Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators

• DOI: 10.48550/arxiv.2107.10699
• Publication Date: 2024-06-12
• Authors: Jianfeng Lu; Kevin D. Stubbs
• Keywords: Condensed Matter - Mesoscale and Nanoscale Physics; Mesoscale and Nanoscale Physics (cond-mat.mes-hall); FOS: Physical sciences; Mathematical Physics (math-ph); 0101 mathematics; 01 natural sciences; Mathematical Physics

12 pages, no figures. We found an error in the previous version of this paper. Because of this, the decay required for our main result is slightly worse ($2 + \epsilon$ instead of finite second moment)<br/>For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to $0$) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis elements satisfy $\int |\boldsymbol{x}|^2 |w(\boldsymbol{x})|^2 \,\textrm{d}{\boldsymbol{x}} < \infty$). In this paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay ($\int |\boldsymbol{x}|^{2+��} |w(\boldsymbol{x})|^2 \,\textrm{d}{\boldsymbol{x}} < \infty$ for any $��> 0$) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.<br/>