Link to HTML version of Power Point Slides for Lecture 16

Reading:Chapter 21 and 22 (also mention ideas in Ch. 23)

It is a great advantage to reformulate the electronic structure problem in terms of localized states. This is widely used in chemistry for practical reasons and provides the basis for "Order-N" methods (Ch. 23, which we will describe briefly). Wannier functions (Ch. 21) provide the fundamental transformation; however, until recently they were used almost exclusively only for formal proofs because of their non-uniqueness. The advent of "maximally-localized" functions provides a practical approach with elegant, rigorous connections to the ideas of polarization and localization (Ch. 22).

The ability to sustain a macroscopic polarization and the lack of conductivity at zero temperature are the defining properties of dielectrics (insulators and semiconductors) distinguishing them from conductors (metals and superconductors). We will review the basic concept of polarization and the subtleties that arise in an infinite solid, which lead to the surprising result that the polarization in the bulk of a dielectric is not directly determined by the bulk electronic density (often stated incorrectly in the well-known textbooks). It is only in the last decade that it has been possible to determine the polarization from the quantum wave function - as a "Berry's phase" in terms of the phases of the Bloch wave functions. Elegant - but surprising - expressions make possible practical calculations now widely used. The expressions can also be understood using the Wannier representation, where the formulas have an intuitive form and provide the correct fundamental expressions that justify the final formulas quoted in textbooks such as the Clausius-Mosotti relations. Finally this leads us to the fundamental definition of "localization" and quantitative expressions for the localization length in an insulator.

References:
Review: "Macroscopic polarization in crystalline dielectrics: the geometric phase approach", R. Resta, Rev. Mod. Phys. 66, 899-915 (1994).
Key papers:
"Theory of the Insulating State", W. Kohn, Phys. Rev. 133, A171-181 (1964)
"Theory of Polarization of crystalline solids", R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
"Polarization and localization in insulators: generating function approach", I. Souza and T. J. Wilkens and R. M. Martin", Phys. Rev. B 62, 1666-1683 (2000).

1. Wannier Functions (Ch. 21)
   • Definition and properties
   • Non-uniqueness
   • "Maximally-projected" Wannier functions - simple examples
   • "Maximally-localized" Wannier functions - invariants
   • Practical formulas for construction
2. Electronic Polarization and Localization in Solids: Berry's phases (Chap. 22)
   • Well-defined expressions for a dipole in a finite system
   • Problem in condensed matter - bulk properties in the limit of an infinite system
   • What to do? How to relate to experiments
   • Experiments measure an integrated current
3. Geometric Berry's phase expression for Polarization
   • Expressions for integrated current from standard perturbation theory
   • Unique expression - but not what one wants because it involves an integral over the history of the material - not a unique function of the state of the material at one time
   • Transformation to form involving a Berry's phase
     • Unique expression - except for "quanta" of polarization
     • The "quanta" have simple interpretation - can be directly relate to experiment
4. Relation to Wannier functions
   • Polarization is an invariant - except for "quanta"
   • Provide the correct quantum formula for local dipole moments in a solid, e.g., the entities in a Classius-Mosotti formula
5. Localization in an insulator
   • Fundamental distinction of insulators and metals
   • A "Berry curvature"
   • Invariant expression for localization length
   • Expression in terms of Maximally localized Wannier functions
6. Conclusions