Representations Of Space Groups Research Articles

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPGandREPRESENTATIONS DSG). The toolDCOMPRELprovides compatibility relations between the irreducible representations of double space groups at differentkvectors of the Brillouin zone when there is a group–subgroup relation between the corresponding little groups. The programDSITESYMimplements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the programBANDREPcalculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results ofBANDREPhave been extensively applied in the description of and the search for topological insulators.

Global band topology of simple and double Dirac-point semimetals

We combine space group representation theory together with scanning of closed subdomains of the Brillouin zone with Wilson loops to algebraically determine global band structure topology. Considering space group #19 as a case study, we show that the energy ordering of the irreducible representations at the high-symmetry points $\{\Gamma,S,T,U\}$ fully determines the global band topology, with all topological classes characterized through their simple and double Dirac-points.

Anisotropic sublattices in cubic perovskite and their symmetry in multidimensional crystal space

Three tetragonal, partially occupied oxygen sublattices with a translational symmetry higher than the lattice translational symmetry can be introduced into the cubic perovskite structure. This higher symmetry is determined in the 9D space constructed on three orthogonal 3D subspaces corresponding to the sublattices. The generating matrices, metric tensors, and point symmetry matrices of this multidimensional space are determined. Irreducible stars of the representations of multidimensional space group are projected onto the irreducible stars of the space crystal group.

Classification of topological crystalline insulators based on representation theory

Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern insulators in the $pm$ or $pmm$ groups, $C_{nv}$ topological insulators in the $p4m$, $p31m$ and $p6m$ groups, and topological nonsymmorphic crystalline insulators in the $pg$ and $pmg$ groups. Aside from these existing results, we also obtain the following new results: (1) there are two integer mirror Chern numbers ($\mathbb{Z}^2$) in the $pm$ group but only one ($\mathbb{Z}$) in the $cm$ or $p3m1$ group for both the spinless and spinful cases; (2) for the $pmm$ ($cmm$) groups, there is no topological classification in the spinless case but $\mathbb{Z}^4$ ($\mathbb{Z}^2$) classifications in the spinful case; (3) we show how topological crystalline insulator phase in the $pg$ group is related to that in the $pm$ group; (4) we identify topological classification of the $p4m$, $p31m$, and $p6m$ for the spinful case; (5) we find topological non-symmorphic crystalline insulators also existing in $pgg$ and $p4g$ groups, which exhibit new features compared to those in $pg$ and $pmg$ groups. We emphasize the importance of the irreducible representations for the states at some specific high-symmetry momenta in the classification of topological crystalline phases. Our theory can serve as a guide for the search of topological crystalline insulator phases in realistic materials.

Generation of Basis Vectors for Magnetic Structures and Displacement Modes

Increasing attention is being focused on the use of symmetry-adapted functions to describe magnetic structures, structural distortions, and incommensurate crystallography. Though the calculation of such functions is well developed, significant difficulties can arise such as the generation of too many or too few basis functions to minimally span the linear vector space. We present an elegant solution to these difficulties using the concept of basis sets and discuss previous work in this area using this concept. Further, we highlight the significance of unitary irreducible representations in this method and provide the first validation that the irreducible representations of the crystallographic space groups tabulated by Kovalev are unitary.

Antisymmetric exchange in La-substituted BiFe0.5Sc0.5O3 system: symmetry adapted distortion modes approach

Abstract Neutron powder diffraction measurements on the 35 % La-substituted Bi1−x La x Fe0.5Sc0.5O3 composition revealed that the samples obtained under high-pressure (6 GPa) and high-temperature (1500 K) conditions crystalize into a distorted perovskite structure with the orthorhombic Pnma symmetry and the unit cell parameters: a o = 5.6745(2) Å, b o = 7.9834(3) Å and c o = 5.6310(2) Å. A long-range magnetic ordering takes place below 220 K and implies a G-type magnetic structure with the moments 4.10(4)μ B per Fe aligned predominately along the orthorhombic c-axis. The space group representation theory using the orthorhombic symmetry yields four bi-linear coupling schemes for the magnetic order parameters imposed by antisymmetric exchange interactions. The couplings are analysed based on symmetry adapted distortion modes defined in respect of the undistorted cubic perovskite structure. The approach allows a quantitative estimation of the coupling strength. It is shown that the experimentally found spin configuration combines the magnetic order parameters coupled by the atomic displacement modes with the largest amplitudes. The results indicate that the antisymmetric exchange is the dominant anisotropic term which fully controls the direction of the Fe3+ spins in the distorted perovskite lattice.

