Arbitrary Symmetry Research Articles

Hydrodynamic equations for a U(N) invariant superfluid

Abstract In this paper, we develop an appropriate set of hydrodynamic equations for a U(N) invariant superfluid that couple the dynamics of superflow and magnetization. In the special case when both the superfluid and normal velocities are zero, the hydrodynamic equations reduce to a generalized version of Landau-Lifshitz equation for ferromagnetism with U(N) symmetry. When both velocities are non-zero, there appears couplings between the superflow and magnetization dynamics, and the superfluid velocity no longer satisfies the irrotational condition. On the other hand, the magnitude of magnetization is no longer a constant of motion as was the case for the standard Landau-Lifshitz theory. In comparison with the simple superfluid, the first and second sounds are modified by a non-zero magnetization through various thermodynamic functions. For U(2) invariant superfluid, we get both (zero-) sound wave and a spin wave at zero temperature. It is found that the dispersion of spin wave is always quadratic, which is consistent with microscopic analysis. In the Appendix, we
show that the hydrodynamic equation for a U(N) invariant superfluid can be obtained from the general hydrodynamic equation with arbitrary internal symmetries.

Tensorial harmonic bases of arbitrary order with applications in elasticity, elastoviscoplasticity and texture-based modeling

In continuum mechanics, one regularly encounters higher-order tensors that require tensorial bases for theoretical or numerical calculations. By using results from [Formula: see text] representation theory, we present a method to derive tensorial harmonic bases which unifies existing approaches in one framework. Applying this convention to symmetric second-order tensors results in a second-order harmonic basis which, for example, simplifies the depiction and numerical handling of stiffness tensors for most common material symmetries and reduces the computational effort involved in implementations of incompressible elastoviscoplastic material laws. The same convention can be applied to texture analysis to yield tensorial texture coefficients for both polycrystals and fiber-reinforced composites. Rotations of higher-order tensors are particularly efficient in harmonic bases as both material and index symmetries can be exploited. While special attention is given here to the examples of small-strain material laws and texture analysis, the framework is entirely general and can be used to simplify calculations in other physical contexts involving tensors of arbitrary order and symmetry. A Python implementation of the harmonic basis convention defined in this work is available as a Supplementary Material.

Nano and micro-structural complexity of nematic liquid crystal configurations

Of our interest are frustration-driven pattern generating mechanisms in systems which in bulk equilibrium display spatially homogeneous long-range orientational order in absence of perturbations. As testbed material, we select thermotropic nematic liquid crystals. In bulk, they exhibit weakly discontinuous order–disorder phase transformation on varying temperature where the ordered nematic phase features spatially uniform axial order along an arbitrary symmetry breaking direction. However, due to continuous symmetry breaking (CSB) the established order is extremely susceptible to various perturbations which are in real systems in general always present. We theoretically illustrate how diverse complex patterns could be excited. Particularly intriguing configurations could appear if topological defects are present that could be generated via CSB. Our analysis is based on a relatively simple Lebwohl-Lasher-type model in which we could get analytical insight into phenomena of our interest. Using it we illustrate history dependent early stage isotropic-nematic phase evolution and final patterns in presence of “impurities” (e.g., nanoparticles). We show how characteristic effective interaction characteristics predict qualitatively different emerging patterns. Our analysis is based on CSB which is ubiquitous in nature. Consequently, demonstrated mechanisms are expected to manifest also in other condensed matter systems whose ordered phase is formed via CSB. We illustrate how kinetics and impurities could impact key structural properties of the systems of our interest.

Study of the anisotropy of α-<sup>33</sup>S single crystal thermal expansion

A technique for arbitrary symmetry crystals unit cell parameters refining on modern single-crystal diffractometers is described. The technique is based on the 2D detector position calibration. The elementary orthorhombic unit cell parameters of the α-33S single crystal have been refined. The anisotropy of parameters changes in the range of 90–350 K has been studied. It is shown that the relative increase in parameter c is 6.4%. The obtained dependences are approximated by second–third degree polynomials. The absolute increase in cell volume is 138.4 A3, and the relative increase is 4.3%. The temperature dependencies of the thermal expansion tensor elements has been refined. The coefficients of α-33S thermal expansion tensor at the room temperature are: α11 = 15.35 × 10–5, α22 = 8.56 × 10–5, α33 = 9.12 × 10–5 К–1.

