Matter Modeling Asked on December 2, 2021

I read in David Vanderbilt’s book named "Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators" the definition of Berry curvature: "Berry curvature $Omega(mathbf{lambda})$ is simply defined as the Berry phase per unit area in ($lambda_x,,lambda_y$) space".

Berry Curvature is defined by:

begin{equation}

Omega_{n,munu}(mathbf{k})=partial_{mu}A_{nnu}(mathbf{k})-partial_{nu}A_{nmu}(mathbf{k})tag{1}

end{equation}

where $A_{nmu}(mathbf{k})=langle u_{nmathbf{k}}|ipartial_{mu}u_{nmathbf{k}}rangle$ and $A_{nnu}(mathbf{k})=langle u_{nmathbf{k}}|ipartial_{nu}u_{nmathbf{k}}rangle$ are Berry connections.

Berry’s curvature has the following property: $Omega_{n,munu}=-Omega_{n,numu}$.

How is this property mathematically demonstrated?

You can just exchange the $mu,nu$ indices to verify the antisymmetry: $$ Omega_{n,munu}(mathbf{k})=partial_{mu}A_{nnu}(mathbf{k})-partial_{nu}A_{nmu}(mathbf{k})\ Rightarrow Omega_{n,numu}(mathbf{k})=partial_{nu}A_{nmu}(mathbf{k})-partial_{mu}A_{nnu}(mathbf{k}) = - left( partial_{mu}A_{nnu}(mathbf{k})-partial_{nu}A_{nmu}(mathbf{k}) right) = -Omega_{n,munu}(mathbf{k}). $$

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