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Triangular and honeycomb lattices are dual to each other -- if we punctureholes into a featureless plane in a regular triangular alignment, the remainingbody looks like a honeycomb lattice, and vice versa, if the holes are in aregular honeycomb alignment, the remaining body has a feature of triangularlattice. In this work, we reveal that the electronic states in graphene sheetswith nano-sized holes in triangular and honeycomb alignments are also dual toeach other in a topological sense. Namely, a regular hole array perforated ingraphene can open a band gap in the energy-momentum dispersion of relativisticelectrons in the pristine graphene, and the insulating states induced bytriangular and honeycomb hole arrays are distinct in topology. In a graphenepatchwork with regions of these two hole arrays put side by sidecounterpropagating topological currents emerge at the domain wall. Thisobservation indicates that the cerebrated atomically thin sheet is wheretopological physics and nanotechnology meet.

Topological Aspect of Graphene Patchwork with Regular Arrays of Nano Holes