Indexing neutron transmission spectra of a rotating crystal.

Neutron time-of-flight transmission spectra of mosaic crystals contain Bragg dips, i.e., minima at wavelengths corresponding to diffraction reflections. The positions of the dips are used for investigating crystal lattices. By rotating the sample around a fixed axis and recording a spectrum at each rotation step, the intensity of the transmitted beam is obtained as a function of the rotation angle and wavelength. The questions addressed in this article concern the determination of lattice parameters and orientations of centrosymmetric crystals from such data. It is shown that if the axis of sample rotation is inclined to the beam direction, the reflection positions unambiguously determine reciprocal-lattice vectors, which is not the case when the axis is perpendicular to the beam. Having a set of such vectors, one can compute the crystal orientation or lattice parameters using existing indexing software. The considerations are applicable to arbitrary Laue symmetry. The work contributes to the automation of the analysis of diffraction data obtained in the neutron imaging mode.

A hyper-Kähler metric on the moduli spaces of monopoles with arbitrary symmetry breaking

We construct the hyper-Kähler moduli space of framed monopoles over R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} for any connected, simply connected, compact, semisimple Lie group and arbitrary mass and charge, and hence arbitrary symmetry breaking. In order to do so, we define a configuration space of pairs with appropriate asymptotic conditions and perform an infinite-dimensional quotient construction. We make use of the b and scattering calculuses to study the relevant differential operators.

Bending and twisting rigidities of 2D materials

A novel torus-based surface parametrization is proposed and used to evaluate the bending and twisting rigidities of 2D materials of arbitrary symmetries. A generalized constitutive model is used to capture the effects of simultaneous bending and twisting curvatures on the strain energy density of a 2D material. The flexural rigidities are evaluated by subjecting the 2D material specimens to different curvature states, including simultaneous bending and twisting, evaluating the corresponding strain energy densities and curve fitting the constitutive model to the data. The proposed methodology is independent of the method used to evaluate the strain energy of the 2D materials, e.g., density functional theory or classical interatomic potentials. The flexural rigidities of graphene, molybdenum disulfide, and black phosphorus are evaluated and compared to those published in the literature. The directional dependence of the flexural rigidities is investigated. Dimensionless coefficients are presented for quantifying the bidirectional bending and bending-twisting coupling effects in 2D materials. Lastly, this study includes an analysis of the flexural anisotropy and flexure-thickness distension coefficients of black phosphorus.

Dualities among massive, partially massless and shift symmetric fields on (A)dS

We catalog all the electromagnetic-like dualities that exist between free dynamical bosonic fields of arbitrary symmetry type and mass on (anti-) de Sitter space in all dimensions, including dualities among the partially massless and shift symmetric fields. This generalizes to all these field types the well known fact that a massless p-form is dual to a massless (D − p − 2)-form in D spacetime dimensions. In the process, we describe the structure of the Weyl modules (the spaces of local operators linear in the fields and their derivative relations) for all the massive, partially massless and shift symmetric fields.

Extending intergranular normal-stress distributions using symmetries of linear-elastic polycrystalline materials

Intergranular normal stresses (INS) are critical in the initiation and evolution of grain boundary damage in polycrystalline materials. To model the effects of such microstructural damage on a macroscopic scale, knowledge of INS is usually required statistically at each representative volume element subjected to various loading conditions. However, calculating INS distributions for different stress states can be cumbersome and time-consuming. This study proposes a new method to extend the existing INS distributions to arbitrary loading conditions using the symmetries of a polycrystalline material composed of randomly oriented linear-elastic grains with arbitrary lattice symmetry. The method relies on a fact that INS distributions can be accurately reproduced from the first (typically) ten statistical moments, which depend trivially on just three stress invariants and a few material invariants due to assumed isotropy and material linearity of the polycrystalline model. While these material invariants are complex averages, they can be extracted numerically from a few existing INS distributions and tabulated for later use. Practically, only three such INS distributions at properly selected loadings are required to provide all relevant material invariants for the first 11 statistical moments, which can then be used to reconstruct the INS distribution for arbitrary loading conditions. The proposed approach is demonstrated to be accurate and feasible for an arbitrarily selected linear-elastic material under various loading conditions.

