Link to HTML version of Power Point Slides for Lecture 22

In previous lectures we described the idea of quasiparticles, self energies and properties in systems that are called "Normal Fermi Liquids". A Fermi liquid is one in which the self-energy is finite and has finite derivatives at the Fermi energy: then there is a jump in the occupation n(k) at the Fermi surface just as a non-interacting system, but with a reduced value in the jump. A remarkable result proved by Luttinger is that IF the self-energy has this form, then the volume of the Fermi surface is exactly the same as a non-interacting system. This is a key concept that enables us to pose sharp questions about correlated systems. It is useful to note [Martin, 1982] that the Luttinger sum rule on the Fermi surface in crystals is completely analogous to the Friedel sum rule for impurities, in which cases one does have exact solutions that have Fermi liquid behavior. In this course, we do not have time to derive the formalism but we will only describe the results and leave it to students (possibly for projects) to explore further. More details are in the notes than in the lectures.

References for Strongly Correlated Systems:
Most complete review to date: M. Imada, A. Fujimori, and Y. Tokura, "Metal Insulator Transitions", "Rev. Mod. Phys. 70, 1039 (1998), which reviews theory and experiment on correlated-electron systems, especially transition metal oxides.
See also E. Dagotto, "Correlated Electrons in High Temperature Superconductors", Rev. Mod. Phys. 66, 763 (1994).
Photoemission: Damascelli, et al., Rev. Mod. Phys. 75, 473 (2003)

References for Luttinger theorem:
J. M. Luttinger, Phys. Rev. 119, 1153 (1960); J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960).
R. M. Martin, Phys. Rev. Letters 48, 362 (1982)

1. Strongly Correlated Systems
1. Examples of Strongly Correlated Systems
2. Intra-atomic (interactions) vs. Interatomic (bands) effects
3. Localized magnetic moments
4. Magnetic order
5. Enhancements
6. Metal-insulator transitions
2. Recall properties of Green's functions and self energies
1. The Green's Function G(l,E) and self energy S^*(l,E) can be labeled by any properly conserved quantities in the many-body system
2. In a crystal - crystal momentum k is conserved
3. An impurity in an isotropic medium conserves angular momentum L,m
3. Luttinger "Theorem" on the volume enclosed by the Fermi surface
1. Non-interacting case
2. Existence of Surface in interacting Fermi liquid
3. Luttinger's theorem in terms of Green's function evaluated only at the Fermi energy which the self energy is purely real
4. Isotropic case - interacting surface must equal noninteracting one
5. Anisotropic case - volume must be the same
4. Applications to crystals
1. IF Fermi liquid theory applies then any system with odd number of

electrons per cell MUST be a metal
2. IF Fermi liquid theory applies then any system with even number of

electrons per cell MAY be an insulator
3. Applies to magnetic systems separately to each spin if spin-orbit interactions can be neglected