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Localization of Generalized Wannier Bases Implies Chern Triviality in Non-periodic Insulators
Giovanna Marcelli1, Massimo Moscolari2, Gianluca Panati3
1Mathematics Area, SISSA, Via Bonomea 265, 34136 Trieste, Italy.
Annales Henri Poincare
|March 23, 2023
Summary
Localized Wannier bases in 2D gapped quantum systems indicate vanishing Chern character, linking localization to topological properties like the Hall conductivity. This suggests a dichotomy in localization for gapped systems.
Area of Science:
- Condensed Matter Physics
- Quantum Mechanics
- Materials Science
Background:
- Topological phases of matter, such as Chern insulators and quantum Hall systems, exhibit unique electronic properties governed by topology.
- Generalized Wannier bases provide a framework for understanding the localization properties of electronic wavefunctions in these systems.
- Disordered backgrounds and magnetic fields are crucial factors influencing the behavior of electrons in two-dimensional quantum systems.
Purpose of the Study:
- To investigate the relationship between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems.
- To establish a connection between the existence of localized Wannier bases and the vanishing of topological invariants like the Chern character.
- To propose a general conjecture regarding the localization behavior in non-periodic gapped quantum systems.
Main Methods:
- Theoretical analysis of generalized Wannier bases in two-dimensional gapped quantum systems.
- Mathematical proof establishing the implication of Wannier basis localization on the Chern character.
- Formulation of a localization dichotomy conjecture for general non-periodic gapped quantum systems.
Main Results:
- The existence of a well-localized generalized Wannier basis for the Fermi projection is proven to imply the vanishing of the Chern character.
- The Chern character is shown to be proportional to the Hall conductivity in the linear response regime.
- A localization dichotomy conjecture is proposed for general non-periodic gapped quantum systems.
Conclusions:
- The localization of generalized Wannier bases is intrinsically linked to the topological properties of gapped quantum systems.
- Vanishing Chern character, and thus zero Hall conductivity, is a direct consequence of the existence of localized Wannier bases.
- The proposed localization dichotomy conjecture offers a new perspective on the behavior of topological states in quantum systems.
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