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We describe in terms of generators and relations the ring structure of the

R O (C_{2})

-graded

C_{2}

-equivariant stable stems

{π<#comment/>}_{⋆<#comment/>}^{C_{2}}

modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational

C_{2}

-equivariant sphere

{π<#comment/>}_{⋆<#comment/>}^{C_{2}} (S_{Q})

Revealing a distortive polar order buried in the Fermi sea

https://doi.org/10.1126/sciadv.adn0929

Shi, Jiaojian; You, Wenjing; Li, Xian; Gao, Frank Y; Peng, Xinyue; Zhang, Shangjie; Li, Jiangxu; Zhang, Yang; Fu, Liang; Taylor, Patrick J; et al (July 2024, Science Advances)

Polar metals are challenging to identify spectroscopically because the fingerprints of electric polarization are often obscured by the presence of screening charges. Here, we unravel unambiguous signatures of a distortive polar order buried in the Fermi sea by probing the nonlinear optical response of materials driven by tailored terahertz fields. We apply this strategy to investigate the topological crystalline insulator Pb_1−xSn_xTe, tracking its soft phonon mode in the time domain and observing the occurrence of inversion symmetry breaking as a function of temperature. By combining measurements across the material’s phase diagram with ab initio calculations, we demonstrate the generality of our approach. These results highlight the potential of terahertz driving fields to reveal polar orders coexisting with itinerant electrons, thus opening additional avenues for material discovery.

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