Spin-charge induced spontaneous scalarization of Kerr-Newman black holes

We investigate the tachyonic instability of Kerr-Newman (KN) black holes in Einstein-Maxwell-scalar (EMS) theory with a positive coupling parameter $\ensuremath{\alpha}$. This corresponds to exploring onset spontaneous scalarization for KN holes. For this purpose, we use hyperboloidal foliation method solve linearized scalar equation numerically. obtain 3D graph [${\mathrm{log}}_{10}\ensuremath{\alpha}(a,Q)$] which indicates surface EMS theory. find that there is no lower bound on rotational $a$, but its upper given by ${M}^{2}\ensuremath{-}{Q}^{2}\ensuremath{\ge}{a}^{2}$. Also, confirm high rotation enhances