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Twisting moir\'e heterostructures to the flatband regime allows for theformation of strongly correlated quantum states, since the dramatic reductionof the bandwidth can cause the residual electronic interactions to set theprincipal energy scale. An effective description for such correlated moir\'eheterostructures, derived in the strong-coupling limit at integer filling,generically leads to spin-valley Heisenberg models. Here we explore theemergence and stability of spin liquid behavior in anSU(2)$^{\mathrm{spin}}\otimes$SU(2)$^{\mathrm{valley}}$ Heisenberg model uponinclusion of Hund's-induced and longer-ranged exchange couplings, employing apseudofermion functional renormalization group approach. We consider twolattice geometries, triangular and honeycomb (relevant to different moir\'eheterostructures), and find, for both cases, an extended parameter regimesurrounding the SU(4) symmetric point where no long-range order occurs,indicating a stable realm of quantum spin liquid behavior. For large Hund'scoupling, we identify the adjacent magnetic orders, with both antiferromagneticand ferromagnetic ground states emerging in the separate spin and valleydegrees of freedom. For both lattice geometries the inclusion of longer-rangedexchange couplings is found to have both stabilizing and destabilizing effectson the spin liquid regime depending on the sign of the additional couplings.

Emergence and stability of spin-valley entangled quantum liquids in moir\'e heterostructures