Relativistic and nonrelativistic spin splitting above and below the Fermi level in a $g$-wave altermagnet

TL;DR

This study combines advanced spectroscopic techniques and first-principles calculations to map and distinguish relativistic and nonrelativistic spin splitting in a layered altermagnet, revealing complex spin textures across energy levels.

Contribution

It provides the first complete momentum-resolved mapping of both relativistic and nonrelativistic spin splitting in an altermagnet using combined spin-ARPES and spin-ARRES techniques.

Findings

Distinct momentum-dependent spin splitting phenomena identified

NRSS and RSS exhibit different symmetry and temperature dependence

The work demonstrates the effectiveness of combined spectroscopies in spin texture analysis

Abstract

Nonrelativistic spin splitting (NRSS) challenges conventional wisdom about antiferromagnets by allowing spin-split electronic bands even in collinear orders with zero net magnetization. This sub-class of antiferromagnets, recently dubbed "altermagnets," enforces distinctive spin textures via spin-group symmetries in the crystal. However, direct experimental evidence for such symmetry-driven magnetism remains scarce, and distinguishing it from relativistic spin splitting presents additional challenges. Here, we combine first-principles calculations, symmetry analysis, and two spin-resolved spectroscopies--angle-resolved photoemission (spin-ARPES) and our newly developed spin- and angle-resolved electron reflection spectroscopy (spin-ARRES)--to achieve the first complete momentum-resolved mapping of relativistic (RSS) and nonrelastivistic (NRSS) spin splitting in CoNb$_4$Se$_8$. By probing both the occupied (spin-ARPES) and unoccupied (spin-ARRES) electronic states in a single experiment, we uncover a series of momentum-dependent spin splitting phenomena each of which switch sign under sixfold rotations and persists far above and below the Fermi level. Crucially, distinct nodal planes in momentum space distinguish NRSS from RSS features. Additionally, the observed collapse of NRSS and the persistence of RSS above the N\'eel temperature, distinguishes a genuine magnetic phase transition from inversion symmetry breaking. Our work demonstrates, for the first time, the combined power of spin-ARPES and spin-ARRES in capturing the full spin texture across an extended energy range, positioning CoNb$_4$Se$_8$ as a prototype for exploring spin-group-based phenomena. These findings open new routes for engineering spin-based functionalities ranging from neuromorphic computing to unconventional superconductivity in layered antiferromagnets.