Symmorphic Space Groups Research Articles

Generalized Bloch boundary conditions based on a symmorphic space group and the finite-element implementation in photonic crystal

We studied generalized Bloch boundary conditions and their finite element implementation within the theoretical framework of a symmorphic space group. By combining translation symmetry operations with mirror and rotational symmetry operations, we developed a procedure for implementing generalized Bloch boundary conditions in the finite element method (FEM) for periodic photonic structures. First, we lay out the theoretical foundation and numerical implementation of generalized Bloch boundary conditions in FEM. We illustrate the proposed method via 2D/3D periodic photonic structures. Without a loss of generality, we calculate the band structures of 2D/3D photonic crystals using our proposed generalized Bloch boundary conditions and benchmark the results against the conventional Bloch boundary conditions. The comparisons show that band structure and eigenmode yield excellent agreement with the results obtained from conventional Bloch boundary conditions. However, our method has improved the computational efficiency by at least twofold. We further elaborate the comparisons with computation errors, memory efficiency, and computation times, all of which show that our proposed method outperforms the conventional one due to careful consideration of the mirror and rotational symmetry operation, apart from the translation symmetry. In addition, our method can easily be extended to other methods such as FDTD and transfer matrix.

General Theory of Momentum-Space Nonsymmorphic Symmetry

As a fundamental concept of all crystals, space groups are partitioned into symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains glide reflections or screw rotations with fractional lattice translations, which are absent in symmorphic groups. Although nonsymmorphic groups ubiquitously exist on real-space lattices, on the reciprocal lattices in momentum space, the ordinary theory only allows symmorphic groups. In this work, we develop a novel theory for momentum-space nonsymmorphic space groups ($k$-NSGs), utilizing the projective representations of space groups. The theory is quite general: Given any $k$-NSGs in any dimensions, it can identify the real-space symmorphic space groups ($r$-SSGs) and construct the corresponding projective representation of the $r$-SSG that leads to the $k$-NSG. To demonstrate the broad applicability of our theory, we show these projective representations and therefore all $k$-NSGs can be realized by gauge fluxes over real-space lattices. Our work fundamentally extends the framework of crystal symmetry, and therefore can accordingly extend any theory based on crystal symmetry, for instance, the classification crystalline topological phases.

A periodic-like table of space groups.

There are about 100 chemical elements, and 200 space groups, rounding to the nearest hundreds. The elements, by virtue of the iconic periodic table, which hangs in schoolrooms worldwide, are part of our common culture. Few on the outside of solid-state science and discrete mathematics know of the remarkable enumeration of the space groups and how they organize solids, the chemical elements included, that can be conformed to translationally periodic lattices in three dimensions. Perhaps the comparative inaccessibility of space groups to non-specialists can be attributed to the absence of a single chart that captures and organizes the space groups in a glance, as does the periodic table for the chemical elements? Here is designed a periodic-like table of the space groups. The symmorphic space groups are organized as the coordinates of two non-linear, non-orthogonal axes that capture the point group symmetry and the order of the group, the multiplicity of general positions. The related non-symmorphic groups are treated as 'isotopes' of the symmorphic groups and are enumerated with a subscript. The intent is to inspire better designs and to bring the crystallographic space groups to the attention of a wider audience.

Atiyah-Hirzebruch spectral sequence in band topology: General formalism and topological invariants for 230 space groups

We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant K-theory in the context of band theory. Various notions in the band theory such as irreducible representations at high-symmetric points, the compatibility relation, topological gapless and singular points naturally fits into the AHSS. As an application of the AHSS, we get the complete list of topological invariants for 230 space groups without time-reversal or particle-hole invariance. We find that a lot of torsion topological invariants appear even for symmorphic space groups.

