Spin-orbit and exchange proximity couplings in graphene/1T-TaS$_2$ heterostructure triggered by a charge density wave

Proximity-induced fine features and spin-textures of the electronic bands in graphene-based van der Waals heterostructures can be explored from the point of tailoring a twist angle. Here we study spin-orbit coupling and exchange coupling engineering of graphene states in the proximity of 1T-TaS$_2$ not triggering the twist, but a charge density wave in 1T-TaS$_2$-a realistic low-temperature phase. Using density functional theory and effective model we found that the emergence of the charge density wave in 1T-TaS$_2$ significantly enhances Rashba spin-orbit splitting in graphene and tilts the spin texture by a significant Rashba angle-in a very similar way as in the conventional twist-angle scenarios. Moreover, the partially filled Ta $d$-band in the charge density wave phase leads to the spontaneous emergence of the in-plane magnetic order that transgresses via proximity from 1T-TaS$_2$ to graphene, hence, simultaneously superimposing along the spin-orbit also the exchange coupling proximity effect. To describe this intricate proximity landscape we have developed an effective model Hamiltonian and provided a minimal set of parameters that excellently reproduces all the spectral features predicted by the first-principles calculations. Conceptually, the charge density wave provides a highly interesting knob to control the fine features of electronic states and to tailor the superimposed proximity effects-a sort of twistronics without twist.