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Considered are eighty sets of layer groups, each set consisted of four groups: ordinary single and double, and gray single and double layer group. Structural properties of layer groups (factorization onto cyclic subgroups and existence of grading according to the sequence of halving subgroups) enable efficient symbolic computation (by POLSym code) of the relevant properties, real and complex irreducible and allowed (half-)integer (co-)representations in particular. This task includes, as the first step, classification of the irreducible domains based on the group action in Brillouin zone combined with torus topology. Also, the band (co-)representations induced from the irreducible (co-)representations of Wyckoff position stabilizers (site symmetry groups) are decomposed onto the irreducible components. These, and other layer group symmetry related theoretical data relevant for physics, layered materials in particular, are tabulated and made available through the web site., Comment: 12 pages, 2 figures

We present the full in-plane phonon dispersion of graphite obtained from inelastic x-ray scattering, including the optical and acoustic branches, as well as the mid-frequency range between the $K$ and $M$ points in the Brillouin zone, where experimental data have been unavailable so far. The existence of a Kohn anomaly at the $K$ point is further supported. We fit a fifth-nearest neighbour force-constants model to the experimental data, making improved force-constants calculations of the phonon dispersion in both graphite and carbon nanotubes available., Comment: 7 pages; submitted to Phys. Rev. B

Rolling up of a layer with arbitrary lattice gives nanotube with very reach symmetry, described by a line group which is found for arbitrary diperiodic group of the layer and chiral vector. Helical axis and pure rotations are always present, while the mirror and glide planes appear only for specific chiral vectors in rhombic and rectangular lattices. Nanotubes are not translationally periodical unless layer cell satisfies very specific conditions. Physical consequences, including incommensurability of carbon nanotubes, are discussed., Comment: 5 pages, 1 table, 1 figure

Various properties of the energy band structures (electronic, phonon, etc.), including systematic band degeneracy, sticking and extremes, following from the full line group symmetry of the single-wall carbon nanotubes are established. The complete set of quantum numbers consists of quasi momenta (angular and linear or helical) and parities with respect to the z-reversal symmetries and, for achiral tubes, the vertical plane. The assignation of the electronic bands is performed, and the generalized Bloch symmetry adapted eigen functions are derived. The most important physical tensors are characterized by the same set of quantum numbers. All this enables application of the presented exhaustive selection rules. The results are discussed by some examples, e.g. allowed interband transitions, conductivity, Raman tensor, etc., Comment: 11 pages, 2 figures, 2 tables; pdf available from: http://www.ff.bg.ac.yu/qmf/qsg_e.htm

The friction between the walls of multi-wall carbon nanotubes is shown to be extremely low in general, with important details related to the specific choice of the walls. This is governed by a simple expression revealing that the phenomenon is a profound consequence of the specific symmetry breaking: super-slippery sliding of the incommensurate walls is a Goldstone mode. Three universal principles of tribology, offering a recipe for the lubricant selection are emphasized., Comment: 4 pages, 2 figures, 1 table; pdf available from: http://www.ff.bg.ac.yu/qmf/qsg_e.htm

Twin observables, i.e. opposite subsystem observables A+ and A- that are indistinguishable in measurement in a given mixed or pure state W, are investigated in detail algebraicly and geometrically. It is shown that there is a far-reaching correspondence between the detectable (in W) spectral entities of the two operators. Twin observables are state-dependently quantum-logically equivalent, and direct subsystem measurement of one of them ipso facto gives rise to the indirect (i.e. distant) measurement of the other. Existence of nontrivial twins requires singularity of W. Systems in thermodynamic equilibrium do not admit subsystem twins. These observables may enable one to simplify the matrix representing W., Comment: 13 pages

Utilizing the line-group symmetry of single-wall nanotubes, we have assigned their electron-energy bands by the symmetry-based quantum numbers. The selection rules for optical absorption are presented in terms of these quantum numbers. Different interband transitions become allowed as the polarization of incident light is varied, and we predict a substantial optical dichroism. We propose how to observe this effect in experiments on a single nanotube, and how it can be used to control quantum transport in nanotubes to obtain information about the structure., Comment: 4 pages, 3 fiures, to appear in Phy. Rev. B

The general form of the operators commuting with the ground representation (appearing in many physical problems within single particle approximation) of the group is found. With help of the modified group projector technique, this result is applied to the system of identical particles with spin independent interaction, to derive the Dirac-Heisenberg hamiltonian and its effective space for arbitrary orbital occupation numbers and arbitrary spin. This gives transparent insight into the physical contents of this hamiltonian, showing that formal generalizations with spin greater than 1/2 involve nontrivial additional physical assumptions., Comment: 10 pages

Modified group projector technique for induced representations is a powerful tool for calculation and symmetry quantum numbers assignation of a tight binding Hamiltonian energy bands of crystals. Namely, the induced type structure of such a Hamiltonian enables efficient application of the procedure: only the interior representations of the orbit stabilizers are to be considered. Then the generalized Bloch eigen functions are obtained naturally by the expansion to the whole state space. The method is applied to the electronic pi-bands of the single wall carbon nanotubes: together with dispersion relations, their complete symmetry assignation by the full symmetry (line) groups and the corresponding symmetry-adapted eigen function are found., Comment: 10 pages 1 figure

The full geometrical symmetry groups (the line groups) of the monolayered, 2Hb and 3R polytypes of the inorganic MoS2 and WS2 micro- and nanotubes of arbitrary chirality are found. This is used to find the coordinates of the representative atoms sufficient to determine completely geometrical structure of tubes. Then some physical properties which can be deduced from the symmetry are discussed: electron band degeneracies, selection rules, general forms of the second rank tensors and potentials, phonon spectra., Comment: 6 pages 1 figure

For the single-wall carbon nanotubes conducting in the simplest tight binding model, the complete set of line group symmetry based quantum numbers for the bands crossing at Fermi level are given. Besides linear (k), helical (k'} and angular momenta, emerging from roto-translational symmetries, the parities of U axis and (in the zig-zag and armchair cases only) mirror planes appear in the assignation. The helical and angular momentum quantum numbers of the crossing bands never vanishes, what supports proposed chirality of currents. Except for the armchair tubes, the crossing bands have the same quantum numbers and, according to the non-crossing rule, a secondary gap arises, as it is shown by the accurate tight-binding calculation. In the armchair case the different vertical mirror parity of the crossing bands provides substantial conductivity, though kF is slightly decreased., Comment: 6 pages, 2 figures

The full symmetry groups for all single- and multi-wall carbon nanotubes are found. As for the single-wall tubes, the symmetries form nonabelian nonsymorphic line groups, enlarging the groups reported in literature. In the multi-wall case, any type of the line and the axial point groups can be obtained, depending on single-wall constituents and their relative position. Several other consequences are discussed: quantum numbers and related selection rules, electronic and phonon bands, and their degeneracy, application to tensor properties., Comment: 9 pages, 2 figures, revtex

The irreducible representations of all of the 80 diperiodic groups, being the symmetries of the systems translationally periodical in two directions, are calculated. To this end, each of these groups is factorized as the product of a generalized translational group and an axial point group. The results are presented in the form of the tables, containing the matrices of the irreducible representations of the generators of the groups. General properties and some physical applications (degeneracy and topology of the energy bands, selection rules, etc.) are discussed., Comment: 30 pages, 5 figures, 28 tables, 18 refs, LaTex2.09