Development of the Bethe Method for the Construction of Two-Valued Space Group Representations and Two-Valued Projective Representations of Point Groups

A procedure of calculation of two-valued space group representations and two-valued projective representations of point groups is considered. A method of construction of factor systems w2(r2, r1), which reflect the transformations of half-integer spin quantum wave functions and are required in order to find the two-valued irreducible projective representations of the point groups, is presented. This method is based on the introduction of an operation q, firstly used by Bethe, as an additional symmetry element. The pathway of introducing the relations, which permit to make a one-valued algebra of double groups and, particularly, their multiplication tables, is shown by the examples of the 222 (D2) and 32 (D3) groups. The construction of a standard factor-system w′(1)(r2, r1) of the projective class K1 for the group 222 on the base of the discussed relations is presented for the first time. The whole role and the possibilities of Bethe’s method and its modifications for the construction of two-valued representations of the point and space groups are discussed.

Unfolding of electronic structure through induced representations of space groups: Application to Fe-based superconductors

We revisit the problem that relevant parts of bandstructures for a given cell choice can reflect exact or approximate higher symmetries of subsystems in the cell and can therefore be significantly simplified by an unfolding procedure that recovers the higher symmetry. We show that bandstructure unfolding can be understood as projection onto induced irreducible representations of a group obtained by extending the original group of translations with a number of additional symmetry operations. The resulting framework allows us to define a generalized unfolding procedure which includes the point group operations and can be applied to any quantity in the reciprocal space. The unfolding of the Brillouin zone follows naturally from the properties of the induced irreducible representations. In this context, we also introduce a procedure to derive tight-binding models of reduced dimensionality by making use of point group symmetries. Further, we show that careful consideration of unfolding has important consequences on the interpretation of angle resolved photoemission experiments. Finally, we apply the unfolding procedure to various representative examples of Fe-based superconductor compounds and show that the one iron picture arises as an irreducible representation of the glide mirror group and we comment on the consequences for the interpretation of one-iron versus two-iron Brillouin zone representations.

Characters of graphene’s symmetry group Dg80

We have determined the characters of the diperiodic group Dg80—the symmetry group of the honeycomb lattice. We started from the general formula for construction of irreducible representations (irreps) of symmorphic space groups and have found characters by summing diagonal elements of irrep matrices. The result is given for every element of Dg80, not just for generators. We have calculated the Frobenius–Schur indicator for irreps of Dg80 and proved that all of them are equivalent to real representations.

A collection of magnetic structures with cif-like files as a database seed

We report the release within the Bilbao Crystallographic server [1] of a webpage providing detailed quantitative information on a representative set of published magnetic structures. Under the name of MAGNDATA (www.cryst.ehu.es/magndata) more than 140 entries are available. Each magnetic structure has been saved making use of magnetic symmetry, i.e. Shubnikov magnetic groups for commensurate structures, and magnetic superspace groups for incommensurate ones. This ensures a unified communication method and a robust and unambiguous description of both atomic positions and magnetic moments. The origin and main crystallographic axes of the parent phase are usually kept, with the cost of often using a non-standard setting for the magnetic symmetry. The magnetic point group is also given, so that the allowed macroscopic tensor properties can be derived. The fact that magnetic structures are being described according to various methods, often with ambiguous information, has forced an elaborate interpretation and transformation of the original data. For this purpose the freely available internet tools MAXMAGN [1] and ISODISTORT [2] have been our essential tools. Most of the analyzed structures happen to possess maximal magnetic symmetries within the constraints imposed by the magnetic propagation vector, and the relevant model could be derived in a straightforward manner using MAXMAGN [1]. In a few cases a lower symmetry is realized, but then it corresponded to one isotropy subgroup of one or several irreducible representations (irreps) of the paramagnetic grey space group, and ISODISTORT [2] could be applied to model the structure. Although the structure description is done using magnetic groups, the active irrep(s) are also given in most cases. The entries of the collection can be retrieved in a cif-like format, which is supported by internet tools as STRCONVERT [1] and ISOCIF [2], the visualization program VESTA [3], and some refinement programs (JANA2006, FULLPROF). Each entry also includes Vesta files that allow the visualization of a single magnetic unit cell.

