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The Dirac operator at 2 dimension can be written as

$$ D=\sum_{k=1,2}\sigma^{k}D_{k}=\left( \begin{array}{cc} 0 & \partial_{x}-i\partial_{y}-i(A_x-iA_y)\\ \partial_{x}+i\partial_{y}-i(A_x+iA_y) & 0 \end{array} \right) , $$ where $A_i$ is a gauge field.

If we add some other terms to the operator above, such as $\sigma^{3}\Delta$, where $\Delta$ is a constant, is it still a Dirac operator? Can the last term be understood as some kind of gauge field?


Clear the background of the question: I am reading Kane and Mele's paper on $Z_2$ topological order (pdf). For my understanding the general question is how the inner structure (in this case is time reversal symmetry) of a Hamiltonian can be used to classify the band structure. They said in this case it is related with twisted Real K theory.

Is there anyone know how to apply K theory in the graphene case?

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Well, strictly mathematically speaking, only object that can be called a Dirac operator is a thing $D$ that obeys $D^2 = \Delta$ (here $\Delta$ is a Laplacian). This can be relaxed with replacing the Laplacian with an object built out of covariant derivatives (so that your $D$ with gauge $A$ can be also treated as a Dirac operator). But your additional term surely spoils even this weaker definition of Dirac operator. In any case, what is your motivation for this? In other words, how would knowing that this object is a Dirac operator help you? – Marek Mar 18 '11 at 14:06
@ Marek Thanks for your answer. I am reading Kane&Mele(2005) paper on $Z_2$ TI and QSHE. They said that the problem is related with twisted real K theory. I am trying to understand the paper and want to know if it is easier to understand it with those mathematical tools(Atiyah-Singer K theorem etc.). Well in the graphene case, there are spin and pesudospin. So it makes the problem a bit complex. For my understanding if there is no gauge field, K index theory predict that zero mode number $V_R-V_L$ is zero. I am not clear where the Z_2 come from and how Kane&Mele's work related with K theory. – Z.Sun Mar 18 '11 at 14:26
I see; in that case you don't really need it to be a Dirac operator, elliptic operator is good enough. Unfortunately I don't know much about K theory and even much less about its applications to graphene so I can't really help you. But it's a very interesting topic and I'll be looking forward to answers; I am sure we have people here who know something about this stuff. – Marek Mar 18 '11 at 15:17
Ellipticity depends on the 1st-order terms only, so for example you can forget about the zeroeth-order terms involving the connection, A. Now it looks like you are only proposing adding zeroeth-order terms (if I understand your notation correctly), so you would not be changing the ellipticity of the operator. In particular, smoothness results about solutions of differential equations involving your operator will not be affected, nor will arguments about the dimensions of the spaces of zero modes. – Eric Zaslow Mar 18 '11 at 16:59

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