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 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, two characteristic temperatures $T^{*}$ and $T_c$, associated with short- and long-range ordering, respectively, can be identified in a mean-field-motivated Hamiltonian. $T^{*}$ at the same time corresponds to the opening of a gap in the excitation spectrum. In addition to introducing a novel procedure to study strongly coupled $d$-wave superconductors, our results indicate that classical phase fluctuations are not sufficient to explain the pseudogap features of high-temperature superconductors.

Figure 1

(a) The phase correlation functions $S(V)$, $F(V)$ at the shortest and maximum (linear) lattice distance at low $T$ and $⟨n⟩=1$. (b) Same functions as in (a), now for $⟨n⟩\sim 0.18$, leading to (extended) $s$-wave behavior. The symbols stand for $S(1,0)$ (filled squares), $S(5,0)$ (open triangles), and $F(0,0)$ (filled triangles). The statistical error is much smaller than the symbols for $V<4$, and roughly the symbol size for $V\lesssim 6$. (c) The correlation length $\xi (T)$ for $V=5.6$ at $⟨n⟩=1$ from the $H_{\mathrm{SC}}$ model. From the Kosterlitz-Thouless fit (broken line) we obtain $T_c\simeq 0.08$($T_c\simeq 0.17$ for the d-p model at $V=4.8$).

Figure 2

(a) SR and LR phase correlation functions vs $T$ for two different values of $V$, covering the weak- and strong-coupling regime, for $⟨n⟩=1$. Once $V\lesssim 2$, no difference between short- and long-range correlations is observed, and $S(6,0)$ $(V=1.2)$ is not shown for reasons of clarity. Symbol sizes roughly match the errors. (b) The phase diagram for $H_{\mathrm{SC}}$ derived from (a); $T_c$ and $T^{*}$ as explained in the text. (c) shows the phase diagram for the d-p model. Note the differences between (b) and (c). The dPG regime is indicated in both (b),(c). Beyond the dashed lines the $d$-wave character of $H_{\mathrm{SC}}$ is lost, whereas the d-p model incorporates higher harmonics as well.

Figure 3

The $T$-dependent $N(\omega )$ ($12\times 12$) for weak- [(a), $V=1.2$] and intermediate-coupling [(b), $V=4$] for $H_{\mathrm{SC}}$, illustrating the development of the PG as $T$ is lowered. In (a) the system is uncorrelated just above $T_c$, whereas in (b) the PG appears at $T^{*}$, concurrently with SR ordering. For $T=0.06$, the deviation from the $d$-wave state becomes evident, and a gap ${\omega }_0\simeq 0.6$ opens up. A broadening $\gamma =0.05$ was used. For reasons of comparison, the two thin lines represent the d-p model for $T=0.15$ ($V=1.2$) and $T=0.20$ ($V=4$), respectively.