Publisher's Note: Graph theory data for topological quantum chemistry [Phys. Rev. E 96, 023310 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.96.023310.

Author Correction: Topological quantum chemistry.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Topology of Disconnected Elementary Band Representations.

Elementary band representations are the fundamental building blocks of atomic limit band structures. They have the defining property that at partial filling they cannot be both gapped and trivial. Here, we give two examples-one each in a symmorphic and a nonsymmorphic space group-of elementary band representations realized with an energy gap. In doing so, we explicitly construct a counterexample to a claim by Michel and Zak that single-valued elementary band representations in nonsymmorphic space groups with time-reversal symmetry are connected. For each example, we construct a topological invariant to explicitly demonstrate that the valence bands are nontrivial. We discover a new topological invariant: a movable but unremovable Dirac cone in the "Wilson Hamiltonian" and a bent-Z_{2} index.

Graph theory data for topological quantum chemistry.

Topological phases of noninteracting particles are distinguished by the global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties that are local (at high-symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [Bradlyn et al., Nature (London) 547, 298 (2017)NATUAS0028-083610.1038/nature23268] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain "elementary" band structures highlighted in the aforementioned reference, and thus we identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed bandrep program on the Bilbao Crystallographic Server to access the results of our computation.

Topological quantum chemistry.

Since the discovery of topological insulators and semimetals, there has been much research into predicting and experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non-trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more.

The effect of chemical variations on the structural polarity of relaxor ferroelectrics studied by resonance Raman spectroscopy.

Resonance Raman spectroscopy was applied to doped PbSc0.5Ta0.5O3 and PbSc0.5Nb0.5O3 relaxor ferroelectrics, to better understand the effect of composition disorder on the mesoscopic-scale polar order in complex perovskite-type (ABO3) ferroelectrics. The excitation photon energy used was 3.8 eV, which is slightly above the energy gap and corresponds to the maximum of the optical dielectric permittivity. Group-theory analysis reveals that the resonance Raman scattering (RRS) observed under these conditions is allowed only in polar crystal classes. Therefore, RRS is dominated by the atomic dynamics of nanoregions with coherent polar distortions, which considerably facilitates the comparison of polar order in various compounds. The results show that A-site doping (Ba(2+), Sr(2+), La(3+), Bi(3+)) has significantly stronger effect on the structural polarity than the introduction of a third element at the B site (Nb(5+) or Sn(4+) doped in PbSc0.5Ta0.5O3). The A-site substitution by cations that in contrast to Pb(2+) have isotropic outermost electron shells disturbs the system of lone-pair electrons, thus reducing the correlation length of coupled polar distortions and the strength of the electric field associated with the mean polarization of polar nanoregions. A-site doping with larger cations (Ba(2+)) augments the polar deformation of the individual BO6 octahedra due to local elastic fields. As a result, such A-site doping intensifies the initial structural polarity at high temperatures and prevails the enlargement of the polar fraction at low temperatures. A-site doping with smaller cations (Sr(2+), La(3+)), regardless if they are isovalent or aliovalent to Pb(2+), increases the correlation length of antiferrodistortive order (BO6 tilts), which in turn assists the development of double-perovskite structure with coherent local polar distortions. A-site doping with aliovalent cations (Bi(3+)) having the same outermost electron shell and ionic radius as the host A-site Pb(2+) cations leads to stronger coupling along the -B-O-B- bond linkages due to enhanced random local electric fields.

Symmetry conditions for type II multiferroicity in commensurate magnetic structures.

Type II multiferroics are magnetically ordered phases that exhibit ferroelectricity as a magnetic induced effect. We show that in single-k magnetic phases the presence in the paramagnetic phase of non-symmorphic symmetry combined with some specific type of magnetic propagation vector can be sufficient for the occurrence of this type of multiferroic behaviour. Other symmetry scenarios especially favourable for spin driven multiferroicity are also presented. We review and classify known type II multiferroics under this viewpoint. In addition, some other magnetic phases which due to their symmetry properties can exhibit type II multiferroicity are pointed out.

