Coexistence of CDW with staggered superconductivity in a ferromagnetic material

In usual superconductivity (SC), the pairs have zero total momentum irrespective of their symmetry. Staggered SC would involve, instead, pairs with a finite commensurate total momentum, but such exotic states have never been proven to be realized in nature. Here we study for the first time the influence of particle-hole asymmetry on the competition of staggered SC with Charge Density Waves (CDW) in a ferromagnetic medium. We obtain unprecedented situations in which CDW and staggered SC coexist. We also obtain cases of a SC dome near the collapse of a CDW state as well as cascades of transitions that exhibit remarkable similarities with the pressure phase diagram in UGe$_2$ suggesting that SC in this material may be staggered coexisting and competing with a CDW state.