Spin liquids and pseudogap metals in the SU(4) Hubbard model in a moiré superlattice

Motivated by the realization of spin-valley Hubbard model on a triangular moir\'e superlattice in ABC trilayer graphene aligned with hexagon boron nitride (hBN) and possibly also in twisted transition metal dichalcogenide homobilayers, we study possible Mott insulating phases and pseudogap metals based on symmetry constraint and parton mean field theories. First, we show that the Luttinger constraint allows two distinct symmetric and featureless Fermi liquids when there is an intervalley Hund's term breaking $\text{SU}(4)$ spin rotation. Especially, there exists a symmetric and featureless ``pseudogap metal'' with small Fermi surfaces. Then, we suggest to search for such an unconventional metallic state by doping the Mott insulator at ${\ensuremath{\nu}}<sub>T</sub>=2$. For this purpose, we study the ${\ensuremath{\nu}}<sub>T</sub>=2$ Mott insulator using the $\text{SO}(6)$ Schwinger boson or Schwinger fermion parton. At the $\text{SU}(4)$-symmetric point, we find two symmetric ${Z}<sub>2</sub>$ spin liquids. With a large anti-intervalley Hund's term, a featureless Mott insulator is natural. Next, we show that doping the featureless Mott insulator or a ${Z}<sub>2</sub>$ spin liquid can lead to featureless or orthogonal ``pseudogap metal'' with small Fermi surfaces proportional to the doping. Besides, we also provide one scenario for the evolution from pseudogap metal to the conventional Fermi liquid through an intermediate exotic ``deconfined metal'' phase. Last, we give brief comments on the possibility of $\text{U}(1)$ spinon Fermi surface state or ${Z}₄$ spin liquid at ${\ensuremath{\nu}}_T=1$.