Symmetry-resolved topology from interacting Green's functions

Abstract

Topological phases in narrow-gap materials often arise from a delicate interplay of spin-orbit coupling (SOC) and electron correlation, making their prediction sensitive to the underlying electronic-structure approximation. Electronic topology is commonly characterized using effective one-body Hamiltonians, but this approach becomes delicate near gap closings and does not reveal how a band inversion is realized in the interacting spectrum. Here we introduce a multi-messenger analysis that combines three complementary readouts of a single interacting Green's function: the Fu-Kane invariant of the topological Hamiltonian, the symmetry-resolved spectral function, and the symmetry-resolved orbital occupation. Together, they identify the topological class, resolve band connectivity, and track the orbital character of the inversion. We implement this framework within fully self-consistent relativistic GW, treating SOC and electronic correlation on equal footing while eliminating dependence on a density-functional starting point. Applied to strain-tunable $α$-Sn, the approach distinguishes its competing zero-gap, Dirac-semimetal, and topological-insulator regimes and provides an internally consistent description of their common band-inversion mechanism. More broadly, this framework offers a frequency-resolved route to diagnosing correlated topology across both gapped and near-critical states.