Physicists prove hidden equivalence in quantum material blueprints
Researchers have demonstrated that two competing mathematical methods for understanding topological materials are mathematically identical, settling a decade-long implicit assumption in the field. The proof simplifies how engineers can design and verify advanced semiconductors for quantum computing and next-generation electronics without requiring perfect crystal structures.
Originaltitel: Explicit equivalence between the spectral localizer and local Chern and winding markers
Topological band insulators are classified using momentum-space topological invariants, such as Chern or winding numbers, when they feature translational symmetry. The lack of translation symmetry in disordered, quasicrystalline, or amorphous topological systems has motivated alternative, real-space definitions of topological invariants, including the local Chern marker and the spectral localizer invariant. However, the equivalence between these invariants is so far implicit. Here, we explicitly demonstrate their equivalence from a systematic perturbative expansion in powers of the spectral localizer’s parameter \kappa <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>κ</mml:mi> </mml:math> . By leveraging only the Clifford algebra of the spectral localizer, we prove that Chern and winding markers emerge as leading-order terms in the expansion. It bypasses abstract topological machinery, offering a simple approach accessible to a broader physics audience.