Rank-Selective Optical Tomography of Higher-Wave Altermagnetism

Abstract

Identifying the spatial rank of higher-wave altermagnetic order optically is challenging because local electric-dipole response does not uniquely resolve distinct continuum harmonics. We show that finite photon momentum turns one-photon spin-resolved absorption into a rank-selective tomography. For the planar $|m|=\ell$ sector of an even-parity $\ell$-wave component, a joint Fourier projection in polarization and momentum angle isolates $\mathcal T_\ell\proptoη_\ell q^{\ell-2}$, yielding the hierarchy $d:q^0$, $g:q^2$, and $i:q^4$; phase changes track rotations of the selected magnetic harmonic. A Ward-consistent finite-$q$ microscopic calculation reproduces these powers without imposing them. Although discrete crystal symmetry can generate lower-order local aliases, they are orthogonal to the selected momentum harmonic and cannot contaminate it below $q^{\ell-2}$. In MnTe, whose nonrelativistic parent order is three-dimensional $g$ wave while spin--orbit coupling lowers the exact relativistic spin-momentum-locking symmetry, first-principles calculations show that more than $99.9\%$ of the Fourier power of the Néel-projected $A$-region spin-energy contrast remains in the parent $g$-wave-derived $m=3$ harmonic. Structured near fields place the required momentum window within experimental reach.