  Symmetry breaking is an effective tool for tuning the transport and
topological properties of 2D layered materials. Among these materials, twisted
bilayer graphene (TBG) has emerged as a promising platform for new physics,
characterized by a rich interplay between topological features and strongly
correlated electronic behavior. In this study, we utilize time-reversal
symmetry breaking (TRSB) to manipulate the topological properties of TBG. By
varying the strength of TRSB, we discover a topological phase transition
between a topological insulating phase, which exhibits a pair of flat bands
with opposite Chern numbers, and a novel insulating state where the Chern
number, but not the Berry curvature, of the flat bands vanishes. We demonstrate
that this topological transition is mediated by a gap closing at the $\Gamma$
point, and we construct a three-dimensional phase diagram as a function of the
twisting angle, the symmetry-breaking parameter, and the mismatch coupling
between AA and AB stacking regions. Finally, we show that this novel electronic
phase can be identified in the lab by measuring, as a function of the Fermi
energy, its non-quantized anomalous Hall conductivity that is induced by the
Berry dipole density of the lowest flat bands.
