Hao-Jie Lin, Tao Zhu, Jing-Fei Zhang, Xin Zhang

We study spontaneous scalarization of a scalar field in the magnetized Reissner--Nordström spacetime induced by parity-violating and parity-preserving interactions, represented by couplings to the electromagnetic Chern--Simons, gravitational Chern--Simons, and Gauss--Bonnet invariants, respectively. Working in the decoupling limit, we evolve scalar perturbations in the time domain and determine the critical coupling for the onset of tachyonic instability. This allows us to compare, within the same magnetized background, how the external magnetic field affects scalarization induced by parity-violating and parity-preserving interactions. We find that the magnetic field lowers the scalarization threshold in the electromagnetic and gravitational Chern--Simons channels. In the Gauss--Bonnet channel, by contrast, the effect divided into two branches: on the negative-$α$ branch in our convention, corresponding to the standard GB$^{+}$ branch, the magnitude of the critical coupling increases with the magnetic field, whereas on the positive-$α$ branch, corresponding to GB$^{-}$, the critical coupling decreases with the magnetic field but diverges in the limit of vanishing field. The magnetic field also modifies the late-time dynamics and gives rise to Melvin-like modes. When nonlinear couplings are included, the unbounded growth of the linearized theory is replaced by bounded oscillatory evolution. These results show that external magnetic fields affect scalarization induced by parity-violating and parity-preserving interactions in qualitatively different ways, and reveal a pronounced asymmetry between the two Gauss--Bonnet branches.