  In a time-reversal invariant system, while the anomalous Hall effect
identically vanishes in the linear response regime due to the constraint of
time-reversal symmetry on the distribution of Berry curvature, a nonlinear Hall
effect can emerge in the second-order response regime if the inversion symmetry
is broken to allow a nonzero Berry curvature dipole (BCD) on the Fermi surface.
In this work, we study the nonlinear Hall effect of the BCD origin in
two-dimensional doped insulators and semimetals belonging to the symmetry class
AI which has spinless time-reversal symmetry. Despite that the class AI does
not host any strong topological insulator phase in two dimensions, we find that
they can still be classified as topologically obstructed insulators and trivial
insulators if putting certain constraint on the Hamiltonians. When the
insulator gets closer to the phase boundary of the two distinct phases, we find
that the BCDs will become more prominent if the doping level is located near
the band edge. Moreover, when the insulator undergoes a phase transition
between the two distinct phases, we find that the BCDs will dramatically change
their signs. For the semimetals without inversion symmetry, we find that the
BCDs will sharply reverse their signs when the doping level crosses the Dirac
points. With the shift of the locations of Dirac points in energy, the critical
doping level at which the BCDs sharply reverse their signs will accordingly
change. Our study reveals that class AI materials can also have interesting
geometrical and topological properties, and remarkable nonlinear Hall effect
can also appear in this class of materials even though the spin-orbit coupling
is negligible. Our findings broaden the scope of materials to study the
nonlinear Hall effect and provide new perspectives for the application of this
effect.
