The Berry connection is defined as $$A_n(R)=i\left<\psi_n(R)\right|\nabla_R\left|\psi_n(R)\right>$$ and it is mathematically analogous to the vector potential.

We can then naively define the Berry curvature (the analogous magnetic field) by taking the curl. However, by doing so, we rule out the possibility that there is any magnetic charge, as $4\pi\rho_m=\nabla\cdot B=\nabla\cdot \left(\nabla \times A_n(R)\right)$ is always zero.

How can I reconcile this with what I read elsewhere that there is an effective monopole description for the Berry curvature in some materials such as Weyl semimetals?

The Berry connection is defined as $$A_n(R)=i\left<\psi_n(R)\right|\nabla_R\left|\psi_n(R)\right>$$ and it is mathematically analogous to the vector potential.

We can then naively define the Berry curvature (which is analogous to the magnetic field) by taking the curl. However, by doing so, we rule out the possibility that there is any magnetic charge, as $4\pi\rho_m=\nabla\cdot B=\nabla\cdot \left(\nabla \times A_n(R)\right)$ is always zero.

How can I reconcile this with what I read elsewhere that there is an effective monopole description for the Berry curvature in some materials such as Weyl semimetals?