Unconventional Superconductivity in Luttinger Semimetals: Theory of Complex Tensor Order and the Emergence of the Uniaxial Nematic State

TL;DR

This paper develops a theoretical framework for unconventional d-wave superconductivity in Luttinger semimetals, revealing a complex tensor order parameter that leads to a uniaxial nematic state with line nodes, and discusses implications for real materials like YPtBi.

Contribution

It introduces a comprehensive Ginzburg--Landau free energy analysis for complex tensor order in Luttinger semimetals, highlighting the emergence of nematic states and time-reversal symmetry breaking.

Findings

Uniaxial nematic state features line nodes in excitation spectrum.

Genuine complex orders breaking time-reversal symmetry are favored at weak coupling.

The theory relates to experimental observations in YPtBi.

Abstract

We investigate unconventional superconductivity in three-dimensional electronic systems with the chemical potential close to a quadratic band touching point in the band dispersion. Short-range interactions can lead to d-wave superconductivity, described by a complex tensor order parameter. We elucidate the general structure of the corresponding Ginzburg--Landau free energy and apply these concepts to the case of an isotropic band touching point. For vanishing chemical potential, the ground state of the system is given by the superconductor analogue of the uniaxial nematic state, which features line nodes in the excitation spectrum of quasiparticles. In contrast to the theory of real tensor order in liquid crystals, however, the ground state is selected here by the sextic terms in the free energy. At finite chemical potential, the nematic state has an additional instability at weak coupling and low temperatures. In particular, the one-loop coefficients in the free energy indicate that at weak coupling genuinely complex orders, which break time-reversal symmetry, are energetically favored. We relate our analysis to recent measurements in the half-Heusler compound YPtBi and discuss the role of the cubic crystal symmetry.