Critical field theory for ferromagnetic quantum criticality in the strong coupling regime of Hertz-Moriya-Millis theory

We develop U(1) slave spin-rotor theory, suggesting a metal-metal transition from Landau's Fermi-liquid state to a bad metal phase, as U(1) slave charge-rotor theory [Phys. Rev. B {\bf 70}, 035114 (2004)] describes a metal-insulator transition from Landau's Fermi-liquid state to a spin-liquid phase. U(1) slave spin-rotor formulation allows us to generalize Hertz-Moriya-Millis theory for ferromagnetic quantum phase transitions, replacing Landau's Fermi-liquid state with an incoherent metallic phase. As a result, we argue that localized magnetic moments emerge to govern quantum critical physics in bad metals.