Macroscopic Polarization from Nonlinear Gradient Couplings.

We show that a lattice mode of arbitrary symmetry induces a well-defined macroscopic polarization at first order in the momentum and second order in the amplitude. We identify a symmetric flexoelectric-like contribution, which is sensitive to both the electrical and mechanical boundary conditions, and an antisymmetric Dzialoshinskii-Moriya-like term, which is unaffected by either. We develop the first-principles methodology to compute the relevant coupling tensors in an arbitrary crystal, which we illustrate with the example of the antiferrodistortive order parameter in SrTiO_{3}.

Optical second harmonic generation in anisotropic multilayers with complete multireflection of linear and nonlinear waves using ♯SHAARP.ml package

Optical second harmonic generation (SHG) is a nonlinear optical effect widely used for nonlinear optical microscopy and laser frequency conversion. Closed-form analytical solution of the nonlinear optical responses is essential for evaluating materials whose optical properties are unknown a priori. A recent open-source code, ♯SHAARP.si, can provide such closed form solutions for crystals with arbitrary symmetries, orientations, and anisotropic properties at a single interface. However, optical components are often in the form of slabs, thin films on substrates, and multilayer heterostructures with multiple reflections of both the fundamental and up to ten different SHG waves at each interface, adding significant complexity. Many approximations have therefore been employed in the existing analytical approaches, such as slowly varying approximation, weak reflection of the nonlinear polarization, transparent medium, high crystallographic symmetry, Kleinman symmetry, easy crystal orientation along a high-symmetry direction, phase matching conditions and negligible interference among nonlinear waves, which may lead to large errors in the reported material properties. To avoid these approximations, we have developed an open-source package named Second Harmonic Analysis of Anisotropic Rotational Polarimetry in Multilayers (♯SHAARP.ml). The reliability and accuracy are established by experimentally benchmarking with both the SHG polarimetry and Maker fringes using standard and commonly used nonlinear optical materials as well as twisted 2-dimensional heterostructures.

Effective Mechanical Properties of Periodic Cellular Solids with Generic Bravais Lattice Symmetry via Asymptotic Homogenization.

In this paper, the scope of discrete asymptotic homogenization employing voxel (cartesian) mesh discretization is expanded to estimate high fidelity effective properties of any periodic heterogeneous media with arbitrary Bravais's lattice symmetry, including those with non-orthogonal periodic bases. A framework was developed in Python with a proposed fast-nearest neighbour algorithm to accurately estimate the periodic boundary conditions of the discretized representative volume element of the lattice unit cell. Convergence studies are performed, and numerical errors caused by both voxel meshing and periodic boundary condition approximation processes are discussed in detail. It is found that the numerical error in periodicity approximation is cyclically dependent on the number of divisions performed during the meshing process and, thus, is minimized with a refined voxel mesh. Validation studies are performed by comparing the elastic properties of 2D hexagon lattices with orthogonal and non-orthogonal bases. The developed methodology was also applied to derive the effective properties of several lattice topologies, and variation of their anisotropic macroscopic properties with relative densities is presented as material selection charts.

Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur–Weyl duality

It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.

Intrinsic symmetry of nonlocal nonlinear optical susceptibility tensor in degenerate multi-wave mixing

Using energy and momentum conservation laws, we obtained the intrinsic symmetry relations for the nonlocal nonlinear optical susceptibility tensor in lossless nth order nonlinear medium of arbitrary symmetry class for the case when less than n + 1 electromagnetic waves with different frequencies interact. Particular attention is devoted to the relations of the components of this tensor, which cannot be obtained as limiting case from the symmetry relations for the nonlocal nonlinear susceptibility tensor describing interaction of exactly n + 1 waves with different frequencies. The examples of these symmetry relations for degenerate second- and third-order processes often considered are given.