Ideal topological phononic nodal chain in K2O materials class

Nodal-chain semimetals are characterized by linked nodal rings in Brillouin zone and they have not been reported in the phonon system of atomic crystals yet. Here, based on first-principles calculations and tight-binding model analysis, we propose nearly ideal nodal chains can be realized in the phonon modes of experimentally synthesized ionic crystal K2O materials class. Different from all previous cases, the connection points of the nodal chain in K2O are located at the high symmetry points W and the positions are limited by the symmetries. The nodal rings that forming the nodal chain are symmetrically equivalent and protected by the mirror symmetries. The nontrivial topological properties of nodal chains are validated by the nontrivial Berry phase and surface modes. A general discussion about all symmorphic space groups reveals that, if no more than two inequivalent mirrors exist, Fm m (No. 225) is a unique space group for the appearance of the nodal chain connected at the high symmetry points. Our studies demonstrate that K2O materials class is a wonderful implementable platform to study the nontrivial topological nodal chains in a phonon system.

Symmetry-enforced straight nodal-line phonons

Based on first-principles calculations and effective model analysis, all possible straight nodal-line phonons in symmorphic space groups are uncovered. A classification of two-band ${\textbf{k}}\cdot {\textbf{p}}$ models at an arbitrary point on the nodal lines is carried out, and these nodal-line phonons are found to locate along the high symmetry lines of the Brillouin zone. According to our classification, there are two types of nodal-line phonons in symmorphic space groups. The first type possesses a linear dispersion perpendicular to the nodal lines, while the second type exhibits a quadratic dispersion. The underlying mechanisms are revealed by symmetry analysis, and the types of nodal-line phonons can be distinguished through the little group of the high symmetry lines. Our work not only classifies nodal-line phonons but also provides guidance to search for topological nodal lines in bosonic systems.

Higher Chern numbers in multilayer Lieb lattices ( N≥2 ): Topological transitions and quadratic band crossing lines

We consider a hitherto unexplored setting of a stacked multilayer ($\mathcal{N}$) Lieb lattice which undergoes an unusual topological transition in the presence of intralayer spin-orbit coupling (SOC). The specific stacking configuration induces an effective nonsymmorphic two-dimensional lattice structure, even though the constituent monolayer Lieb lattice is characterized by a symmorphic space group. This emergent nonsymmorphicity leads to multiple doubly degenerate bands extending over the edge of the Brillouin zone (i.e., quadratic band crossing lines). In the presence of intralayer SOC, these doubly degenerate bands typically form three $\mathcal{N}$-band subspaces, mutually separated by two band gaps. We analyze the topological properties of these multiband subspaces, using specially devised Wilson loop operators to compute non-Abelian Berry phases in order to show that they carry a higher Chern number $\mathcal{N}$.

Research progress on topological nodal line semimetals

Topological quantum materials, such as topological insulators, topological semimetals, and topological superconductors, are the research focus of modern condensed matter physics and materials science. The purpose of this article is to give a brief introduction on the research progress of topological nodal line semimetal materials in recent years, and to focus on the latest research progress of our group in this field. Topological semimetals can be classified mainly by two aspects: energy band degeneracy and dispersion. In terms of energy band degeneracy, this article will introduce nodal line semimetals protected by different symmetries including nodal line semimetals protected by symmorphic space groups and non-symmorphic space groups in the system with time reversion symmetry, and nodal line semimetals coupled with magnetic and topological orders without time reversion symmetry, respectively. In terms of energy band dispersion, this article will introduce high-order nodal line semimetals and type-II nodal line semimetals.

Systematic study of stacked square nets: From Dirac fermions to material realizations

Non-symmorphic symmetries protect Dirac line nodes in square net materials. This phenomenon has been most prominently observed in ZrSiS. Here, we systematically study the symmetry-protected nodal fermions that result from different ways of embedding the square net into a larger unit cell. Surprisingly, we find that a nonsymmorphic space group is not a necessary condition for a filling enforced semimetal: symmorphic space groups can also host nodal fermions that are enforced by band folding and electron count, that is, a combination of a particular structural motif combined with electron filling. We apply the results of this symmetry analysis to define an algorithm, which we utilize to find square net materials with nodal fermions in specific symmorphic space groups. We highlight one result of this search, the compound ThGeSe, which has not been discussed before in the context of nodal fermions. Finally, we discuss how band folding can impose constraints on band connectivity beyond the connectivity of single elementary band representations.

