1. Electronic instabilities Electron phonon BCS superconductor Localization in 1D - CDW Electron-electron (  ve exchange)d-wave superconductor Localization in 1D - SDW Electron-electron (+ve exchange) p-wave superconductor Itinerant ferromagnetism

2. Why is the Fermi surface so important? Energy conservation Momentum conservation Exclusion principle Interactionmediated by virtual particles: phononsmagnonspolaronsplasmons Pairing or cooper- ative interaction

3. Pairing favors states with opposite momenta within a shell of thickness k B T about the Fermi surface If K = k 1 + k 2 = 0, then k 1 =  k 2 and k´ 1 =  k´ 2 q = 2  k 1  = 2  k 2  Now, k 1 and k 2 can scatter into any states in the inter- action shell with opposite momenta

4. Superconductivity Explains condensation into K = 0 pairs The pair may have orbital momentum namelspin state s  wave:0singlet p  wave:1triplet d  wave:2singlet Electron-phonon coupling favors BCS s  wave Coulomb repulsion favors l  0 states hence p, d wave in strongly correlated systems hence p, d wave in strongly correlated systems

5. So, let’s consider the electron- phonon interaction first

6. What about low-dimensionality? In the limit dimensionality  1 A single q (= ±2k F ) can map one surface onto the other Thus, essentially all exchange phonons have same q

7. One-dimensional Fermi surface not essential All that is needed are reasonable flat and parallel sections of Fermi surface, i.e. an appreciable # of states connected by the same q Under these circumstances, interactions are enhanced further still, becoming singular in one-dimension In BCS case, Interaction renormalizes phonon dispersion  (q)] 2 =   (q)] 2  C  q    (q) is the bare phonon frequency, C is a constant related to density of states

8.  (q)] 2 =   (q)] 2  C  q   q  diverges in 1D as T  0 Kohn anomaly 2k F phonon “softens” As T  0 v g and v   0 Result: Static lattice distortion

9. 2k F Lattice distortion in 1D Consequently, new zone boundary @ k = ±  /2a i.e. at k F For half filled band, k F =  /2a, so q =  /a, = 2a So, the distortion is twice the lattice spacing Charge-Density-Wave (CDW)

10. Metal insulator transition New zone boundary @ ±2  /a

11. The opening of a gap, lowers the energy of the system States at zone boundary lower in energy If energy gain exceeds electrostatic energy increase associated with lattice distortion, then ordered state wins Why does the system do this?

12. In general: deviations from half filling Distortion has: k d = 2k F d = 1/k F d = 1/k F New zone boundary @: ±  / d = ±k F So metal insulator transition still results However, d and a may not be commensurate

13. These lattice distortions are responsible for BCS attraction short lived - long enough to bind As you go towards 1D, interaction becomes stronger. However, eventually the static distortion wins, at which point you get the CDW. In principal this is a macroscopic K=0 state - ought to superconduct. Rigid, or incompressible - one impurity creates threshold against conduction

14. So, e-ph interaction can give rise to superconductivity. However, in the extreme 1D case, it causes locallization. Thus, we begin to see the delicate balance between superconductivity and insulating behavior What about superconductivity and magnetism?

15. Density waves Kohn anomaly, lattice distortion, etc.. is an example of a Peierls instability In the phononic case, this gives rise to: Charge Density Wave In systems with strong Coulomb repulsion this instability is unlikely to occur. However, a magnetic equivalent can occur: Spin Density Wave Spin Density Wave

16. Spin Density Waves Magnons, or spin-wave modes soften  Antiferromagnetic order Weak SDW Incommensurate case Extreme case

17. Metal insulator transition Magneticzoneboundary @ ±2  /a Small gap