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 Středa'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, the adiabatic continuity enables us to regard graphene as a simulator of square lattice with half flux quantum per unit cell, which implies, with a semiclassical treatment around rational fluxes implemented here, 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. Possible experimental setups for observing this topological numbers in cold atoms are proposed as well.

Figure 1

For the square lattice, the band dispersion (with the energy as a horizontal axis), density of states $D\left(E\right)$, Chern number $t_r$, polarization topological number $s_r$ and screened polarization ${\tilde{s}}_r$ are plotted against energy for a small $\varphi =\delta$ (left) or around the $\pi$ flux, $\varphi =1/2+\delta$ (right) with $\delta =1/1223$. Small gaps are neglected. Dashed lines indicate van Hove singularities.

Figure 2

(a) Hofstadter butterfly (one-particle energy spectrum vs magnetic field $\varphi )$ for the tight-binding model for honeycomb (left) or for square (right) lattices. To indicate the correspondence, $1/q\leftrightarrow 1/2+1/2q$, between the two cases, the flux scale is halved with a shift by 1/2 in the honeycomb. Here the butterfly for flux $\varphi =p/q$ with all the prime $q\le 179\left(1987\right)$ are plotted for honeycomb (square). (b) For the honeycomb lattice, the band dispersion (with the energy as a horizontal axis), density of states, Chern number $t_r$, the polarization topological number $s_r$ and reduced polarization ${\tilde{s}}_r$ are plotted against energy for a weak $\varphi =\delta (=1/107)$. Small gaps are neglected. Dashed lines indicate van Hove singularities.

Figure 3

The adiabatic connection between graphene with $\varphi =1/q$ and the square lattice with $\Phi =(q+1)/2q$ around $\pi$ flux (with the diagonal hopping $t^{\prime }=-1$ in the inset), here displayed in terms of the gaps in the energy spectrum (bold lines).