Polarization as a topological quantum number in graphene

Graphene, with its quantum Hall topological (Chern) number reflecting the massless Dirac particle, is shown to harbor yet another topological quantum number. This is obtained by combining Streda's general formula for the polarization associated with a second topological number in the Diophantine equation for the Hofstadter problem, and the adiabatic continuity, earlier shown to exist between the square and honeycomb lattices by Hatsugai et al. Specifically, we can regard, from the adiabatic continuity, graphene as a ``simulator" of square lattice with half flux quantum per unit cell, which implies that the polarization topological numbers in graphene in weak magnetic fields is 1/2 per Dirac cone for the energy region between the van Hove singularities, signifying a new quantum number characterizing graphene.