Singular Geometry of the Momentum Space: From wire networks to quivers and monopoles

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

Journal of Singularities
volume 15 (2016), 53-79

Received: 30 September 2014. Received in revised form: 19 May 2015.

DOI: 10.5427/jsing.2016.15d

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Abstract:

A new nano-material in the form of a double gyroid has motivated us to study (non)-commutative C^* geometry of periodic wire networks and the associated graph Hamiltonians.

Here we present a general more abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. The resulting effective C^*-geometry is that of the momentum space, which parameterizes the quasi--momenta.

This geometry is usually singular, where the singularities describe so-called band intersections in physics. We give geometric and algebraic methods to study these intersections; their origin being singularity theory and representation theory. A technique we newly apply to this situation is the use of topological invariants, which we formalize and explain in the paper. This uses K-theory and Chern classes as well as "slicing methods" for their computation. In this method the invariants can be computed using Berry's connection in the momentum space. This brings monopole charges and issues of topological stability into the picture.

Adding a constant magnetic field or allowing projective representations makes the C^* geometry non-commutative. In this case, we can also use K-theory, albeit in a different way, to make statements about the band structure using gap labeling.

Author(s) information:

Ralph M. Kaufmann	Sergei Khlebnikov	Birgit Wehefritz-Kaufmann
Department of Mathematics	Department of Physics and Astronomy	Dept. of Mathematics and Dept. of Physics and Astronomy
Purdue University	Purdue University	Purdue University
West Lafayette, IN 47907, USA	West Lafayette, IN 47907, USA	West Lafayette, IN 47907, USA
email: rkaufman@math.purdue.edu	email: skhleb@physics.purdue.edu	email: ebkaufma@math.purdue.edu