Twisted bilayer BC$_3$: Valley interlocked anisotropic flat bands

Here we propose BC$_3$, a graphene derivative that has been synthesized, as a platform to realize exotic quantum phases by introducing a moir\'e pattern with mismatched stacking. In twisted bilayer BC$_3$, it is shown that a crossover from two-dimensional to quasi one-dimensional band structure takes place with the twist angle as a control parameter. This is a typical manifestation of the guiding principle in van der Waals stacked systems: the quantum interference between Bloch wave functions in adjacent layers has a striking effect on the effective interlayer tunneling. Interestingly, quasi one-dimensionalization happens in a valley dependent manner. Namely, there is interlocking between the valley index and the quasi-1D directionality, which makes BC$_3$ a plausible candidate for valleytronics devices. In addition, the strongly correlated regime of the valley interlocked quasi-1D state reduces to an interesting variant of the Kugel-Khomskii model where intertwined valley and spin degrees of freedom potentially induces exotic quantum phases. Notably, this variant of the Kugel-Khomskii model cannot be realized in conventional solids due to the three fold valley degeneracy.