  We show that when the electron and hole densities are unbalanced, observable
nonlinear anomalous Hall effect and current-induced orbital magnetization
appear at zero magnetic field in the weak charge ordering (CO) state of an
organic two-dimensional Dirac fermion system, a-(BEDT-TTF)2I3. These
current-induced phenomena are caused by a finite Berry curvature dipole
resulting from inversion symmetry breaking and Dirac cone tilting. In the
actual system, however, these effects are canceled out between different types
of inversion asymmetric CO domains. To avoid the cancellation, we propose a new
experimental method to realize the selective formation of a single type of
domain using the current-induced magnetization.
