# Superconductivity near a ferroelectric quantum critical point in ultralow-density Dirac materials

## TL;DR

This paper proposes a new mechanism for superconductivity in ultralow-density Dirac materials near a ferroelectric quantum critical point, highlighting the role of phonon-mediated interactions and Coulomb screening.

## Contribution

It introduces a theoretical framework showing how proximity to a ferroelectric quantum critical point enhances superconductivity in Dirac materials without requiring retardation.

## Key findings

- Coulomb repulsion is strongly screened near the critical point.
- Electron-phonon interaction dominates the effective pairing.
- Transition temperature is significantly enhanced near the quantum critical point.

## Abstract

The experimental observation of superconductivity in doped semimetals and semiconductors, where the Fermi energy is comparable to or smaller than the characteristic phonon frequencies, is not captured by the conventional theory. In this paper, we propose a mechanism for superconductivity in ultralow-density three-dimensional Dirac materials based on the proximity to a ferroelectric quantum critical point. We derive a low-energy theory that takes into account both the strong Coulomb interaction and the direct coupling between the electrons and the soft phonon modes. We show that the Coulomb repulsion is strongly screened by the lattice polarization near the critical point even in the case of vanishing carrier density. Using a renormalization group analysis, we demonstrate that the effective electron-electron interaction is dominantly mediated by the transverse phonon mode. We find that the system generically flows towards strong electron-phonon coupling. Hence, we propose a new mechanism to simultaneously produce an attractive interaction and suppress strong Coulomb repulsion, which does not require retardation. For comparison, we perform same analysis for covalent crystals, where lattice polarization is negligible. We obtain qualitatively similar results, though the screening of the Coulomb repulsion is much weaker. We then apply our results to study superconductivity in the low-density limit. We find strong enhancement of the transition temperature upon approaching the quantum critical point. Finally, we also discuss scenarios to realize a topological $p$-wave superconducting state in covalent crystals close to the critical point.