Abstract 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 condensate fraction scaling as a highly efficient tool to achieve that and accordingly propose efficient schemes to accurately determine the associated Berezinskii−Kosterlitz−Thouless (BKT) transitions. Using the 2D attractive Fermi-Hubbard model as a testbed and applying numerically exact auxiliary-field quantum Monte Carlo simulations, we access unprecedented system sizes (up to 64 × 64 = 4096 lattice sites) and perform a comprehensive analysis for the temperature dependence and finite-size scaling of condensate fraction across the BKT transition. We demonstrate that this quantity exhibits algebraic scaling below the transition and exponential scaling above it, with significantly reduced finite-size effects comparing to the extensively studied on-site pairing correlator. This greatly improves the determination of BKT transition using moderate system sizes. We also extract finite-size BKT transition temperature from condensate fraction, and confirm its logarithmic correction on system size. Based on the accurately determined transition, we reveal 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.
We study the three-dimensional (3D) attractive Hubbard model by means of the determinant quantum Monte Carlo method. This model is a prototype for the description of the smooth crossover between BCS superconductivity and Bose-Einstein condensation. By detailed finite-size scaling we extract the finite-temperature phase diagram of the model. In particular, we interpret the observed behavior according to a scenario of two fundamental temperature scales;
T∗ associated with Cooper pair formation and
Tc with condensation (giving rise to long-range superconducting order). Our results also indicate the presence of a recently conjectured phase transition hidden by the superconducting state. A comparison with the 2D case is briefly discussed, given its relevance for the physics of high-
Tc cuprate superconductors.
An attempt is made to analyze the state-of-the-art in the description of the superconducting phase in 2D metals with an arbitrary carrier density. It is shown that a decrease in the density leads to a nontrivial equation for chemical potential μ, in particular, to the inequality μ ≠ εF including μ < 0 (εF is the Fermi energy). In this case, the region of μ ≃ 0 corresponds to a transition (crossover) from Cooper pairing (with strongly overlapping pairs) and from the description of superconductivity in terms of the BCS–Bogoliubov theory to local (nonoverlapping) pairs and their Bose–Einstein condensation. Some examples of such a crossover are discussed.