Vestigial $d$-wave charge-$4e$ Superconductivity from Bidirectional Pair Density Waves

We analyze the leading vestigial instability due to the melting of a bidirectional pair-density-wave state in two dimensions. In a previous work by one of the authors, it was found that the interplay between pair-density-wave fluctuations with ordering momenta along the $x$ and $y$ directions can provide a strong attractive interaction for charge-$4e$ superconductivity in the $d$-wave channel. In this work, we go beyond the artificial large-$M$ mean-field theory previously adopted and compute the phase diagram by incorporating phase fluctuations of the pair-density-wave order parameters. By investigating the relevance of various topological defects, we show that the interaction in the $d$-wave channel, together with the strong anisotropy of phase fluctuations around the pair-density-wave ordering momenta, favors a vestigial charge-$4e$ superconducting order at intermediate temperatures. By contrast, a competing charge-density-wave vestigial order does not develop, due to the suppression of its stiffness.