Symmetry constraints in frustrated magnets with strong magnetoelastic coupling

Geometrical frustration, related to the specific topology of certain crystal structures, plays a crucial role in forming exotic magnetic ground states. The presence of frustrated spins often leads to the suppression of long-range magnetic ordering and promotes short-range correlations due to fluctuations between nearly or totally degenerate ground states. The well-known structural topologies causing the presence of geometrical frustration are the three-dimensional pyrohlore and two-dimensional Kagome lattices. Compounds whose structural motif embraces these lattices are of great interest as model systems and have been the focus of numerous studies. In some cases, frustration is partially or entirely released by structural distortions through a strong magnetoelastic coupling and long-range magnetic order is established at a finite temperature. In the resulting distorted phases, complex noncollinear or partially disordered spin configurations can be observed. The phase transitions to the ordered state are quite often first order and may involve several irreducible representations of the paramagnetic space group and sometimes, like in the case of ZnCr2O4, even several propagation vectors which do not belong to the same star. The approach to determine magnetic structures in these systems, based on representation theory, should take into account the coupling free-energy invariants relating the magnetic and structural order parameters. Application of magnetic space groups and superspace groups is especially useful and can be efficiently combined with the representation theory. Based on specific examples, I will demonstrate how both approaches can be combined to provide symmetry constraints sufficient to solve complex magnetic structures in some geometrically frustrated systems.

Symmetry analysis of magnetic structures possible in TbNi2Si2-TbMn2Si2 solid solutions

TbNi2Si2-TbMn2Si2 solid solutions have been studied by neutron diffraction to reveal magnetic structures with wave vectors {k 8} = μ(b 1–b 2) + b 3/2 = 2π [(1–2μ)/2, (1 + 2μ)/2, 0]/a, {k 13} = b 3/2 = 2π(1/2, 1/2, 0)/a, and {k 14} = 0. The method of calculation is described and the results of calculations are given for the basis functions of irreducible representations of the space group D 4h 17 (14/mmm) that are used for the most rational search for a real model of the magnetic structure of these solid solutions. The compositions of the magnetic representations for each of the above-mentioned wave vectors and the possible variants of the mutual orientation of atomic magnetic moments in the crystallographic positions in which the magnetically active atoms (Tb, Ni, Mn) are located have been determined.

Brillouin-zone database on theBilbao Crystallographic Server

The Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.

Opening of a Gap in Graphene Due to Supercell Potential: Group Theory Point of View

We analyze in the framework of the space group theory the change of the dispersion law in grapenein the vicinity of the (former) Dirac points due to application of supercell potential with the $\sqrt{3}\times\sqrt{3}$ space periodicity and the same point symmetry as graphene.

Representation of Crystallographic Subperiodic Groups in Clifford’s Geometric Algebra

This paper explains how, following the representation of 3D crystallographic space groups in Clifford's geometric algebra, it is further possible to similarly represent the 162 so called subperiodic groups of crystallography in Clifford's geometric algebra. A new compact geometric algebra group representation symbol is constructed, which allows to read off the complete set of geometric algebra generators. For clarity moreover the chosen generators are stated explicitly. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford monomials, Lipschitz elements). Keywords: Subperiodic groups, Clifford's geometric algebra, versor representation, frieze groups, rod groups, layer groups .

