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 study, 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. 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.