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While numerically playing with the 2-level Haldane model recently, I wondered how I could analytically calculate the z-component of the Berry curvature $F$. I framed my problem as needing an expression for

$\sigma_zF_{\mu\nu}$.

I am looking for an expression involving the Berry connections $A_\nu$, $A_\mu$, but I am having some trouble doing so.

I tried using the following expression for $F$ from Fruchart et al.:

$$ F = \frac { 1 } { 4 } \epsilon ^ { i j k } \| h \| ^ { - 3 } h _ { i } \frac { \partial h _ { j } } { \partial k _ { a } } \frac { \partial h _ { k } } { \partial k _ { b } } \mathrm { d } k _ { a } \wedge \mathrm { d } k _ { b } = \frac { 1 } { 2 } \frac { \vec { h } } { \| h \| ^ { 3 } } \cdot \left( \frac { \partial \vec { h } } { \partial k _ { x } } \times \frac { \partial \vec { h } } { \partial k _ { y } } \right) \mathrm { d } k _ { x } \wedge \mathrm { d } k _ { y } $$

I think that applying $\sigma_z$ to the right hand side above gives:

$\frac{1}{2}\frac{h_z}{||h||^3}\cdot(\frac{\partial h_z}{\partial k_x}\times\frac{\partial h_z}{\partial k_y}) dk_x \wedge dk_y$,

with $||h||^3$ remaining the square root of the sum of all components of $h$.

How do I proceed from there if I am to arrive at an analytical expression in terms of the Berry connections $A_\mu$ and $A_\nu$, where they adopt the following form:

$A_\mu=\begin{bmatrix} \langle \phi_+|\partial_\mu\partial_+\rangle & \langle \phi_+|\partial_\mu\partial_-\rangle \\ \langle \phi_-|\partial_\mu\partial_+\rangle & \langle \phi_-|\partial_\mu\partial_-\rangle \end{bmatrix}$?

I don't really have to use the $F$ from Fruchart et al. above, but I felt that was the most manageable. I also tried starting with:

$F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu + i[A_\mu,A_\nu]$,

but I wasn't sure how to take the z-component from there. I looked into writing the Berry curvature in terms of the Hamiltonian (so that I could multiply $\sigma_z$ in), but the way forward was not too obvious.

Ultimately, I am striving to get an analytic expression for $Tr(\sigma_z)F_{\mu\nu}$ pertaining to the Haldane model, and then to calculate this quantity numerically using the Fukui-Hatsugai method on a discretized grid (J. Phys. Soc. Jpn. 74 pp. 1674-1677, 2005). The given resource does not allow for two 'expressions' for the Hamiltonian (i.e. one for $||h||$ and one for $h_z$). So, I could use some advice there too.

Please bear with me if this question is trivial, for I am only an undergraduate. However, I really could use some advice/references.

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  • $\begingroup$ Oh and, any alternative forms for $F$ would be appreciated! $\endgroup$ – TribalChief Jan 19 '19 at 13:33

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