Re-Gauging Groupoid, Symmetries and Degeneracies for Graph Hamiltonians and Applications to the Gyroid Wire Network

Kaufmann, Ralph M.; Khlebnikov, Sergei; Wehefritz-Kaufmann, Birgit

doi:10.1007/s00023-015-0443-8

Journal Article

PUBDB-2018-00564

Re-Gauging Groupoid, Symmetries and Degeneracies for Graph Hamiltonians and Applications to the Gyroid Wire Network

Kaufmann, R. M. (Corresponding author) ; Khlebnikov, S. ; Wehefritz-Kaufmann, B.

2016
Birkhäuser Basel

Annales Henri Poincaré 17(6), 1383 - 1414 (2016) [10.1007/s00023-015-0443-8]

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Please use a persistent id in citations: doi:10.1007/s00023-015-0443-8 doi:10.3204/PUBDB-2018-00564

Report No.: DESY-12-133; arXiv:1208.3266

Abstract: We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)-commutative geometries. By selecting gauging data, these geometries are realized by matrices through an explicit construction or a Kan extension. We describe the changes in gauge via the action of a re-gauging groupoid. It acts via matrices that give rise to a noncommutative 2-cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of re-gaugings. In the commutative case, we deduce that the extended symmetries act via a projective representation. This yields isotypical decompositions and super-selection rules. We apply these results to the primitive cubic, diamond, gyroid and honeycomb wire networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid) and the honeycomb systems.

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