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I need your help.

I consider this Hamiltonian

$H=v(\xi k_y \sigma_x+k_x\sigma_y)+m\sigma_z $

$\xi$ is valley index. It is +1 (at K' valley) or -1 (at K valley ). I want to know Chern number of this massive Dirac Hamiltonian. So I calculated Eigen values and Eigen vectors.

$E_\pm=\pm\sqrt{v^2(k_x^2+k_y^2)+m^2}=\pm|E|$

$|\phi_+> = \frac{1}{\sqrt{2E(E-m)}}(\matrix{v(\xi k_y-ik_x) \\ E-m})$

$|\phi_-> = \frac{1}{\sqrt{2E(E-m)}}(\matrix{-E+m \\ -v(\xi k_y+i k_x)})$

And I calculated Berry connection $\vec{A_\pm}=i<\phi_\pm|\nabla_k|\phi_\pm>$ and curvature $\vec{B}=\nabla \times \vec{A_\pm}$.

$A_{\pm x}=\pm \frac{\xi k_y v^2}{2E(E-m)}$

$A_{\pm y}=\mp \frac{\xi k_x v^2}{2E(E-m)}$

$A_{\pm z}= 0 $

$\vec{B}_{\pm}=(0,0,\mp \frac{\xi v^2}{2E^3})$

I think these calculations are correct. But I don't know how to calculate Chern number.

$v_{\pm} = \int_{BZ}\frac{d^2\vec{k}}{2\pi}\vec{B}_{\pm} $

I think I should use stokes theorem. BZ means Brillouin Zone.

$v_{\pm} = \int_{BZ}\frac{d^2\vec{k}}{2\pi}\cdot \nabla \times \vec{A_\pm} $

$=\int_{\partial(BZ)} \frac{d\vec{k}}{2\pi}\cdot \vec{A_\pm}$

My question:

  1. What should I do next to know Chern number? I think the singularity of the wave functions at $\vec{k}=(0,0)$ is important.

  2. And I know the Chern number should be Integer, but I heard the answer is 1/2. Why??

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  • $\begingroup$ The answer is $1/2$ since the Hamiltonian you are using is a low energy approximation and not the full lattice Hamiltonian. Notice that the Hamiltonian doesn't have the periodicity of a lattice. Although calculating the Berry phase in this way doesn't give you the Chern number, you can actually use it to see how the Chern number changes after a top. phase transition. $\endgroup$ – honey.mustard May 5 '17 at 17:17

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