Moir\'e $M$-valley bilayers: quasi-one-dimensional physics, unconventional spin textures and twisted van Hove singularities

Motivated by the discovery of quasi-two-dimensional kagome metals AV$_3$Sb$_5$, we consider the theory of twisted bilayers in which the Fermi surface is near the $M$-point. Surprisingly, unlike twisted bilayers of graphene or transition metal dichalcogenides, the moir\'e potential is quasi-one-dimensional: at each $M$-valley, the potential flattens the dispersion strongly along one direction, and weakly along the perpendicular direction. The combination of spin-orbit coupling and twist-induced broken inversion symmetry results in a similarly anisotropic `$\textit{moir\'e-Rashba}$' potential, which spin-splits the dispersion into coexisting two-dimensional and quasi-one-dimensional bands. We discuss novel aspects of the interplay between mixed dimensionality and spin textures in this platform. First, an applied electric field produces spin polarisation which can be tuned by doping, suggesting potential spintronics applications. Secondly, an in-plane magnetic field momentum- and spin-polarises the Fermi surfaces, producing unconventional spin density waves. Thirdly, in the small-twist-angle limit, the large density of states due to a twisted van Hove singularity near $M$ results in a dense energy spectrum. Our results demonstrate a new variation of moir\'e bandstructure engineering, instigating the study of spin-textured one-dimensional physics in moir\'e materials.