# Bundle of Dirac (Elliptic) operator on $T^2$

I am a physicist with weak math background and am reading a paper (pdf) on the electronic structure of graphene(wiki) (a 2D carbon sheet with hexagonal lattice structure). In the paper they claim it is related with twisted Real K theory which makes me curious.

I will try to describe the problem as following. In the effective theory, the momentum space $\vec{k}$ is a 2-torus $T^2$. Also in graphene electron has spin and pesudospin which both have $SU(2)$ symmetry. Also there are too valleys. The Hamiltonian $H$ is a first order elliptic operator with some kind of inner symmetry.

The equation for a single valley should be $H\psi=E\psi$, where $\psi=(\psi_{1Au}, \psi_{1Ad}, \psi_{1Bu}, \psi_{1Bd}, \psi_{2Au}, \psi_{2Ad}, \psi_{2Bu}, \psi_{2Bd})$ is a spinor and $A,B$ are subindex for pesudospin and $u,d$ for spin, $1,2$ for two valleys.

A example of the Hamiltonian looks like this: $$-i\hbar v_F \begin{pmatrix} c & \partial_x -i\partial_y & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \partial_x +i\partial_y & -c & 0 & 0 & 0 & 0 & 0 & 0\\\\ 0 & 0 & -c & \partial_x -i\partial_y & 0 & 0 & 0 & 0\\\\ 0 & 0 & \partial_x +i\partial_y& c & 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0 & -c & \partial_x -i\partial_y & 0 & 0\\\\ 0 & 0 & 0 & 0 & \partial_x +i\partial_y & c & 0 & 0\\\\ 0 & 0 & 0 & 0 & 0 & 0 & c & \partial_x -i\partial_y\\\\ 0 & 0 & 0 & 0 & 0 & 0 & \partial_x +i\partial_y & -c \end{pmatrix}$$ where $c$ is a constant.

Can someone briefly show me how to apply K theory in this system to get the $Z_2$ classification of the bundle on $T^2$?

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I can suggest a superficial explanation which has nothing to do with your specific setup, but which nevertheless is probably related. An elliptic operator on a smooth manifold gives rise to a class in K-homology (dual to K-theory), and there is a natural pairing between K-homology and K-theory which, roughly speaking, associates to (elliptic operator, vector bundle) the Fredholm index of the elliptic operator twisted by the vector bundle. This also works (with some modifications) in real K-homology / K-theory. –  Paul Siegel Mar 19 '11 at 12:40