Modified group projectors: tight-binding method

Abstract

Abstract. 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 π-bands of the single wall nanotubes: together with dispersion relations, their complete symmetry assignation by the full symmetry (line) groups and the corresponding symmetry-adapted eigen function are found.

Carbon nanotubes band assignation, topology, Bloch states, and selection rules

“…They correspond to different parameterizations of the irreducible representations 7,8 of the line groups L C . The inherent physical meaning of the quantum numbers makes one or another choice more suitable for different physical considerations.…”

“…7. Besides the pair (k, m) of linear and z-component of the angular momenta (or (k,m) of helical and pure angular momenta), some of the irreducible representations carry also the quantum numbers of parities with respect to the U axis, and mirror planes σ v or σ h .…”

“…Knowing this representation the eigen states (in the form of the generalized Bloch functions) and the eigen energies (organized as the energy bands) may be easily found 6,7 . Analysis of the linear and helical symmetry based quantum numbers and their mutual relations is performed in section II; this enable to emphasize a priori properties of the band structures, referring to any subsystem (electrons, phonons, etc.).…”

Symmetry Based Fundamentals of Carbon Nanotubes

“…However, for crystalline systems, due to the infinite symmetry group and the infinite dimensional state, this procedure, involving summation of D( ) over group elements, is inappropriate at least numerically. These deficiencies are cured by the modified group projector technique (MGPT) [17,18]. It is fully implemented into the polymer symmetry simulation package POLSym devoted to the physics of the quasi-1D crystals [19], which is used in the forthcoming calculations.…”

“…This manifests that m is not a conserved quantum number: it is related to the isogonal group (2.6), also containing rotations that are not isolated (but only combined with translations) symmetries of NTs. Therefore, sometimes it is plausible to use the alternative set of conserved quantum numbers [5,18]. These helical quantum numbers [20] are the helical momentumk, including linear and part of the angular momentum, and the remaining angular momentumm, related to k and m as:…”