Group Theoretical Symmetry Research Articles

Polarization-selective excitation of multiband quasibound states in the continuum supported by symmetry-perturbed silicon metasurface

Bound states in the continuum (BICs) possessing a typical characteristic of high quality (Q) factor allow for strong light–matter interactions and boost various research fields including optical detecting and nonlinear photonics. In this work, a spatial symmetry-perturbed metasurface composed of a period array of single-notched silicon nanodisk is proposed to support multiband quasi-BICs (QBICs). After group theory analysis of eigenmodes and group-theoretical symmetry arguments describing the symmetric matching relationship between BICs and external radiation, we numerically demonstrate that a magnetic quadrupole BIC and an electric dipole BIC are excited by x-polarized plane wave, and an electric quadrupole BIC and a magnetic quadrupole BIC are excited by y-polarized plane wave, achieving the polarization-selective excitation of multiband QBICs by adjusting the light polarization. The Q-factors of these four QBICs exhibit the inverse quadratic dependence on two different asymmetric parameters, which provides two independent channels for controlling the coupling to BICs. This new device featuring a simple bulk geometry will become an extremely important platform to obtain more flexible and stronger electromagnetic wave manipulation at the nanoscale, which facilitates large-scale and highly integrated electromagnetic applications in non-linear devices, high-throughput sensors, optical communications, and so on.

Energetic Couplings in Ferroics

AbstractFerroics include diverse degrees of freedom (such as structural distortions and magnetic moments) among which cross couplings occur, rendering a large variety of interesting phenomena. Determining such couplings, based on symmetry analysis, is not only important to interpret observed phenomena but can also result in novel predictions to be then experimentally checked. Often, such energetic couplings are difficult to construct without a deep knowledge of group theoretical symmetry principles. In the present review, a crash course toward the derivation of energetic couplings, without using much group theoretical language, is provided. Rather, the present approach relies on a graphical technique and suitable symbolic language, which naturally yields some known couplings (resulting in, e.g., spin/dipole canting, magnetically driven polarization, and antipolar/antiferroelectric states). This review also reports and discusses other symmetry‐allowed energetic terms, including some leading to the occurrence of an electric polarization in a variety of materials, and “exotic” ones that generate complex phases and phenomena in, for example, nanostructures and heterostructures.

Dzyaloshinskii-Moriya-like interaction in ferroelectrics and antiferroelectrics.

The Dzyaloshinskii-Moriya interaction (DMI) between two magnetic moments mi and mj is of the form [Formula: see text]. It originates from spin-orbit coupling, and is at the heart of fascinating phenomena involving non-collinear magnetism, such as magnetic topological defects (for example, skyrmions) as well as spin-orbit torques and magnetically driven ferroelectricity, that are of significant fundamental and technological interest. In sharp contrast, its electric counterpart, which is an electric DMI characterized by its [Formula: see text] strength and describing an interaction between two polar displacements ui and uj, has rarely been considered, despite the striking possibility that it could also generate new features associated with non-collinear patterns of electric dipoles. Here we report first-principles simulations combined with group theoretical symmetry analysis which not only demonstrate that electric DMI does exist and has a one-to-one correspondence with its magnetic analogue, but also reveals a physical source for it. These findings can be used to explain and/or design phenomena of possible technological importance in ferroelectrics and multiferroics.

Hybrid density functional theoretical study of NASICON-type NaxTi2(PO4)3 (x = 1-4).

