Exploring supersolids of single-microwave shielded molecules via exact and mean-field theories

Ultracold polar molecular gases provide a powerful platform for exploring quantum many-body physics with strong, long-range, and anisotropic interactions. In this work, we develop an extended Gross-Pitaevskii approach tailored to bosonic dipolar molecules under single-microwave shielding, incorporating their effective interactions and adapting the quantum fluctuation corrections. We benchmark this beyond-mean-field theory against exact path-integral Quantum Monte Carlo simulations. Focusing on the regime of positive scattering lengths, we find excellent agreement across a range of quantum phases, including superfluid, supersolid, and droplet states. We show that elliptic microwave polarization induces anisotropic superfluidity with direction-dependent sound velocities along each spatial axis—an effect absent in atomic dipolar gases. A quasi-one-dimensional theory captures roton softening and predicts roton instabilities tunable via ellipticity. While most experiments rely on double-microwave shielding to reduce losses, we demonstrate that single-shielded molecules already support rich and tunable many-body behavior. Our framework is readily extendable to the double-shielded case. This work establishes a versatile theoretical foundation for ultracold molecular gases and opens the door to future studies with more advanced shielding protocols.