Geometry of the momentum space: From wire networks to quivers and monopoles

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 the general 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. In these geometric situations, the non- commutativity is introduced by a constant magnetic field and the theory splits into two pieces: commutative and non-commutative, both of which are governed by a $C^*$ geometry. In the non-commutative case, we can use tools such as K-theory to make statements about the band structure. In the commutative case, we give geometric and algebraic methods to study band intersections; these methods come from singularity theory and representation theory. We also provide new tools in the study, using $K$-theory and Chern classes. The latter can be computed using Berry connection in the momentum space. This brings monopole charges and issues of topological stability into the picture.Comment: 31 pages, 4 figure

Grain boundaries in graphene grown by chemical vapor deposition

The scientific literature on grain boundaries (GBs) in graphene was reviewed. The review focuses mainly on the experimental findings on graphene grown by chemical vapor deposition (CVD) under a very wide range of experimental conditions (temperature, pressure hydrogen/hydrocarbon ratio, gas flow velocity and substrates). Differences were found in the GBs depending on the origin of graphene: in micro-mechanically cleaved graphene (produced using graphite originating from high-temperature, high-pressure synthesis), rows of non-hexagonal rings separating two perfect graphene crystallites are found more frequently, while in graphene produced by CVD—despite the very wide range of growth conditions used in different laboratories—GBs with more pronounced disorder are more frequent. In connection with the observed disorder, the stability of two-dimensional amorphous carbon is discussed and the growth conditions that may impact on the structure of the GBs are reviewed. The most frequently used methods for the atomic scale characterization of the GB structures, their possibilities and limitations and the alterations of the GBs in CVD graphene during the investigation (e.g. under e-beam irradiation) are discussed. The effects of GB disorder on electric and thermal transport are reviewed and the relatively scarce data available on the chemical properties of the GBs are summarized. GBs are complex enough nanoobjects so that it may be unlikely that two experimentally produced GBs of several microns in length could be completely identical in all of their atomic scale details. Despite this, certain generalized conclusions may be formulated, which may be helpful for experimentalists in interpreting the results and in planning new experiments, leading to a more systematic picture of GBs in CVD graphene

From Graphene constrictions to single carbon chains

We present an atomic-resolution observation and analysis of graphene constrictions and ribbons with sub-nanometer width. Graphene membranes are studied by imaging side spherical aberration-corrected transmission electron microscopy at 80 kV. Holes are formed in the honeycomb-like structure due to radiation damage. As the holes grow and two holes approach each other, the hexagonal structure that lies between them narrows down. Transitions and deviations from the hexagonal structure in this graphene ribbon occur as its width shrinks below one nanometer. Some reconstructions, involving more pentagons and heptagons than hexagons, turn out to be surprisingly stable. Finally, single carbon atom chain bridges between graphene contacts are observed. The dynamics are observed in real time at atomic resolution with enough sensitivity to detect every carbon atom that remains stable for a sufficient amount of time. The carbon chains appear reproducibly and in various configurations from graphene bridges, between adsorbates, or at open edges and seem to represent one of the most stable configurations that a few-atomic carbon system accomodates in the presence of continuous energy input from the electron beam.Comment: 12 pages, 4 figure

Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on diluted bipartite lattices

The classic combinatorial construct of {\em maximum matchings} probes the random geometry of regions with local sublattice imbalance in a site-diluted bipartite lattice. We demonstrate that these regions, which host the monomers of any maximum matching of the lattice, control the localization properties of a zero-energy quantum particle hopping on this lattice. The structure theory of Dulmage and Mendelsohn provides us a way of identifying a complete and non-overlapping set of such regions. This motivates our large-scale computational study of the Dulmage-Mendelsohn decomposition of site-diluted bipartite lattices in two and three dimensions. Our computations uncover an interesting universality class of percolation associated with the end-to-end connectivity of such monomer-carrying regions with local sublattice imbalance, which we dub {\em Dulmage-Mendelsohn percolation}. Our results imply the existence of a monomer percolation transition in the classical statistical mechanics of the associated maximally-packed dimer model and the existence of a phase with area-law entanglement entropy of arbitrary many-body eigenstates of the corresponding quantum dimer model. They also have striking implications for the nature of collective zero-energy Majorana fermion excitations of bipartite networks of Majorana modes localized on sites of diluted lattices, for the character of topologically-protected zero-energy wavefunctions of the bipartite random hopping problem on such lattices, and thence for the corresponding quantum percolation problem, and for the nature of low-energy magnetic excitations in bipartite quantum antiferromagnets diluted by a small density of nonmagnetic impurities.Comment: minor typos and errors fixed; further clarifications added. no substantive changes in result

Artificial flat band systems: from lattice models to experiments

Certain lattice wave systems in translationally invariant settings have one or more spectral bands that are strictly flat or independent of momentum in the tight binding approximation, arising from either internal symmetries or fine-tuned coupling. These flat bands display remarkable strongly-interacting phases of matter. Originally considered as a theoretical convenience useful for obtaining exact analytical solutions of ferromagnetism, flat bands have now been observed in a variety of settings, ranging from electronic systems to ultracold atomic gases and photonic devices. Here we review the design and implementation of flat bands and chart future directions of this exciting field.Comment: 14 pages, 5 figures, to appear in Adv. Phys.:

Liftings and stresses for planar periodic frameworks

We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201