I'm struggling to reconcile two concepts and understand if the Berry curvature in graphene is zero or non-zero. Following the reference here, given a generic two-level Hamiltonian (eqn 1.15)
$$H=\boldsymbol{\sigma}\cdot\mathbf{h}$$
where $\boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ is the vector of pauli matrices, the Berry curvature in vector form is (eqn 1.20)
$$\boldsymbol{\Omega} = \frac{1}{2}\frac{\mathbf{h}}{h^3}$$
Thus it appears that the low energy Hamiltonian of graphene $H=\sigma_x k_x + \sigma_y k_y$ must be non-zero and $\mathbf{k}$-dependent, actually is it not $\boldsymbol{\Omega} = \mathbf{k}/2k^3$?
However in the same reference (eqn 3.22) it goes on to say that in graphene (same Hamiltonian as above) "the Berry curvature vanishes everywhere except at the Dirac points where it diverges", i.e. it is zero almost everywhere. These two assertions seem contradictory. I would appreciate help in understanding what I misunderstanding here.
Best Answer
The precise statement should be:
Why do we need to care about the third-component of the Berry curvature? Because it is the only component that contributes to the Chern number $C$ of a 2D band structure $$C=\frac{1}{2\pi}\int_\text{BZ}\mathrm{d}^2\boldsymbol{k}\; \Omega_3(\boldsymbol{k}).$$ Therefore $\Omega_3$ is also named as the Berry flux density or the Chern density. The Berry curvature near the Dirac point is indeed given by $\boldsymbol{\Omega}=\boldsymbol{k}/2k^3$, which is non-vanishing. But since the momentum $\boldsymbol{k}=(k_x,k_y,0)$ lies in the $xy$-plane and has no third component, so the Berry flux density $\Omega_3$ vanishes everywhere except at the origin.
To see what happens at the origin, we need to regularize the problem with a small mass. Consider $$H=k_x\sigma^x+k_y\sigma^y+m\sigma^z.$$ One finds $$\Omega_3(\boldsymbol{k})=\frac{m}{2(\boldsymbol{k}^2+m^2)^{3/2}}.$$ As $m\to0$, one can see that $\Omega_3\sim m/k^3\to0$ vanishes everywhere (as long as $k\neq0$). But at the Dirac point where $k=0$, $\Omega_3\sim \pm1/m^2\to\pm\infty$ diverges to either $+\infty$ or $-\infty$ depending on the sign of the mass $m$. In this case, since the band gap vanishes, the Chern number is not well defined, so usually, it is also not meaningful to talk about the Berry flux density at the Dirac point.