Sodium Super Ionic Conductor (NASICON) structured phosphate framework compounds represent a very attractive class of materials for their use as Na-ion battery electrodes. A series of NASICON-structured NaxTi2(PO4)3 compounds corresponding to varying degrees of sodiation (x = 1-4) have been investigated using high-level hybrid density functional theory calculations using the Linear Combination of Atomic Orbitals and Gaussian-type basis set formalism together with hybrid B1WC and HSE06 exchange-correlation functionals. Using primitive cells of NaxTi2(PO4)3 compounds with different stoichiometry, sodium sublattice structure and titanium oxidation states are constructed and analyzed using group theoretical symmetry considerations. The existence of mixed titanium oxidation states for x = 4 (Ti2+/Ti3+) and x = 2 (Ti3+/Ti4+) and a single oxidation state for x = 1 (Ti4+) and x = 3 (Ti3+) has been demonstrated. The results show a necessary set of symmetry reductions taking place due to the highest possible sodium/vacancy and titanium charge ordering with changing x. For each composition, an electroneutrality condition for the oxidation states of all atoms was applied which led to the discovery of several energy minima corresponding to different electronic configurations as identified by different Ti magnetic moments. An interesting relation between the bulk electronic properties of NaxTi2(PO4)3 compounds and the variation of sodium content was also found. In addition to sodium and titanium oxidation state charge ordering, the existence of large differences between the origin and the size of the band gap is shown. The band gap changes from the 4.05 eV 2p-3d gap in Na1Ti2(PO4)3 to the 0.59 eV 3d-3d gap in Na4Ti(PO4)3 with extra states due to mixed titanium valence. These results serve as an important electronic structure benchmark for further studies of such polyanion materials and help to explain some important properties of these systems relevant to battery applications.

Combined use of translational and spin-rotational invariance for spin systems

Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry $C_N$ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2) can be used, where the Hamiltonian matrix is block-structured according to the total spin- and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both $C_N$ and SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.

Microscopic modeling of the effect of phonons on the optical properties of solid-state emitters

Understanding the effect of vibrations in optically active nano systems is crucial for successfully implementing applications in molecular-based electro-optical devices, quantum information communications, single photon sources, and fluorescent markers for biological measurements. Here, we present a first-principles microscopic description of the role of phonons on the isotopic shift presented in the optical emission spectrum associated to the negatively charged silicon-vacancy color center in diamond. We use the spin-boson model and estimate the electron-phonon interactions using a symmetrized molecular description of the electronic states and a force-constant model to describe molecular vibrations. Group theoretical arguments and dynamical symmetry breaking are presented in order to explain the optical properties of the zero-phonon line and the isotopic shift of the phonon sideband.

Continuum-based free vibration of circular trigonal and isotropic plates

The free vibration behavior of completely unrestrained elastic circular plates with trigonal and isotropic material symmetry is studied with an approach involving approximate continuum solutions to the three-dimensional theory of linear elasticity. Of primary interest are (1) the influence of trigonal material symmetry on the modes of free vibration and (2) the accuracy of thin plate theory relative to the more exact three-dimensional theory. Resonant frequencies are calculated from the weak form of the equations of motion for the plate through the use of approximation functions and the Ritz method formulated in cylindrical coordinates. This approach enables the resulting eigenvalue problem to be split through group-theoretical symmetry analysis. Representative examples are given and quantitative limits are discussed.

Data structure and movement for lattice-based simulations

We show that for the lattice Boltzmann model, the existing paradigm in computer science for the choice of the data structure is suboptimal. In this paper we use the requirements of physical symmetry necessary for recovering hydrodynamics in the lattice Boltzmann description to propose a hybrid data layout for the method. This hybrid data structure, which we call a structure of an array of structures, is shown to be optimal for the lattice Boltzmann model. Finally, the possible advantages of establishing a connection between group theoretic symmetry requirements and the construction of the data structure is discussed in the broader context of grid-based methods.

Second-order Raman scattering in CuO

Polarized second-order Raman scattering spectra of CuO single crystals are reported. It is shown that for some scattering geometries the second-order processes dominate the inelastic light scattering spectra. Group-theoretical symmetry analysis of the selection rules for the first- and second-order scattering processes is performed and phonon dispersion relations are calculated within density functional theory. The main spectral features of the two-phonon spectra are assigned to overtones of the vibrational branches at various special points across the Brillouin zone.

Spin electric effects in molecular antiferromagnets

Molecular nanomagnets show clear signatures of coherent behavior and have a wide variety of effective low-energy spin Hamiltonians suitable for encoding qubits and implementing spin-based quantum information processing. At the nanoscale, the preferred mechanism for control of quantum systems is through application of electric fields, which are strong, can be locally applied, and rapidly switched. In this work, we provide the theoretical tools for the search for single molecule magnets suitable for electric control. By group-theoretical symmetry analysis we find that the spin-electric coupling in triangular molecules is governed by the modification of the exchange interaction, and is possible even in the absence of spin-orbit coupling. In pentagonal molecules the spin-electric coupling can exist only in the presence of spin-orbit interaction. This kind of coupling is allowed for both $s=1/2$ and $s=3/2$ spins at the magnetic centers. Within the Hubbard model, we find a relation between the spin-electric coupling and the properties of the chemical bonds in a molecule, suggesting that the best candidates for strong spin-electric coupling are molecules with nearly degenerate bond orbitals. We also investigate the possible experimental signatures of spin-electric coupling in nuclear magnetic resonance and electron spin resonance spectroscopy, as well as in the thermodynamic measurements of magnetization, electric polarization, and specific heat of the molecules.

