Physics Asked on January 25, 2021
I see at least two different definitions of Berry curvature in literature. One is given at Wikipedia
$$
Omega_n = isum_{nne n’}frac{ langle n|partial_{k_x} H|n’rangle langle n’|partial_{k_y} H|nrangle – langle n|partial_{k_y} H|n’rangle langle n’|partial_{k_x} H|nrangle}{(E_n-E_{n’})^2}
$$
The other one is given in this paper (also given some other articles)
$$
Omega_n = -2:Imagsum_{nne n’}frac{ langle n|partial_{k_x} H|n’rangle langle n’|partial_{k_y} H|nrangle}{(E_n-E_{n’})^2}
$$
I wonder what exactly is the difference between these two definitions.
Edit: The article that I mentioned above actually defines the Berry curvature similar to Wikipedia (as pointed out by @NDewolf). One article that defines Berry curvature differently is this one (paragraph below eq24)
Answering my own question.
Note from the first equation that $$ [langle n|partial_{k_x} H|n'rangle langle n'|partial_{k_y} H|nrangle]^*=langle n|[partial_{k_y} H]^*|n'rangle langle n'|[partial_{k_x} H)]^*|nrangle $$ as $partial_i H=frac{1}{hbar} v_i$, and velocity operator $v_i$ are Hermition so $[partial_i H]^*=partial_i H$. So, $$ [langle n|partial_{k_x} H|n'rangle langle n'|partial_{k_y} H|nrangle]^*=langle n|partial_{k_y} H|n'rangle langle n'|partial_{k_x} H)|nrangle $$ and
$$ Omega_n = isum_{nne n'}frac{ langle n|partial_{k_x} H|n'rangle langle n'|partial_{k_y} H|nrangle - [langle n|partial_{k_x} H|n'rangle langle n'|partial_{k_y} H|nrangle]^*}{(E_n-E_{n'})^2} = isum_{nne n'}frac{ 2i:Imag[langle n|partial_{k_x} H|n'rangle langle n'|partial_{k_y} H|nrangle]}{(E_n-E_{n'})^2} = -2:Imagsum_{nne n'}frac{ langle n|partial_{k_x} H|n'rangle langle n'|partial_{k_y} H|nrangle}{(E_n-E_{n'})^2} $$
Correct answer by Luqman Saleem on January 25, 2021
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