Theoretical results for the Gyroid wire system
ORAL
Abstract
the complements of triply periodic surfaces. Our predominant example is the
gyroid surface and its complementary wire network which can be fabricated on a
nanoscale, but we also worked on wire networks stemming from other CMC
surfaces. For all these geometries, we study the Harper Hamiltonian and its band
structure in the commutative and non-commutative cases. In the latter, an
ambient magnetic field destroys the commutativity of the relevant algebras. We
have successfully applied methods from singularity theory, representation theory
and topology to these materials. In this setting the gyroid geometry can be seen as
the 3d generalization of graphene. We will discuss the most important results of our work.
* I acknowledge suport from the NSF under the grant PHY-1255409.
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Publication: 1.) The geometry of the Double Gyroid wire network: Quantum and
Classical, J. Noncomm. Geom. 6 (2012) 623-664.
2.) The noncommutative geometry of wire networks from triply
periodic surfaces, J. Phys.: Conf. Ser. 343 (2012), 012054.
3.) Singularities, swallowtails and Dirac points. An analysis for families
of Hamiltonians and applications to wire networks, especially the
Gyroid, Ann. Phys. 327 (2012) 2865-2884.
4.) Projective representations from quantum enhanced graph
symmetries, J. Phys.: Conf. Ser. 597 (2015), 012048.
5.) Re-gauging groupoid, symmetries and degeneracies for Graph
Hamiltonians and applications to the Gyroid wire network,
Annales Henri Poincare 17 (2016) 1383, 1414
6.) Singular geometry of the momentum space: From wire networks to
quivers and monopoles, J. Sing. Theory 15 (2016) 53, 80
7.) Local models and global constraints for degeneracies and band
crossings, J. Geom. and Phys. 158 (2020) 103892-103901
Presenters
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Birgit B Kaufmann
Purdue University
Authors
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Birgit B Kaufmann
Purdue University
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Ralph Kaufmann
Purdue University
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Sergei Khlebnikov
Purdue University