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Multi-step phase transition provides a paradigm in which a broken symmetryduring phase transition can be restored, enriching the phenomena of both darkmatter and baryon asymmetry. We study the dynamics of the multi-step phasetransition in the standard model extension with additional isospin $N$-pletscalar field $\Phi_2$ under a discrete $\mathbb{Z}_2$ symmetry. We find thatthe multi-step phase transition could be triggered if there is a moderatelylarge coupling between the Higgs and the $\Phi_2$ and this coupling is requiredto be larger as the mass of the $\Phi_2$ and/or isospin increase. Thefirst-order phase transition at the first (second) step can be realized by thethermal loop (tree-level barrier) effects. Thus it is more likely that adetectable spectrum of gravitational waves can be produced at the second stepof the phase transition.

Multi-step phase transition and gravitational wave from general $\mathbb{Z}_2$ scalar extensions