Magnetic superspace groups and symmetry constraints in incommensurate magnetic phases.

Superspace symmetry has been for many years the standard approach for the analysis of non-magnetic modulated crystals because of its robust and efficient treatment of the structural constraints present in incommensurate phases. For incommensurate magnetic phases, this generalized symmetry formalism can play a similar role. In this context we review from a practical viewpoint the superspace formalism particularized to magnetic incommensurate phases. We analyse in detail the relation between the description using superspace symmetry and the representation method. Important general rules on the symmetry of magnetic incommensurate modulations with a single propagation vector are derived. The power and efficiency of the method is illustrated with various examples, including some multiferroic materials. We show that the concept of superspace symmetry provides a simple, efficient and systematic way to characterize the symmetry and rationalize the structural and physical properties of incommensurate magnetic materials. This is especially relevant when the properties of incommensurate multiferroics are investigated.

Mode crystallography of distorted structures.

The description of displacive distorted structures in terms of symmetry-adapted modes is reviewed. A specific parameterization of the symmetry-mode decomposition of these pseudosymmetric structures defined on the setting of the experimental space group is proposed. This approach closely follows crystallographic conventions and permits a straightforward transformation between symmetry-mode and conventional descriptions of the structures. Multiple examples are presented showing the insight provided by the symmetry-mode approach. The methodology is shown at work, illustrating its various possibilities for improving the characterization of distorted structures, for example: detection of hidden structural correlations, identification of fundamental and marginal degrees of freedom, reduction of the effective number of atomic positional parameters, quantitative comparison of structures with the same or different space group, detection of false refinement minima, systematic characterization of thermal behavior, rationalization of phase diagrams and various symmetries in families of compounds etc. The close relation of the symmetry-mode description with the superspace formalism applied to commensurate superstructures is also discussed. Finally, the application of this methodology in the field of ab initio or first-principles calculations is outlined. At present, there are several freely available user-friendly computer tools for performing automatic symmetry-mode analyses. The use of these programs does not require a deep knowledge of group theory and can be applied either a posteriori to analyze a given distorted structure or a priori to parameterize the structure to be determined. It is hoped that this article will encourage the use of these tools. All the examples presented here have been worked out using the program AMPLIMODES [Orobengoa et al. (2009). J. Appl. Cryst. 42, 820-833].

Symmetry analysis of extinction rules in diffuse-scattering experiments.

Structured diffuse-scattering intensities, whether of compositional or of pure displacive origin, static or dynamic, contain important information about the symmetry of the individual compositional and/or displacive modes responsible for the observed intensities. However, the interpretation of the experimental data is very often impeded by the lack of a symmetry-based approach to the analysis of the structured diffuse-scattering distributions. Recently, we have demonstrated the existence of systematic phonon selection rules for diffuse scattering that depend on the symmetries of the mode and the scattering vector, and not on the specific structure. Here, we show that such symmetry analysis can be successfully extended and also applied to structure-dependent diffuse scattering associated with 'disordered' materials: the combination of theoretically determined, diffuse-scattering extinction conditions with the concept of non-characteristic orbits proves to be very useful in the interpretation of the observed diffuse-scattering extinctions. The utility of this approach is illustrated by the analysis of diffuse-scattering data from ThAsSe, FeOF and FeF(2). The essential part of the associated calculations are performed by the computer programs NEUTRON (systematic phonon extinction rules in inelastic scattering) and NONCHAR (non-characteristic orbits of space groups) that are available on the Bilbao crystallographic server (http://www.cryst.ehu.es).

Experimental study of the ferroelastic phase transition in urea/n-heptadecane composite.

An experimental study of the ferroelastic phase transition in urea/n-heptadecane CO(NH2)2/C17H40 composite around the structural phase transition undergone by this crystal at 159 K is presented. The temperature dependence of the macroscopic spontaneous strain and the optical birefringence around this temperature has been determined. A phenomenological model limited to the hexagonal-orthorhombic change of the urea sublattice leads to a linear relation between these quantities and the phase transition entropy. The experimental data agree with this description exception made of the near vicinity of the phase transition, where the influence of the alkane chain ordering processes cannot be excluded.

