Berezinskii-Kosterlitz-Thouless transition with enhanced phase stiffness in $d$-wave strongly coupled two-dimensional superconductors

We reveal the key role of the $d$-wave symmetry of the superconducting gap in strongly coupled two-dimensional superconductors in determining the properties of the Berezinskii-Kosterlitz-Thouless (BKT) transition, associated with a sizable enhancement of the phase stiffness compared to nodeless-gap superconductors. The enhanced stiffness originates from extended regions of vanishing gap around the nodal lines of the Brillouin zone (BZ). Our study, based on mean-field and BKT theory, presents a comparative analysis of $s$-wave and $d$-wave scenarios, highlighting the features of the latter that boost the stiffness and the BKT transition temperature (T$_{BKT}$). The comparison focuses on two quantities: the mean-field critical temperature and the maximum superconducting gap related to the pairing strengths. We present a phase diagram showing the scaling of T$_{BKT}$ with respect to the mean-field critical temperature across the BCS-BEC crossover and the evolution of the pseudogap. We also present a zero-temperature phase-stiffness intensity map over the Brillouin zone, displaying a two-component structure consisting of low- and high-stiffness regions whose extent depends on microscopic parameters. These results identify the nodal gap structure of strongly coupled two-dimensional superconductors as a key mechanism enabling enhanced stiffness and elevated T$_{BKT}$ compared to their $s$-wave counterparts.