# Property related with Berry curvature: $Omega_{n,munu}=-Omega_{n,numu}$

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: $$Omega_{n,munu}(mathbf{k})=partial_{mu}A_{nnu}(mathbf{k})-partial_{nu}A_{nmu}(mathbf{k})\ Rightarrow Omega_{n,numu}(mathbf{k})=partial_{nu}A_{nmu}(mathbf{k})-partial_{mu}A_{nnu}(mathbf{k}) = - left( partial_{mu}A_{nnu}(mathbf{k})-partial_{nu}A_{nmu}(mathbf{k}) right) = -Omega_{n,munu}(mathbf{k}).$$