What is the correct form of the Berry curvature formula?

I am studying Berry curvature for a specific material and faced different types of the Berry curvature formula. Some papers use only valence eigenstates (u1) like this
i∗(<(∂U1/∂kx)|(∂U1/∂ky)>−<(∂U1/∂ky)|(∂U1/∂kx)>)

and someone uses summation on all the conduction and valence bands like this
i∗(<(∂U1/∂kx)|(∂U2/∂ky)>−<(∂U1/∂ky)|(∂U2/∂kx)>).
I'm confused about which one should I use for my calculations.

Presumably the second form is more general. The users of the first form probably restricted the sum because they could argue that those contributions were small when summed. Could be geometry, could be something else.......you have not given much info here. Good Luck.

For those that wondered how the equations look when Latexified:
$$i\left(\left\langle\frac{\partial U_1}{\partial k_x}\bigg|\frac{\partial U_1}{\partial k_y}\right\rangle - \left\langle\frac{\partial U_1}{\partial k_y}\bigg|\frac{\partial U_1}{\partial k_x}\right\rangle\right)$$

and the second one:
$$
i\left(\left\langle\frac{\partial U_1}{\partial k_x}\bigg|\frac{\partial U_2}{\partial k_y}\right\rangle - \left\langle\frac{\partial U_1}{\partial k_y}\bigg|\frac{\partial U_2}{\partial k_x}\right\rangle\right)$$

hutchphd said:

Presumably the second form is more general. The users of the first form probably restricted the sum because they could argue that those contributions were small when summed. Could be geometry, could be something else.......you have not given much info here. Good Luck.

Thank you.
My case here is a quantum spin Hall insulator named PbBiI that has two valences and two conduction bands that we can name ##v_1, v_2, c_1, c_2##. Do you mean that I can consider the Berry curvature of the valence bands as ##i(\langle \frac{\partial v_1}{\partial k_x}|\frac{\partial v_2}{\partial k_y}\rangle-\langle \frac{\partial v_1}{\partial k_y}|\frac{\partial v_2}{\partial k_x}\rangle)## and neglect intra-band transitions like ##v_1## to ##c_1## and ##c_2##, ##i(\langle \frac{\partial v_1}{\partial k_x}|\frac{\partial c_1}{\partial k_y}\rangle-\langle \frac{\partial v_1}{\partial k_y}|\frac{\partial c_1}{\partial k_x}\rangle)## & ##i(\langle \frac{\partial v_1}{\partial k_x}|\frac{\partial c_2}{\partial k_y}\rangle-\langle \frac{\partial v_1}{\partial k_y}|\frac{\partial c_2}{\partial k_x}\rangle)## and also for ##v_2## to the ##c_1## and ##c_2## transitions?

Apparently one of your sources thinks so. Did they provide justifiction?
I have no working knowledge of this system by perhaps there are symmetries at worrk here. Please cite the source where they ignore the terms.

hutchphd said:

Apparently one of your sources thinks so. Did they provide justifiction?
I have no working knowledge of this system by perhaps there are symmetries at worrk here. Please cite the source where they ignore the terms.

Here are some papers that have ignored the intra-band contributions.

Mohammad-gl said:

Here are some papers that have ignored the intra-band contributions.

Can you please cite the specific equations in question for each paper...I really don't feel like plowing through them all ab initio.

hutchphd said:

Can you please cite the specific equations in question for each paper...I really don't feel like plowing through them all ab initio.

Excuse me for late reply. Can you tell me generally when we can ignore inter-band transitions (valence to conduction)? For example, can we ignore inter-band transitions for large gap materials and consider only intra-band (valence to valence) transitions?