Structure and classification of band representations

Michel, Louis



Abstract : Band representaitons in solids are investigated in the general framework of induced representations by using the concepts of orbits (stars) and strata (Wyckoff positions) in their construction and classification. The connection between band representations and irreducible representations of space groups is established by reducing the former in the basis of quasi-Bloch functions wich are eigenfunctions of translations but ar not, in general, eigenfunctions of the Hamiltonian. While irreducible representations of space group ar fnite-dimensional and are induced from infinite-order little groups Gk for vectors K in the Brillouin zone, band representations are infinite-dimensional adn are induced from finite-order little groups Gr for vectors r in the Wigner-Seitz cell. This connection between irreductible representations and band representations of space groups shedd new light on the duality properties of the Brillouin zone and the Wigner-Seitz cell. As an introduction to band representations the induced representations of point groups wich id applied to the investigation of th equivalency of band representations. Based on this connection and on the properties of the crystallographic point groups a necessary condition id established for the inequivalency of band representations induced from maximal isotropy groups. For using this condition there is need fro the induced representaitons of oint groups and a full list of them is given in the paper. One is especially interested in irreducible-band representations which form the elementary building bricks for band representations. From the point of view of the physics, irreducible-band representations correspond to energy bands with minimal numbers of branches. A method id developped for finding all the inequivalent irreducible-band representations of space groups by using the induction from maximal isotropy groups. As a rule the latter leads to inequivalent irreducible-band representations. There are, however, few exceptions to this rule. A full list of such exceptions is tabulated in the paper. With this list at hand one can construct all the different irreducible-band representations of 2-dimensional space groups. For them we list the continuity chords of all their irreducible-band representations.




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MICHEL, BACRY, et ZAK, “Structure and classification of band representations,” Archives de l'IHES, consulté le 22 juin 2024,