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

Ethan Huecker, Yuxuan Wang

TL;DR

The paper investigates how phase fluctuations of a bidirectional PDW state in two dimensions can generate a vestigial $d$-wave charge-$4e$ superconducting order. By building a Ginzburg-Landau description, deriving a non-linear sigma model, and mapping to a dual sine-Gordon theory, it identifies topological defects and their proliferation as the mechanism behind KT-like transitions. A key result is the emergence of an intermediate $d$-wave $4e$-SC phase stabilized by anisotropic PDW phase stiffness, while vestigial CDW order is suppressed due to small stiffness and cross-gradient terms. The findings provide a controlled framework for vestigial orders in intertwined orders and suggest experimental relevance to cuprate-like systems and other bidirectional PDW materials.

Abstract

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.

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

TL;DR

The paper investigates how phase fluctuations of a bidirectional PDW state in two dimensions can generate a vestigial -wave charge- superconducting order. By building a Ginzburg-Landau description, deriving a non-linear sigma model, and mapping to a dual sine-Gordon theory, it identifies topological defects and their proliferation as the mechanism behind KT-like transitions. A key result is the emergence of an intermediate -wave -SC phase stabilized by anisotropic PDW phase stiffness, while vestigial CDW order is suppressed due to small stiffness and cross-gradient terms. The findings provide a controlled framework for vestigial orders in intertwined orders and suggest experimental relevance to cuprate-like systems and other bidirectional PDW materials.

Abstract

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 and directions can provide a strong attractive interaction for charge- superconductivity in the -wave channel. In this work, we go beyond the artificial large- 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 -wave channel, together with the strong anisotropy of phase fluctuations around the pair-density-wave ordering momenta, favors a vestigial charge- superconducting order at intermediate temperatures. By contrast, a competing charge-density-wave vestigial order does not develop, due to the suppression of its stiffness.

Paper Structure

This paper contains 13 sections, 57 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model and Methods
  3. Ginzburg-Landau Free Energy
  4. Non-Linear Sigma Model
  5. Topological Defects
  6. Sine-Gordon Model
  7. Vestigial and primary orders
  8. Phase Diagram
  9. Additional Interactions
  10. Concluding Remarks
  11. Details of the Ginzburg-Landau Theory
  12. Details of the Analysis with κ_4,κ_5 Interactions
  13. Proof of Intermediate Representation Eq. (\ref{['nlsmphi4']})

Figures (6)

  • Figure 1: Filled Fermi sea (orange) with $C_4$ rotation symmetry, and PDW momenta $\bm{Q}_i$ with internal fermions situated on the Fermi surface. Ellipses (red) represent anisotropic PDW fluctuations with longitudinal and transverse amplitudes that scale as $1/\sqrt{\alpha_1}$ and $1/\sqrt{\alpha_2}$ respectively.
  • Figure 2: Log-log plot of the phase transitions Eqs. (\ref{['doubledisc']}-\ref{['singledisc']}). The red line signifies the proliferation of double CDW dislocations, whereas the purple line signifies the proliferation of HVSDs.
  • Figure : (i)
  • Figure : (i)
  • Figure : (ii)
  • ...and 1 more figures