Phase Fluctuations in Strongly Coupled $d$-Wave Superconductors

We present a numerically exact solution for the BCS Hamiltonian at any temperature, including the degrees of freedom associated with classical phase, as well as amplitude, fluctuations via a Monte Carlo (MC) integration. This allows for an investigation over the whole range of couplings: from weak attraction, as in the well-known BCS limit, to the mainly unexplored strong-coupling regime of pronounced phase fluctuations. In the latter, for the first time two characteristic temperatures $T^\star$ and $T_c$, associated with short- and long-range ordering, respectively, can easily be identified in a mean-field-motivated Hamiltonian. $T^\star$ at the same time corresponds to the opening of a gap in the excitation spectrum. Besides introducing a novel procedure to study strongly coupled d-wave superconductors, our results indicate that classical phase fluctuations are not sufficient to explain the pseudo-gap features of high-temperature superconductors (HTS).