response of each grating can be described by a single resonator model. By changing the interlayer distance between the gratings to tune the propagation phase shift related to wave vectors, the position of the accidental BIC can be arbitrarily controlled in momentum space. Moreover, we present a general method and rigorous numerical analyses for extracting the polarization vector fields to observe the topological properties of BICs from the polarization-resolved transmission spectra. Finally, an application of the highly efficient second harmonic generation assisted by quasi-BIC is demonstrated. Our work provides a straightforward strategy for manipulating BICs and studying their topological properties in momentum space.

Momentum-Space Properties for Ground and Rydberg States of Lithium Atom

Momentum-Space Properties for Ground and Rydberg States of Lithium Atom

System response to the initial energy-momentum tensor in relativistic heavy-ion collisions

The evolution of a relativistic heavy-ion collision is typically understood as a process that transmutes the initial geometry of the system into the final momentum distribution of observed hadrons, which can be described via a cumulant expansion of the initial distribution of energy density and is represented at leading order as the well-known eccentricity scaling of anisotropic flow. We summarize a proposed extension of this framework to include the contribution from initial momentum-space properties, as encoded in other components of the energy-momentum tensor. Numerical tests validate this proposal.

Three interaction energy scales in the single-layer high- Tc cuprate HgBa2CuO4+δ

The lamellar cuprate superconductors exhibit the highest ambient-pressure superconducting transition temperatures (T$_C$) and, after more than three decades of extraordinary research activity, continue to pose formidable scientific challenges. A major experimental obstacle has been to distinguish universal phenomena from materials- or technique-dependent ones. Angle-resolved photoemission spectroscopy (ARPES) measures momentum-dependent single-particle electronic excitations and has been invaluable in the endeavor to determine the anisotropic momentum-space properties of the cuprates. HgBa$_2$CuO$_{4+\delta}$ (Hg1201) is a single-CuO$_2$-layer cuprate with a particularly high optimal T$_C$ and a simple crystal structure; yet there exists little information from ARPES about the electronic properties of this model system. Here we present an ARPES study of doping-, temperature-, and momentum-dependent systematics of near-nodal dispersion anomalies in Hg1201. The data reveal a hierarchy of three distinct energy scales -a sub-gap low-energy kink, an intermediate-energy kink near 55 meV, and a peak-dip-hump structure. The first two features are attributed to the coupling of electrons to Ba-derived optical phonons and in-plane bond-stretching phonons, respectively. The nodal peak-dip-hump structure appears to have a common doping-dependence in several single-layer cuprates, and is interpreted as a manifestation of pseudogap physics at the node. These results establish several universal phenomena, both in terms of connecting multiple experimental techniques for a single material, and in terms of connecting comparable spectral features in multiple structurally similar cuprates.

Mind the angle

Magnetism Interplay between real- and momentum-space properties of materials can lead to exotic phenomena. Suzuki et al. studied electrical transport in the presence of a magnetic field in cerium-aluminum-germanium, a Weyl semimetal that also harbors magnetism (see the Perspective by Hassinger and Meng). As they varied the orientation of the applied field, they noticed spikes of resistivity sharply centered around the high symmetry axes of the material. The spikes were a consequence of the small overlap of Fermi surfaces—which “live” in momentum space—on either side of magnetic domain walls, which occur in real space. This extreme angular sensitivity may be useful in practical applications. Science , this issue p. [377][1]; see also p. [324][2] [1]: /lookup/doi/10.1126/science.aat0348 [2]: /lookup/doi/10.1126/science.aay3059

Group-theoretical classification of multipole order: Emergent responses and candidate materials

The multipole moment is an established concept of electrons in solids. Entanglement of spin, orbital, and sublattice degrees of freedom is described by the multipole moment, and spontaneous multipole order is a ubiquitous phenomenon in strongly correlated electron systems. In this paper, we present group-theoretical classification theory of multipole order in solids. Intriguing duality between the real space and momentum space properties is revealed for odd-parity multipole order which spontaneously breaks inversion symmetry. Electromagnetic responses in odd-parity multipole states are clarified on the basis of the classification theory. A direct relation between the multipole moment and the magnetoelectric effect, Edelstein effect, magnetopiezoelectric effect, and dichromatic electron transport is demonstrated. More than 110 odd-parity magnetic multipole materials are identified by the group-theoretical analysis. Combining the list of materials with the classification tables of multipole order, we predict emergent responses of the candidate materials.

