Wannier decay and the Thouless conjecture

2025/05/04 by Becker, Simon, Tao, Zhongkai, Yang, Mengxuan
#FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci) #Mathematical Physics (math-ph) #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2505.01999

Abstract

Non-trivial Chern classes pose an obstruction to the existence of exponentially decaying Wannier functions which provide natural bases for spectral subspaces. For non-trivial Bloch bundles, we obtain decay rates of Wannier functions in dimensions d=2,3. For d=2, we construct Wannier functions with full asymptotics and optimal decay rate O(|x|^-2) as conjectured by Thouless; for d=3, we construct Wannier functions with the uniform decay rate O(|x|^-7/3).

Citations

• Fragile topology on solid grounds: a mathematical perspective
• Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension
• Constraints on real space representations of Chern bands
• Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators
• On Parseval frames of exponentially decaying composite Wannier functions
• An overview of periodic elliptic operators
• Maximally localized Wannier functions: Theory and applications
• Maximally-localized generalized Wannier functions for composite energy bands

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