Generalized homology and Atiyah–Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

Abstract We propose that symmetry-protected topological (SPT) phases with crystalline symmetry are formulated by an equivariant generalized homology $h^G_n(X)$ over a real space manifold X with G a crystalline symmetry group. The Atiyah–Hirzebruch spectral sequence unifies various notions in crystalline SPT phases, such as the layer construction, higher-order SPT phases, and Lieb–Schultz–Mattis-type theorems. This formulation is applicable to not only free fermionic systems but also interacting systems with arbitrary onsite and crystal symmetries.

Renormalized Energy of a Dislocation Loop in a 3D Anisotropic Body

AbstractThe paper presents a rigorous analysis of the singularities of elastic fields near a dislocation loop in a body of arbitrary material symmetry that extends over the entire three-space. Explicit asymptotic formulas are given for the stress, strain and the incompatible distortion near the curved dislocation. These formulas are used to analyze the main object of the paper, the renormalized energy. The core-cutoff method is used to introduce that notion: first, a core in the form of a curved tube along the dislocation loop is removed; then, the energy of the complement is determined (= the core-cutoff energy). As in the case of a straight dislocation, the core-cutoff energy has a singularity that is proportional to the logarithm of the core radius. The renormalized energy is the limit, as the radius tends to 0, of the core-cutoff energy minus the singular logarithmic part. The main result of the paper are novel formulas for the coefficient of logarithmic singularity (the ‘prelogarithmic energy factor’) and for the renormalized energy.

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

Use of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%.

Theoretical and numerical modeling of Rayleigh wave scattering by an elastic inclusion

This work presents theoretical and numerical models for the backscattering of two-dimensional Rayleigh waves by an elastic inclusion, with the host material being isotropic and the inclusion having an arbitrary shape and crystallographic symmetry. The theoretical model is developed based on the reciprocity theorem using the far-field Green's function and the Born approximation, assuming a small acoustic impedance difference between the host and inclusion materials. The numerical finite element (FE) model is established to deliver a relatively accurate simulation of the scattering problem and to evaluate the approximations of the theoretical model. Quantitative agreement is observed between the theoretical model and the FE results for arbitrarily shaped surface/subsurface inclusions with isotropic/anisotropic properties. The agreement is excellent when the wavelength of the Rayleigh wave is larger than, or comparable to, the size of the inclusion, but it deteriorates as the wavelength gets smaller. Also, the agreement decreases with the anisotropy index for inclusions of anisotropic symmetry. The results lay the foundation for using Rayleigh waves for quantitative characterization of surface/subsurface inclusions, while also demonstrating its limitations.

Spin wave dispersion relation engineering by magnonic crystals with arbitrary symmetry

The use of metasurfaces to engineer the response of magnetic materials is of utmost importance in the field of magnon-spintronics. Here, we demonstrate a method to fabricate one- and two-dimensional magnonic crystals with arbitrary symmetry and use it to engineer the amplitude-frequency characteristic of magnetostatic surface spin waves excited in a magnetic material. The technique is based on the gentle microablation of the sample surface by focused femtosecond laser pulses. Tightly focused illumination allows using modest pulse energy while achieving micrometer precision. By raster scanning the incident laser spot on the sample surface, we control the shape and size of the building blocks constituting the unit cell of the crystal along with its symmetry and lattice parameter. Remarkable and controlled changes in the measured transmission characteristics reveal the strong and complex symmetry-dependent interaction of the spin waves with Bravais and non-Bravais lattices. The described single-step microfabrication method facilitates and speeds up the realization of integrated spintronics components and provides an efficient tool to explore complex magnetic dynamics in scattering lattices.

First-Principles Calculation of Ligand Field Parameters for L-Edge Spectra of Transition Metal Sites of Arbitrary Symmetry

Recently we have proposed a simple method for obtaining the parameters of a ligand field multiplet model for L-edge spectra calculations from density functional theory. Here we generalize the method to systems where the metal site has arbitrary point symmetry. The ligand field-induced splitting of the metal d-level becomes a hermitian matrix with cross-terms between the different d-orbitals. The anisotropy of the covalency is fully taken into account and it rescales the electron–electron interaction and the oscillator strength in an orbital-dependent way. We apply the method to polarization-dependent V L-edge spectra of vanadium pentoxide and obtain very good agreement with the experiment.