  Physics arising from two-dimensional~(2D) Dirac cones has been a topic of
great theoretical and experimental interest to studies of gapless topological
phases and to simulations of relativistic systems. Such $2$D Dirac cones are
often characterized by a $\pi$ Berry phase and are destroyed by a perturbative
mass term. By considering mean-field nonlinearity in a minimal two-band Chern
insulator model, we obtain a novel type of Dirac cones that are robust to local
perturbations without symmetry restrictions. Due to a different pseudo-spin
texture, the Berry phase of the Dirac cone is no longer quantized in $\pi$, and
can be continuously tuned as an order parameter. Furthermore, in an
Aharonov-Bohm~(AB) interference setup to detect such Dirac cones, the adiabatic
AB phase is found to be $\pi$ both theoretically and computationally, offering
an observable topological invariant and a fascinating example where the Berry
phase and AB phase are fundamentally different. We hence discover a
nonlinearity-induced quantum phase transition from a known topological
insulating phase to an unusual gapless topological phase.