Momentum-Space Topological Effects of Nonreciprocity

The connection between topology and nonreciprocity in photonic systems is reviewed. Topological properties such as Chern number, and momentum-space properties such as Berry phase and Berry connection, are used to explain back-scattering immune edge states and their topological protection. We consider several examples to illustrate the role of momentum-space topology on wave propagation, and discus recent magnet-less approaches.

PAREMD: A parallel program for the evaluation of momentum space properties of atoms and molecules

The present work describes a code for evaluating the electron momentum density (EMD), its moments and the associated Shannon information entropy for a multi-electron molecular system. The code works specifically for electronic wave functions obtained from traditional electronic structure packages such as GAMESS and GAUSSIAN. For the momentum space orbitals, the general expression for Gaussian basis sets in position space is analytically Fourier transformed to momentum space Gaussian basis functions. The molecular orbital coefficients of the wave function are taken as an input from the output file of the electronic structure calculation. The analytic expressions of EMD are evaluated over a fine grid and the accuracy of the code is verified by a normalization check and a numerical kinetic energy evaluation which is compared with the analytic kinetic energy given by the electronic structure package. Apart from electron momentum density, electron density in position space has also been integrated into this package. The program is written in C++ and is executed through a Shell script. It is also tuned for multicore machines with shared memory through OpenMP. The program has been tested for a variety of molecules and correlated methods such as CISD, Møller–Plesset second order (MP2) theory and density functional methods. For correlated methods, the PAREMD program uses natural spin orbitals as an input. The program has been benchmarked for a variety of Gaussian basis sets for different molecules showing a linear speedup on a parallel architecture. Program summaryProgram Title: PAREMDProgram Files doi:http://dx.doi.org/10.17632/gcr9gmh6zv.1Licensing provisions: GPLv3Programming language: C, CshExternal routines/libraries: GSL[1], BLAS[2,3], OpenMP[4], GAMESS[5,6] and GAUSSIAN[7].Nature of problem: Momentum space properties for a multi-electron system.Solution method: Analytic Fourier transformation of Gaussian basis in position space to get momentum space basis followed by EMD evaluation on spherical or Cartesian grids. A numerical integration procedure is implemented for evaluating moments of EMD and Shannon information entropy on a polar grid.[1] Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.[2] L. S. Blackford, J. Demmel, J. Dongarra, I. Duff, S. Hammarling, G. Henry, M. Heroux, L. Kaufman, A. Lumsdaine, A. Petitet, R. Pozo, K. Remington, R. C. Whaley, An Updated Set of Basic Linear Algebra Subprograms (BLAS), ACM Trans. Math. Soft., 28-2 (2002), pp. 135–151.[3] J. Dongarra, Basic Linear Algebra Subprograms Technical Forum Standard, International Journal of High Performance Applications and Supercomputing, 16(1) (2002), pp. 1–111, and International Journal of High Performance Applications and Supercomputing, 16(2) (2002), pp. 115–199.[4] OpenMP Architecture Review Board, ”OpenMP Application Program Interface, Version 4.5”, November 2015.[5] ”General Atomic and Molecular Electronic Structure System” M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S.J. Su, T.L. Windus, M. Dupuis, J.A. MontgomeryJ. Comput. Chem. 14 1347–1363, 1993.[6] “Advances in electronic structure theory: GAMESS a decade later M.S. Gordon, M.W. Schmidt pp. 1167–1189, in Theory and Applications of Computational Chemistry: the first forty years C.E. Dykstra, G. Frenking, K.S. Kim, G.E. Scuseria (editors), Elsevier, Amsterdam, 2005.[7] Gaussian 09, Revision C.01, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2016.