Suppressed topological phase transitions due to nonsymmorphism in SnTe stacking

We combine first principles calculations with a group theory analysis to investigate topological phase transitions in the stacking of SnTe monolayers. We show that distinct finite stacking yields different symmetry-imposed degeneracy, which dictates the hybridization properties of opposite surface states. For SnTe aligned along the [001] direction, an (even) odd number of monolayers yields a (non)symmorphic space group. For the symmorphic case, the hybridization of surface states lead to band inversions and topological phase transitions as the sample height is reduced. In contrast, for a nonsymmorphic stacking, an extra degeneracy is guaranteed by symmetry, thus avoiding the hybridization and topological phase transitions, even in the limit of a few monolayers. Our group theory analysis provide a clear picture for this phenomenology and matches well the first principles calculations.

Interplay of nonsymmorphic symmetry and spin-orbit coupling in hyperkagome spin liquids: Applications to Na4Ir3O8

Na$_4$Ir$_3$O$_8$ provides a material platform to study three-dimensional quantum spin liquids in the geometrically frustrated hyperkagome lattice of Ir$^{4+}$ ions. In this work, we consider quantum spin liquids on hyperkagome lattice for generic spin models, focusing on the effects of anisotropic spin interactions. In particular, we classify possible $\mathbb{Z}_2$ and $U(1)$ spin liquid states, following the projective symmetry group analysis in the slave-fermion representation. There are only three distinct $\mathbb{Z}_2$ spin liquids, together with 2 different $U(1)$ spin liquids. The non-symmorphic space group symmetry of hyperkagome lattice plays a vital role in simplifying the classification, forbidding "$\pi$-flux" or "staggered-flux" phases in contrast to symmorphic space groups. We further prove that both $U(1)$ states and one $Z_2$ state among all 3 are symmetry-protected gapless spin liquids, robust against any symmetry-preserving perturbations. Motivated by the "spin-freezing" behavior recently observed in Na$_4$Ir$_3$O$_8$ at low temperatures, we further investigate the nearest-neighbor spin model with dominant Heisenberg interaction subject to all possible anisotropic perturbations from spin-orbit couplings. We found a $U(1)$ spin liquid ground state with spinon fermi surfaces is energetically favored over $Z_2$ states. Among all spin-orbit coupling terms, we show that only Dzyaloshinskii-Moriya (DM) interaction can induce spin anisotropy in the ground state when perturbing from the isotropic Heisenberg limit. Our work paves the way for a systematic study of quantum spin liquids in various materials with a hyperkagome crystal structure.

A practical approach to symmorphism

A common misunderstanding in the nature of symmorphic space groups, often incorrectly defined as devoid of screw axes and glide planes, is pointed out and the origin of this misunderstanding is critically analysed. The features of symmorphic space groups are illustrated with concrete examples and a detailed classification is presented.

Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts

AbstractPoint groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct product of pure translations and pure point subgroups. In contrast, Bieberbach groups consist of pure translations, screws, and glides. These “torsion-free” space groups are rarely mentioned as being a special class outside of the mathematics literature. Every space group can be thought of as lying along a spectrum with the symmorphic case at one extreme and Bieberbach space groups at the other. The remaining nonsymmorphic space groups lie somewhere in between. Many of these can be decomposed into semi-direct products of Bieberbach subgroups and point transformations. In particular, we show that those 3D Sohncke space groups most populated by macromolecular crystals obey such decompositions. We tabulate these decompositions for those Sohncke groups that admit such decompositions. This has implications to the study of packing arrangements in macromolecular crystals. We also observe that every Sohncke group can be written as a product of Bieberbach and symmorphic subgroups, and this has implications for new nomenclature for space groups.

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.

Infrared spectra of bismuth family of superconductors Bi2Sr2Can-1CunO4+2n+y

FTIR spectra of bismuth family of superconductors are studied. Polycrystalline samples of bismuth superconductors are prepared choosing an off-stoichiometric composition for phase stabilisation and reproducibility. Vibrational modes are strongly affected by the holes introduced into the CuO planes. Mid IR spectra of the Bismuth system was recorded using Bruker IFS 66v and the high frequency oxygen vibrations of the system is discussed based on the factor group analysis for both symmorphic and non-symmorphic space groups I4/mmm, Fmmm and Amaa of different structures based on the modulated superstructure.