Tabulation of irreducible representations of the crystallographic space groups and their superspace extensions

New tables of irreducible representations (IRs) are introduced for the 230 crystallographic space groups (SGs) in three-dimensional space, at both special and non-special k vectors, and for their extensions to (3 + d)-dimensional superspace (`superspace-extended SGs' or SSESGs). Neither a tabulation of SG IR matrices for non-special k vectors nor a tabulation of SSESG IR matrices for d > 1 have been previously published. These tabulations are made possible by a new form in which the IR matrices of SGs are separated as a product of a translation part T and a point-operation part P, and where the IR matrices of SSESGs are separated as a product of a phase-shift part Q and a point-operation part P(s). Both T and Q have a simple prescribed form that does not need to be tabulated. Also, the new IR matrices are in a convenient block form which allows one to see by inspection which parts of the matrices and the associated order parameters belong to which arm of the star of k. In addition to complex IR matrices, real physically irreducible representation (PIR) matrices are tabulated. The new IR and PIR tables are available on the ISO-IR website (http://stokes.byu.edu/iso/irtables.php) in both convenient human-readable and computer-readable forms.

Nonreciprocal optical response of helical periodic structures of plasma spheres in a static magnetic field

A detailed and rigorous theoretical investigation of the optical properties of a generic three-dimensional chiral structure of plasma spheres, without and under the action of an external static uniform magnetic field, is presented. Corresponding photonic band diagrams in conjunction with relevant transmission spectra, calculated by the full electrodynamic layer-multiple-scattering method properly extended to the case of gyrotropic spherical scatterers, are discussed in the light of the theory of nonsymmorphic space groups. This analysis provides a consistent interpretation of some remarkable features and effects like Dirac points, polarization-dependent transmission, as well as band splitting and non-reciprocal optical response that emerge as a result of time-reversal-symmetry breaking, induced by the external static magnetic field, and the lack of space-inversion symmetry in the crystal.

Symmetry analysis of the magnetic structure of Mn3Sb

To estimate the reliability of the previously published results concerning the magnetic structure of Mn3Sb, a symmetry analysis of possible magnetic structures characterized by a wave vector k 12 = 0 has been performed. In this work, we quite briefly describe and apply the previously developed technique for the calculations of the basis functions of the irreducible representations of the space group O h 1 (Pm \(\bar 3\) m) entering into the composition of the magnetic representation with the above wave vector. Results of the calculations of these basis functions for the Mn atoms located in 3c positions are given and discussed in detail. A conclusion is made that the magnetic moments of Mn atoms in this intermetallic compound can be oriented along any crystallographic direction with the formation of collinear or canted magnetic structures. The total magnetic moments of separate Mn atoms in the unit cell of the crystal structure can coincide or differ in magnitude.

CHIRALITY-ASYMMETRY MACROSCOPIC FORCE INDUCED BY AXION BETWEEN THE α-QUARTZ AND COPPER BLOCK

A macroscopic force induced by the effective potential suggested by Moody and Wilczek between α-quartz crystal and copper block is studied in detail. The matrix elements in the force formula are analyzed according to representations of the non-symmorphic space group P 3121 of α-quartz crystal. The asymmetry distribution of valence electrons and the selection rules do not cancel the force. We conclude that there exists a CP-violating macroscopic force. According to the limit of the production of the axion coupling constants gs gp/ℏc ≤ 10-26 at the Compton wavelength λ = 10-3 m, the macroscopic force between a 0.08 × 0.08 × 0.02 m3 block of α-quartz and a 0.08 × 0.08× 0.01 m3 copper block with a separation being 0.5 × 10-3 m in between, is estimated at less than 3.09 × 10-26 N.

Theory of neutron scattering in magnets

We review theoretical studies of magnetically ordered materials using slow neutron scattering. We consider methods for decoding magnetic structures using symmetry theory (irreducible representations of space groups) and the theory of small-angle neutron scattering by large-scale inhomogeneities in magnetic materials.