Curved momentum spaces from quantum groups with cosmological constant

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant Λ. In particular, the momentum space associated to the κ-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson–Lie group manifold parametrized by Λ. Such momentum space includes both the momenta associated to spacetime translations and the ‘hyperbolic’ momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the κ-Poincaré algebra are smoothly recovered in the limit Λ→0, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.

Lattice distortion inducing electron-hole pair condensation in two dimensional systems

A stability of the electron-hole pair condensation at low temperature has been addressed by involving the coupling of exciton to vibrational degrees of freedom in two-dimensional two-band f – c electron system. By mean of the unrestricted Hartree-Fock approximation, we find a formation of the insulating state typifying an excitonic condensate accompanied by a finite lattice distortion if the exciton-phonon coupling is large enough. As functions of temperature both excitonic condensation order parameter and lattice distortion behave in a same way which manifests the continuous transition in analogy to the superconductivity in the BCS theory. Inspecting to the microscopic properties in momentum space we strongly specify the BCS type of the excitonic condensation driven by the exciton-phonon interaction at low temperature.

Three-Dimensional Single Gyroid Photonic Crystals with a Mid-Infrared Bandgap

A gyroid structure is a distinct morphology that is triply periodic and consists of minimal isosurfaces containing no straight lines. We have designed and synthesized amorphous silicon (a-Si) mid-infrared gyroid photonic crystals that exhibit a complete bandgap in infrared spectroscopy measurements. Photonic crystals were synthesized by deposition of a-Si/Al2O3 coatings onto a sacrificial polymer scaffold defined by two-photon lithography. We observed a 100% reflectance at 7.5 μm for single gyroids with a unit cell size of 4.5 μm, in agreement with the photonic bandgap position predicted from full-wave electromagnetic simulations, whereas the observed reflection peak shifted to 8 μm for a 5.5 μm unit cell size. This approach represents a simulation-fabrication-characterization platform to realize three-dimensional gyroid photonic crystals with well-defined dimensions in real space and tailored properties in momentum space.

Structural investigations of fat fractals using small-angle scattering

Experimental small-angle scattering (SAS) data characterized, on a double logarithmic scale, by a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). Here we present a theoretical model for fat fractals and show how one can extract structural information about the underlying fractal using SAS method, for the well known fractals existing in the literature: Vicsek and Menger sponge. We calculate analytically the fractal structure factor and study its properties in momentum space. The models allow us to obtain the fractal dimension at each structural level inside the fractal, the number of particles inside the fractal and about the most common distances between the center of mass of the particles.

Small-angle scattering from fat fractals

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.

Magnetic order in the Hubbard model in three dimensions and the crossover to two dimensions

Systems of fermions described by the three-dimensional (3D) repulsive Hubbard model on a cubic lattice have recently attracted considerable attention due to their possible experimental realization via cold atoms in an optical lattice. Analytical and numerical results are limited away from half-filling. We study the ground state of the doped system from weak to intermediate interaction strengths within the generalized Hartree–Fock approximation. The exact solution to the self-consistent-field equations in the thermodynamic limit is obtained and the ground state is shown to exhibit antiferromagnetic order and incommensurate spin-density waves (SDW). At low interaction strengths, the SDW has unidirectional character with a leading wavevector along the 〈100〉-direction, and the system is metallic. As the interaction increases, the system undergoes a simultaneous structural and metal-to-insulator transition to a unidirectional SDW state along the 〈111〉-direction, with a different wavelength. We systematically determine the real and momentum space properties of these states. The crossover from 3D to two dimensions (2D) is then studied by varying the inter-plane hopping amplitude, which can be experimentally realized by tuning the distance between a stack of square-lattice layers. Detailed comparisons are made between the exact numerical results and predictions from the pairing model, a variational ansatz based on the pairing of spins in the vicinity of the Fermi surface. Most of the numerical results can be understood quantitatively from the ansatz, which provides a simple picture for the nature of the SDW states.

