Hidden antiferromagnetism, persistent valley fluctuations, and $U(6)$ crossovers in triangular-lattice M-point moir\'e materials via determinantal quantum Monte Carlo

A new moir'e material platform was recently proposed based on twisting two-dimensional atomic monolayers whose low-energy states lie at the three M-points of the Brillouin Zone. Continuum and ab initio modeling suggest that electrons in the conduction bands of these materials realize three-valley Hubbard models with valley-selective, quasi-one-dimensional hopping. Remarkably, the onsite Hubbard repulsion is almost $U(6)$-symmetric without fine-tuning. Here, we show that this class of systems naturally admits sign-free determinantal Quantum Monte Carlo simulations at a filling of three electrons per moir'e unit cell. We use these to explore the phase diagram for interactions of various strengths and $U(6)$-breaking anisotropies. We show that for near-isotropic interactions as relevant to, e.g., AA-stacked twisted SnSe${}_2$, the system exhibits an extended intermediate-coupling regime in which local-moment formation and itinerancy compete, and the crossover to a putative low-temperature ordered state can be understood in terms of fluctuating $U(6)$ local moments. We argue that many of these features persist beyond the idealized sign-problem-free limit.