  Topological physics and in particular its connection with artificial gauge
fields is a forefront topic in different physical systems, ranging from cold
atoms to photonics and more recently semiconductor dressed exciton-photon
states, called polaritons. Engineering the energy dispersion of polaritons in
microcavities through nanofabrication or exploiting the intrinsic material and
cavity anisotropies has demonstrated many intriguing effects related to
topology and emergent gauge fields. Here, we show that we can control the Berry
curvature distribution of polariton bands in a strongly coupled
organic-inorganic 2D perovskite single crystal. The spatial anisotropy of the
perovskite crystal combined with photonic spin-orbit coupling make emerge two
Hamilton's diabolical points in the dispersion. The application of an external
magnetic field breaks time reversal symmetry thanks to the exciton Zeeman
splitting. It splits the diabolical points degeneracy. The resulting bands show
non-zero integral Berry curvature which we directly measure by state
tomography. Crucially, we show that we can control the Berry curvature
distribution in the band, the so-called band geometry, within the same
microcavity.
