Link to HTML version of Power Point Slides for Lecture 15

Reading:Chapter 19, App. D (also related to App. E)

Today marks a turning point in the course. Although the "many-body" parts of the course are listed on the schedule as starting with lecture 18, response functions are especially important in explicit many-body theoretical methods. The theory of response functions will be useful later, even though the examples at this point all involve independent-particle methods.

Response functions are the "bread and butter of theoretical physics directly related to important properties and experimental measurements. Elegant textbook formulations reveal key ideas, and equally elegant reformulations of the problem yield efficient methods that have made possible new advances in or ability to predict properties ad phenomena. Lattice vibrations provide a key example. The advances in algorithms have led to entirely new abilities to predict phonon dispersions, anharmonic terms, etc.
Key References:
Review: "Phonons and related properties of extended systems from density-functional perturbation theory", S. Baroni and S. de Gironcoli and A. Dal Corso", Rev. Mod. Phys. 73,515-562 (2001).
Green's-function approach to linear response in solids, S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58, 1861-1864 (1987); Also P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, Phys. Rev. B 43, 7231-7242(1991).
Density-functional approach to nonlinear-response coefficients of solids, X. Gonze and J.-P. Vigneron, Phys. Rev. B 39, 13120-13128 (1989).

1. Response functions and perturbation theory (App. D)
   • General definition of response function
   • Non-interacting particles
   • Expressions involving sums over excited (unoccupied) states follow from Standard perturbation theory
   • Self-consistent field theories - useful in general contexts - many-body theory (later) - Kohn-Sham equations
   • Example of dielectric response function (App. E)
2. Lattice vibrations and phonon dispersion curves (Chap. 19)
   • Harmonic approximation
   • Another example of Bloch theorem - phonon dispersion curves
   • General form shown by models - famous work on Born and von Karmen
3. Lattice vibrations with forces determined from the electrons, e.g., Kohn-Sham DFT
   • Standard perturbation expressions - elegant -useful in simple cases - cumbersome and costly in general
   • "Brute Force" methods - full self-consistent calculation as a function of positions of atoms
   • Total energy calculations
     • Very useful - essential for full information
     • Examples of phase transitions
     • Possible to identify harmonic terms
   • Force theorem gives the force on all atoms - extrapolate the linear regime to get force constants
4. The breakthrough - New approach for linear (and non-linear) response functions
   • Solve as coupled equations in perturbation theory explicitly to first order for the wave functions
   • Use iterative methods to solve equations
   • Expressions give directly phonon dispersion curves for any wavevector k
   • Examples: GaAs, Ni, . . .
5. Further developments
   • "2n+1 theorem" (recall from before)
   • Extend to calculation of non-linear response functions
   • Odd order for free - third order energy from first order wave functions!
   • Variational from for energies at even order (2n)
6. Conclusions