Quantum Phases for Finite-Temperature Gases of Bosonic Polar Molecules Shielded by Dual Microwaves

We investigate the finite-temperature phase diagram of polar molecules shielded by dual microwave fields using the path integral Monte Carlo method combined with the worm algorithm. We determine the critical temperature <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>T</a:mi><a:mi>c</a:mi></a:msub></a:math> for Bose-Einstein condensations (BECs) and identify two distinct phases below <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>T</c:mi><c:mi>c</c:mi></c:msub></c:math>: the expanding gas (EG) phase and the self-bound gas (SBG) phase. We further analyze the temperature and interaction-strength dependence of the condensate and superfluid fractions. Notably, in contrast to dilute atomic BECs, the SBG phase displays a low condensate fraction and a high superfluid fraction, resembling the behavior of strongly correlated <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi/><e:mn>4</e:mn></e:msup><e:mi>He</e:mi></e:math> superfluids. These significant many-body correlations arise from the interplay between long-range dipole-dipole interactions and the short-range shielding potential. Furthermore, we demonstrate that the aspect ratio of the gas provides a characteristic geometric signature to accurately determine the EG-to-SBG transition, robust against external trapping potentials. Our findings provide unbiased and numerically exact results to guide upcoming experiments with polar molecules.