Condensate Fraction Scaling and Specific Heat Anomaly around Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity

Characterizing the superconducting and superfluid transitions in two-dimensional (2D) many-body systems is of broad interest and remains a fundamental issue. In this study, we establish the {\it condensate fraction} as a highly effective tool to achieve that and accordingly propose efficient schemes for accurately determining the transitions, via numerically exact quantum Monte Carlo simulations. Using the 2D attractive Fermi-Hubbard model as a testbed, we access unprecedented system sizes (up to 4096 lattice sites) and perform a comprehensive analysis for the temperature dependence and finite-size scaling of {\it condensate fraction} across the Berezinskii-Kosterlitz-Thouless (BKT) transition. We demonstrate that this quantity exhibits algebraic scaling below the transition and exponential scaling above it, with significantly smaller finite-size effect comparing to the extensively studied on-site pairing correlator. This greatly improves the determination of BKT transition with moderate system sizes. We also extract the finite-size BKT transition temperature from condensate fraction, and confirm its logarithmic correction on system size. Furthermore, we find that the specific heat displays an anomaly, showing a peak at a temperature slightly above BKT transition. Our findings should be generally applicable to 2D fermionic and bosonic systems hosting superconductivity or superfluidity.