Spin triplet nematic pairing symmetry and superconducting double transition in U$_{1-x}$Th$_{x}$Be$_{13}$

TL;DR

This paper develops a theoretical framework to explain the double superconducting transition in U$_{1-x}$Th$_{x}$Be$_{13}$, identifying specific spin triplet pair symmetries and phases consistent with experimental data.

Contribution

The study introduces a symmetry-based theory identifying the pairing symmetry and phases responsible for the double transition in U$_{1-x}$Th$_{x}$Be$_{13}$, including the cyclic p-wave and nematic phases.

Findings

Identification of the low-temperature cyclic p-wave phase.

Recognition of the high-temperature biaxial nematic phase.

Explanation of the double transition via symmetry breaking and degeneracy splitting.

Abstract

Motivated by a recent experiment on U$_{1-x}$Th$_{x}$Be$_{13}$ with $x=3\%$, we develop a theory to narrow down the possible pair symmetry to consistently describe the double transition utilizing various theoretical tools; group theory and Ginzburg-Landau theory. It is explained in terms of the two dimensional representation E$_{\rm u}$ with spin triplet. A symmetry breaking causes the degenerate $T_{\rm c}$ to split into the two. The low temperature phase is identified as the cyclic $p$ wave: $\vec {d}({\bf k})={\hat x}k_x+\varepsilon{\hat y}k_y+\varepsilon^2{\hat z}k_z$ with $\varepsilon^3=1$ while the biaxial nematic phase: ${\vec d}({\bf k})={\sqrt 3}({\hat x}k_x-{\hat y}k_y$) is the high temperature one. This allows us to simultaneously identify the uniaxial nematic phase: ${\vec d}({\bf k})=2{\hat z}k_z-{\hat x}k_x-{\hat y}k_y$ for UBe$_{13}$, which breaks spontaneously cubic symmetry of the system. Those pair functions are fully consistent with the above and existing data. We comment on the accidental scenario in addition to this degeneracy scenario and the intriguing topological nature hidden in this long-known material.