Theory of the cooperative evaporation of volatile droplets.

A theoretical approach to the cooperative process of evaporation of volatile liquid droplet arrays is proposed which combines the equation of the liquid-gas interface with a Landau-type description. The spatial and temporal behaviors of the process are shown to be governed by the spontaneous symmetry breaking mechanism of a parent droplet configuration. The applicability of the approach is illustrated by the theoretical description of recent experiments in closed and open systems.

Prediction of new displacive ferroelectrics through systematic pseudosymmetry search. Results for materials with Pba2 and Pmc21 symmetry.

Polar structures with pseudosymmetry related to a hypothetical non-polar configuration are considered as good candidates for ferroelectrics. Recently, a procedure has been developed for a systematic pseudosymmetry search among structures with a given space-group symmetry. The aim of this paper is the extension of the pseudosymmetry procedure to the case of structures with polar symmetry and its application in the search for new ferroelectrics. The results obtained by the generalized pseudosymmetry search among the compounds with symmetries Pba2 and Pmc2(1) listed in the Inorganic Crystal Structure Database are discussed. The calculations have been performed by the program PSEUDO, which forms part of the Bilbao Crystallographic Server (http://www.cryst.ehu.es). In addition, an empirical relation between the atomic displacements necessary to reach the non-polar structure and the transition temperature is proposed and compared with the Abrahams-Kurtz-Jamieson relation.

Circles: the replication-recombination-chromosome segregation connection.

Crossing over by homologous recombination between monomeric circular chromosomes generates dimeric circular chromosomes that cannot be segregated to daughter cells during cell division. In Escherichia coli, homologous recombination is biased so that most homologous recombination events generate noncrossover monomeric circular chromosomes. This bias is lost in ruv mutants. A novel protein, RarA, which is highly conserved in eubacteria and eukaryotes and is related to the RuvB and the DnaX proteins, gamma and tau, may influence the formation of crossover recombinants. Those dimeric chromosomes that do form are converted to monomers by Xer site-specific recombination at the recombination site dif, located in the replication terminus region of the E. coli chromosome. The septum-located FtsK protein, which coordinates cell division with chromosome segregation, is required for a complete Xer recombination reaction at dif. Only correctly positioned dif sites present in a chromosomal dimer are able to access septum-located FtsK. FtsK acts by facilitating a conformational change in the Xer recombination Holliday junction intermediate formed by XerC recombinase. This change provides a substrate for XerD, which then completes the recombination reaction.

Ferroelectric--paraelectric phase transitions with no group--supergroup relation between the space groups of both phases?

The structures of Sr(3)(FeF(6))(2), beta-NbO(2), TlBO(2) and CrOF(3), previously reported as possible ferroelectrics with no group-supergroup relation between the ferroelectric and the paraelectric symmetries, have been carefully studied. We could not confirm any structural pseudosymmetry with respect to a space group which is not a supergroup of their room-temperature polar space group. In all cases, pseudosymmetry was indeed detected, but only for non-polar supergroups of the actual space groups of the structures. In this sense, the four compounds are possible ferroelectrics, but fulfilling the usual group-supergroup relation between the phase symmetries.

X-ray diffraction study of the phase transition of K2Mn2(BeF4)3: a new type of low-temperature structure for langbeinites.

