  The Berry curvature (BC) - a quantity encoding the geometric properties of
the electronic wavefunctions in a solid - is at the heart of different
Hall-like transport phenomena, including the anomalous Hall and the non-linear
Hall and Nernst effects. In non-magnetic quantum materials with acentric
crystalline arrangements, local concentrations of BC are generally linked to
single-particle wavefunctions that are a quantum superposition of electron and
hole excitations. BC-mediated effects are consequently observed in
two-dimensional systems with pairs of massive Dirac cones and three-dimensional
bulk crystals with quartets of Weyl cones. Here, we demonstrate that in
materials equipped with orbital degrees of freedom local BC concentrations can
arise even in the complete absence of hole excitations. In these solids, the
crystals fields appearing in very low-symmetric structures trigger BCs
characterized by hot-spots and singular pinch points. These characteristics
naturally yield giant BC dipoles and large non-linear transport responses in
time-reversal symmetric conditions.
