Universal Bound States with Bose-Fermi Duality in Microwave-Shielded Polar Molecules

We investigate universal bound states of microwave-shielded ultracold polar molecules. Under a highly elliptic microwave field, few-molecule scatterings in three dimension are shown to be governed by effective one-dimensional (1D) models. These models well reproduce the tetratomic (two-molecule) bound state and the Born-Oppenheimer potential in three-molecule sector. For hexatomic systems comprising three identical molecules, we find the lowest bound state emerge concurrently with tetratomic state, with binding energy exceeding twice of the latter. Strikingly, all these bound states display Bose-Fermi duality, i.e., they share identical energies and spatial densities in both bosonic and fermionic molecular systems. Universal features of these bound states are supported by the 1D nature of effective scattering and a large repulsive core in the reduced effective potential. For large molecule ensembles, our results suggest the formation of elongated self-bound droplets with crystalline patterns in both bosonic and fermionic polar molecules.