The quantum anomalous Hall effect in kagomé lattices

The quantum anomalous Hall (QAH) effect in kagomé lattices is investigated in the presence of both Rashba spin–orbit coupling and an exchange field. In addition to the gap at the Dirac points as found in graphene, a new topological energy gap is opened at the Γ point. With the Fermi energy lying in the first gap, the Chern number \U0001d49e = 2 as in graphene, whereas with it lying in the second one, \U0001d49e = 1. The distribution of Berry curvature is obtained to reveal the nontrivial topological properties in momentum space. For stripes with ‘armchair’ and ‘zigzag’ edges, the topological characteristics of gapless edge states on the genus g = 2 Riemann surface are studied. The obtained nonzero winding numbers also demonstrate the QAH effect.

Momentum distributions in time-dependent density-functional theory: Product-phase approximation for nonsequential double ionization in strong laser fields

We investigate the possibility to deduce momentum space properties from time-dependent density functional calculations. Electron and ion momentum distributions after double ionization of a model helium atom in a strong few-cycle laser pulse are studied. We show that, in this case, the choice of suitable functionals for the observables is considerably more important than the choice of the correlation potential in the time-dependent Kohn-Sham equations. By comparison with the solution of the time-dependent Schroedinger equation, the insufficiency of functionals neglecting electron correlation is demonstrated. We construct a functional of the Kohn-Sham orbitals, which in principle yields the exact momentum distributions of the electrons and the ion. The product-phase approximation is introduced, which reduces the problem of approximating this functional significantly.

Elastic electron scattering at high momentum transfer: A possible new analytic tool

Recently it has become clear that for large-angle elastic scattering of keV electrons, the energy of the scattered electrons is reduced. This energy loss depends on the mass of the atom it scattered from, as well as its momentum. Hence, we can study the atomic composition of samples as well as their vibrational properties in momentum space. Here we present some examples of spectra obtained by this technique and discuss possible applications in the field of scanning electron microscopy.

Exact Kohn-Sham versus Hartree-Fock in momentum space: Examples of two-fermion systems

The question of how density functional theory (DFT) compares with Hartree-Fock (HF) for the computation of momentum-space properties is addressed in relation to systems for which (near) exact Kohn-Sham (KS) and HF one-electron matrices are known. This makes it possible to objectively compare HF and exact KS and hence to assess the potential of DFT for momentum-space studies. The systems considered are the Moshinsky [Am. J. Phys. 36, 52 (1968)] atom, Hooke's atom, and light two-electron ions, for which expressions for correlated density matrices or momentum densities have been derived in closed form. The results obtained show that it is necessary to make a distinction between true and approximate DFTs.

Electron momentum distribution in multiply-ionized atoms as a function of atomic number and degree of ionization

We present momentum-space properties of multiply ionized atoms as a function of atomic number Z and the degree of ionization of the atom. In particular, we have calculated the Compton profiles of all possible electronic configurations of He, Li, Be, B and N atoms as they are progressively ionized with the outer-shell electrons being stripped off. The values of the Compton profiles presented here can be used to deduce doubly differential cross-sections of variously ionized atoms colliding with other atoms and ions. The single-electron radial wavefunctions were obtained from the Hartree–Fock atomic model. Compton profiles of neutral atoms, available in the literature, are in excellent agreement with the present calculation.

Spectroscopic imaging STM studies of high- TC superconductivity

Since its discovery in 1986, understanding both the ground state and excited states of high temperature superconductivity (HTSC) has been one of the main challenges in modern condensed matter physics. One remarkable observation in HTSC cuprates is that the superconductivity appears as the localized electrons of a Mott insulator become itinerant due to carrier doping. Momentum sensitive probes like Angle Resolved Photo-Emission Spectroscopy (ARPES) and Inelastic Neutron Scattering (INS) have played important roles in mapping the momentum-space properties of the cuprate electronic structure. However, considering that cuprate superconductivity develops from atomically localized electrons and shows nanoscale disorder, a purely momentum dependent description is far from sufficient. Here, we describe novel spectroscopic imaging scanning tunneling microscopy (SI-STM) studies, which can probe the real-space electronic structure at the nano-scale, along with momentum-space electronic structure. We discuss a series of recent results from SI-STM studies of Bi 2Sr 2CaCu 2O 8+ x and Ca 2- x Na x CuO 2Cl 2.