Berry curvature as effective magnetic field
Berry curvature plays the role of a magnetic field in parameter or momentum space.
$$\Omega_{ij}=\partial_i\mathcal A_j-\partial_j\mathcal A_i$$
What this means
Berry curvature plays the role of a magnetic field in parameter or momentum space.
Deep dive: mathematical and physical mechanism
The Berry connection is a gauge potential; its antisymmetric derivative gives curvature. For an isolated two-level Hamiltonian $H=\mathbf d\cdot\boldsymbol\sigma$, the lower-band curvature is related to $\hat{\mathbf d}\cdot(\partial_i\hat{\mathbf d}\times\partial_j\hat{\mathbf d})$.
How it appears in quantum phenomena
The anomalous Hall response integrates occupied-band Berry curvature.
Assumptions, caveats and limitations
Berry curvature is not generally an ordinary real-space electromagnetic field.
Connect the ideas
For context, see the Geometric Quantum Mechanics topic guide and the reading references.