Displacement-Field-Driven Transition between Superconductivity and Valley Ferromagnetism in Transition Metal Dichalcogenides

Recent experiments have observed transitions between superconductivity and correlated magnetism in twisted bilayer WSe$_2$ near van-Hove fillings, driven by the displacement field $D$. Motivated by the experiment, we theoretically propose a general mechanism for a $D$-controlled transition between superconductivity and ferromagnetism in two-dimensional (2D) spin-orbit-coupled hexagonal systems, where van Hove singularities (VHS) lie on the Fermi level. We show that such a transition can be naturally captured by a simple VHS-only model without Fermi surface details, where the inter-VHS interactions that govern the Fermi surface instabilities is controlled by $D$ through the band projection of screened Coulomb interaction. By treating this simple model with renormalization group technique beyond mean-field level, we find that a chiral $d/p$-wave superconductivity naturally dominates under a weak displacement field $D<D_c$. At a stronger displacement field $D>D_c$, a \textit{valley ferromagnetic phase} (vFM) takes over, which is spatially non-uniform due to valley-modulated magnetization. Finally, we discuss generic conditions for the predicted superconductivity-to-ferromagnetism transition to take place in the rich family of few-layer hexagonal van der Waals material systems. Taking twisted bilayer WSe$_2$ as a case study, we discuss experimental detections that can falsify our prediction.