Octahedral Transforms for 3-D Image Processing

The octahedral group is one of the finite subgroups of the rotation group in 3-D Euclidean space and a symmetry group of the cubic grid. Compression and filtering of 3-D volumes are given as application examples of its representation theory. We give an overview over the finite subgroups of the 3-D rotation group and their classification. We summarize properties of the octahedral group and basic results from its representation theory. Wide-sense stationary processes are processes with group theoretical symmetries whose principal components are closely related to the representation theory of their symmetry group. Linear filter systems are defined as projection operators and symmetry-based filter systems are generalizations of the Fourier transforms. The algorithms are implemented in Maple/Matlab functions and worksheets. In the experimental part, we use two publicly available MRI volumes. It is shown that the assumption of wide-sense stationarity is realistic and the true principal components of the correlation matrix are very well approximated by the group theoretically predicted structure. We illustrate the nature of the different types of filter systems, their invariance and transformation properties. Finally, we show how thresholding in the transform domain can be used in 3-D signal processing.

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the “commutative case”, has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver’s SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.

Free-Energy Density of the Shape-Memory Alloy AuZn

The shape-memory alloy AuZn transforms martensitically (T M = 65 K) from a cubic B2 (Pm3m) to a rhombohedral R-phase (P3) through softening of the TA 2 [110] phonon branch. We report elastic constants, specific heat, and thermal expansivity measurements through the transition. A large elastic anisotropy, A = 9 at T M , associated with softening of the TA 2 [110] phonon branch accompanied with a volume change ΔV/V of 0.25% characterize the transition. We find that specific heat and thermal expansion are well represented by adding low-energy Einstein modes to the harmonic lattice. On the basis of these measurements, combined with established group-theoretical symmetry criteria, the free-energy density is presented with atomic shuffle displacements as the primary order parameter, Q. We use this free-energy model to explain the atomic displacements in the R-phase.

Antiferromagnetic three-sublattice Tb ordering inTb14Ag51

Bulk magnetic, x-ray, and neutron-diffraction measurements were performed on polycrystalline ${\mathrm{Tb}}_{14}{\mathrm{Ag}}_{51}$ in the temperature range from $1.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ to room temperature. Its chemical ${\mathrm{Gd}}_{14}{\mathrm{Ag}}_{51}$-type structure corresponding to space group $P6∕m$ has been refined at 300 and at $30\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Combined with group-theoretical symmetry analysis, we show that the magnetic structure of this intermetallic compound is of a different $\mathbf{k}=(1∕3,1∕3,0)$ type with three magnetic Tb sublattices ordering simultaneously below ${T}_{\mathrm{N}}=27.5(5)\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ according to the combined irreducible representations ${\ensuremath{\tau}}_{4}$ and ${\ensuremath{\tau}}_{6}$.

Interior symmetry and local bifurcation in coupled cell networks

A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the ‘symmetry groupoid’, has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: ‘interior symmetry’. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for ‘synchrony-breaking’ bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.

Reflections on Parity

The selective synthesis and manipulation of chiral molecules is a major part of stereochemistry. Interactions involving components with “handedness” also occur in nuclear and particle physics. Both fields employ phenomenologies based on group theoretical classifications and symmetry constraints such as the conservation or violation of parity. Smale's discovery of a sphere eversion—homotopic to a central inversion—evades the parity dichotomy by allowing states of opposite chirality to be connected by smooth maps. This transformation demonstrates that parity changes can, in principle, result from the action of sufficiently complex processes, and do not necessarily have to be interpreted as violations of a symmetry.

Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals

In analogy to the Wannier equation for localized electronic impurity states in crystalline materials, a wave equation for the envelope of the resonant optical modes of local defects within two-dimensional periodic dielectric structures is derived. In the case of degenerate satellite extrema, this is generalized to a set of coupled Wannier-like equations for a multienvelope system. The localized Wannier envelope solutions are then used in conjunction with a group theoretical symmetry analysis to determine an approximate form for donor and acceptor modes in a hexagonal photonic crystal. For an effective harmonic potential formed by varying the filling fraction of the lattice, the localized resonant modes are explicitly calculated using the Wannier equation and symmetry analysis, and a comparison to exact numerically computed modes is presented.

Rotational and vibrational dynamics of interstitial molecular hydrogen

The calculation of the hindered roton-phonon energy levels of a hydrogen molecule in a confining potential with different symmetries is systematized for the case when the rotational angular momentum J is a good quantum number. One goal of this program is to interpret the energy-resolved neutron time-of-flight spectrum previously obtained for ${\mathrm{H}}_{2}{\mathrm{C}}_{60}.$ This spectrum gives direct information on the energy-level spectrum of ${\mathrm{H}}_{2}$ molecules confined to the octahedral interstitial sites of solid ${\mathrm{C}}_{60}.$ We treat this problem of coupled translational and orientational degrees of freedom (i) by construction of an effective Hamiltonian to describe the splitting of the manifold of states characterized by a given value of J and having a fixed total number of phonon excitations, (ii) by numerical solutions of the coupled translation-rotation problem on a discrete mesh of points in position space, and (iii) by a group theoretical symmetry analysis. Results obtained from these three different approaches are mutually consistent. The results of our calculations explain several aspects of the experimental observations, but show that a truly satisfactory orientational potential for the interaction of an ${\mathrm{H}}_{2}$ molecule with a surrounding array of C atoms has not yet been developed.

Landau Theory of the Displacive Phase Transformations in Gold-Cadmium and Titanium-Nickel Alloys

On the basis of group theoretical symmetry criteria, the primary and secondary order parameters (OP's) have been identified for the three ferroelastic martensitic transformations that occur in the Au-Cd binary, and in the Ti-Ni-M (M = Fe, At, Cu) pseudo-binary shape-memory alloys, viz (i) from the cubic β 2 austenite parent phase (B2 structure) to the rhombohedral (R) ζ' 2 product phase of P3 symmetry, (ii) from the β 2 to the orthorhombic (γ 2 ) product phase of Pmma symmetry (B19 structure), and (iii) from the B19 to the monoclinic B19'structure of P2 1 /m symmetry. For all three transformations, the Landau free energy and the relations between the primary OP and the atom shuffle displacements are given for the transition to a single product phase variant. For case (i), the 12 experimentally measured structural (shuffle) parameters of the R phase can be accounted for and fitted by only two theoretical model parameters, giving satisfactory agreement with the (only available) room temperature data for Au .505 Cd .495 ; for Ti .4977 Ni .5023 larger discrepancies, but mostly within the relatively large experimental error are found. For the two cases (ii) and (iii), the two shuffle displacements each can be fitted exactly by the two theoretical model parameters required.

Geometrically frustrated magnetic structures of the heavy-fermion compound CePdAl studied by powder neutron diffraction

The heavy-fermion compound CePdAl with ZrNiAl-type crystal structure (hexagonal space group ) was investigated by powder neutron diffraction. The triangular coordination symmetry of magnetic Ce atoms on site 3f gives rise to geometrical frustration. CePdAl orders below with an incommensurate antiferromagnetic propagation vector , and a longitudinal sine-wave (LSW) modulated spin arrangement. Magnetically ordered moments at Ce(1) and Ce(3) coexist with frustrated disordered moments at Ce(2). The experimentally determined magnetic structure is in agreement with group theoretical symmetry analysis considerations, calculated by the program MODY, which confirm that for Ce(2) an ordered magnetic moment parallel to the magnetically easy c-axis is forbidden by symmetry. Further low-temperature experiments give evidence for a second magnetic phase transition in CePdAl between 0.6 and 1.3 K. Magnetic structures of CePdAl are compared with those of the isostructural compound TbNiAl, where a non-zero ordered magnetic moment for the geometrically frustrated Tb(2) atoms is allowed by symmetry.