Symmetrized plane waves: 1. symmorphic space groups

A procedure for systematically constructing symmetrized plane waves for arbitrary symmorphic space groups is given which can also be used to symmetrize arbitrary Bloch-functions. To simplify the technical part, a software package has been developed to compute those specific transformation matrices that are necessary for either determining symmetrized plane waves or symmetrized Bloch-functions.

Symmetrized plane waves: 2. symmetrized Pm3̄ m-plane waves

A procedure for systematically constructing symmetrized plane waves is applied to construct in full detail a complete set of symmetrized plane waves with respect to the symmorphic space group Pm3̄ m.

Comparison of the frequencies of crystallographic space groups in organo-lanthanide and -actinide compounds

Bibliographic and numeric data on 1540 organo-lanthanide and 895 organo-actinide crystalline compounds were derived from the Cambridge Structural Database. After removal of the double entries, the compounds were classified into the 230 crystallographic space groups according to their space-group numbers, and the relative frequency of occurrence of each space group was then determined. 10.0% of the lanthanide and 8.4% of the actinide compounds fall in space groups of tetragonal or higher symmetry. In both cases, only 71 of the space groups have at least one entry. Approximately two-thirds of the compounds crystallize in three space groups: 34.1% of the lanthanides in P2 1/ c (actinides: 38.5%), 22.1% (20.5%) in P 1 and 8.3% (9.0%) in C2/ c; in contrast, there are 46 space groups in the lanthanide group with five or fewer compounds assigned to them (55 for the actinides). The distribution of the lanthanides into the different crystal families corresponds well to that of the actinides. If the data are compared with a study on the space-group frequencies for organic compounds (Mighell et al., Acta Crystallogr., A 39 (1983), 737–740) there are a number of important differences: lanthanides and actinides tend to crystallize more in the triclinic crystal family than the organic compounds (22.6% for lanthanides and 21.0% for actinides vs. 14.8% for organic compounds) and less in the orthorhombic crystal family (15.4% for lanthanides and 17.0% for actinides vs. 25.8% for organic compounds). There is an increase in the number of compounds crystallizing in one of the centrosymmetric space groups (78.5% for lanthanides and 81.7% for actinides vs. 70.4% for organic compounds) and a decrease for the enantiomorphic space groups (11.5% for lanthanides and 10.1% for actinides vs. 23.1% for organic compounds). The number of compounds crystallizing in one of the symmorphic space groups is higher (27.9% for lanthanides and 24.7% for actinides vs. 18.6% for organic compounds), which corroborates the fact that the number of compounds crystallizing in the triclinic system has increased.

Crystallographic viewpoints in the classification of space-group representations

Abstract The available tables of irreducible representations of space groups differ considerably in completeness, kind of listing, and nomenclature. Although published more than 50 years ago, Wintgen's approach making use of the isomorphism between the reciprocal-space groups and the symmorphic space groups has not yet been applied in any of the published listings of irreducible representations of the space groups. In this paper the implications of such a classification scheme are discussed. Use of the International Tables for Crystallography, Vol. A makes such a classification simpler, more transparent, and facilitates some applications. Moreover, the International Tables may serve as a basis for the standardization of the types and symbols of representations.

Symmorphic space groups of finite crystal lattices

The symmorphic space group of a finite n-dimensional crystal lattice is studied and its factorization is presented. This lattice is determined by a direct product of cyclic groups and then its translation, point and space groups are defined as modular images of corresponding ones for a lattice of infinite extent. A factorization of holohedries (of an infinite lattice) is used in order to present a symmorphic finite space group as a direct product. As an example, and as a very special case, a space group in the case of the hypercubic lattice is studied and its irreducible representations are investigated. The results obtained suggest that for vectors lying on a surface of the first Brillouin zone (i.e. for which at least one coordinate is equal to 0 or 1/2) an additional index describing their symmetry properties should be introduced. This enables the authors to make a more detailed classification of states (energy levels). On the other hand, finite symmorphic space groups can be used within, e.g., the so-called finite-lattice approach.