The potassium manganese tetrafluoroberyllate langbeinite compound has been studied in the temperature range 100-300 K. Using DSC measurements, a phase transition has been detected at 213 K. The space group of the low-temperature phase was determined to be P112(1) using X-ray diffraction experiments and optical observations of the domain structure. The b axis is doubled with respect to the prototypic P2(1)3 cubic phase. Lattice parameters were determined by powder diffraction data [a = 10.0690 (8), b = 20.136 (2), c = 10.0329 (4) A, gamma = 90.01 (1) degrees ]. A precise analysis of the BeF(4) tetrahedra in the low-temperature phase shows that two independent tetrahedra rotate in opposite directions along the doubled crystallographic axis. A symmetry mode analysis of the monoclinic distortion is also reported. This is the first report of the existence of such a phase transition in K(2)Mn(2)(BeF(4))(3) and also of a new type of low-temperature structure for langbeinite compounds.

The application of Hermann's group M in group-subgroup relations between space groups.

This paper is devoted to the study of the group-subgroup relations U < G between space groups. A procedure has been developed for the derivation of all subgroups U(j) < G which are obtained from U by a transformation with a translation (T-equivalent subgroups). All T-equivalent supergroups G(k) > U can be determined in the same way from one supergroup G > U. The decisive group in this procedure is the translation part of the (Euclidean) normalizer of Hermann's group M. The group M is the uniquely determined group U < or = M < or = G with the translations of G and the point group of U. The method is particularly useful in the search for supergroups of space groups and is based on several lemmata which are formulated and proven in this paper. The results suggest under special conditions the possibility of a transition with 'region' formation in some analogy to the well known domain formation. This transition could occur from high symmetry to low symmetry or from low symmetry to high symmetry or even both ways.

Characterization of the room-temperature phase of Tl2MoO4: crystal structure, symmetry mode analysis and second-harmonic generation measurements.

The crystal structure of the glaserite-related compound dithallium(I)-molybdate(VI), which at 293 K crystallizes monoclinic, space group C121 with lattice parameters a = 10.565 (3), b = 6.418 (1), c = 8.039 (2) A, beta = 91.05 (4) degrees, has been determined. The structure was refined as an inversion twin to a final R(F(all)) value of 0.0611 for 1006 unique reflections [R(F(obs)) = 0.0285 for 644 observed reflections]. Second-harmonic generation measurements led to a value of d(eff) = 5.5 +/- 0.5 pm V(-1) as an estimation of the second-harmonic conversion efficiency at phase matching. Symmetry mode analysis shows that, in general, primary modes have the highest amplitudes, yet surprisingly some of the secondary modes assume amplitudes of comparable magnitude. A comparison of the phase at 293 K with that at 350 K (space group P3m1) shows that the main change can be described as a rotation of the molybdate tetrahedra around the trigonal a(b) axis. The molybdate tetrahedra as well as the octahedra around one of the symmetry-independent Tl atoms are more strongly distorted in the monoclinic phase. The coordination number for the other two Tl atoms is decreased from 12 and 10 in the high-symmetry phase to 10 and 9 in the monoclinic phase. Furthermore, the number of common edges between the Tl and Mo coordination polyhedra is reduced and the common face which is observed between them in the high-temperature phase is changed to a common edge in the low-temperature phase. The contribution of the primary symmetry modes leads exactly to this change in the coordination spheres of the atoms.

FtsK functions in the processing of a Holliday junction intermediate during bacterial chromosome segregation.

In bacteria with circular chromosomes, homologous recombination can generate chromosome dimers that cannot be segregated to daughter cells at cell division. Xer site-specific recombination at dif, a 28-bp site located in the replication terminus region of the chromosome, converts dimers to monomers through the sequential action of the XerC and XerD recombinases. Chromosome dimer resolution requires that dif is positioned correctly in the chromosome, and the activity of FtsK, a septum-located protein that coordinates cell division with chromosome segregation. Here, we show that cycles of XerC-mediated strand exchanges form and resolve Holliday junction intermediates back to substrate irrespective of whether conditions support a complete recombination reaction. The C-terminal domain of FtsK is sufficient to activate the exchange of the second pair of strands by XerD, allowing both intra- and intermolecular recombination reactions to go to completion. Proper positioning of dif in the chromosome and of FtsK at the septum is required to sense the multimeric state of newly replicated chromosomes and restrict complete Xer reactions to dimeric chromosomes.