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Exponential localization of Wannier functions in insulators
Christian Brouder1, Gianluca Panati, Matteo Calandra
1Institut de Minéralogie et de Physique des Milieux Condensés, CNRS UMR 7590, Universités Paris 6 et 7, IPGP, 140 rue de Lourmel, 75015 Paris, France.
Physical Review Letters
|March 16, 2007
Summary
Wannier functions in insulators with time-reversal symmetry are proven to be exponentially localized in 2D and 3D. This confirms a long-standing conjecture in condensed matter physics.
Area of Science:
- Condensed matter physics
- Quantum mechanics
- Materials science
Background:
- Wannier functions are crucial for understanding electronic properties of materials.
- Their localization properties are key to classifying topological phases.
- A long-standing conjecture proposed exponential localization for certain insulators.
Purpose of the Study:
- To rigorously prove the exponential localization of Wannier functions in 2D and 3D insulators with time-reversal symmetry.
- To investigate the implications of this localization for topological insulators and Chern insulators.
- To identify conditions for the reality of Wannier functions.
Main Methods:
- The study employs a proof based on the equivalence between analytic quasi-Bloch functions and the nullity of Chern numbers.
- This method is applied to insulators exhibiting time-reversal symmetry.
- The implications for systems with non-zero Chern numbers are analyzed.
Main Results:
- Exponential localization of Wannier functions is proven for all insulators with time-reversal symmetry.
- It is demonstrated that Chern insulators cannot possess exponentially localized Wannier functions.
- An explicit condition for the reality of Wannier functions has been identified.
Conclusions:
- The conjecture on exponential Wannier function localization in time-reversal symmetric insulators is settled.
- The findings provide a deeper understanding of topological phases and their characterization.
- The results offer new insights into the nature and conditions of Wannier function localization.
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