Electron-Phonon Interactions and 1D Superconductivity: A Deep Dive into Localization and Pairing

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? Pairing or cooper- ative interaction Energy conservation Momentum conservation Exclusion principle Interactionmediated by virtual particles: phonons magnons polarons plasmons

3. Pairing favors states with opposite momenta within a shell of thickness kBT about the Fermi surface If K = k1 + k2 = 0, then k1 = -k2 and k´1 = -k´2 q = 2k1= 2k2 Now, k1 and k2 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 name l spin state s- wave: 0 singlet p- wave: 1 triplet d- wave: 2 singlet Electron-phonon coupling favors BCS s- wave Coulomb repulsion favors l 0 states 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 (= ±2kF) 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 [w(q)]2 = [wo(q)]2- Cc(q) wo(q) is the bare phonon frequency, C is a constant related to density of states

8. Kohn anomaly [w(q)]2 = [wo(q)]2- Cc(q) c(q) diverges in 1D as T  0 2kF phonon “softens” As T  0 vg and vf 0 Result: Static lattice distortion

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

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

11. Why does the system do this? 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

12. In general: deviations from half filling Distortion has: kd = 2kF ld = 1/kF New zone boundary @: ±p/ld = ±kF So metal insulator transition still results However, ld 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

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

17. Metal insulator transition Magnetic zone boundary @ ±2/a Small gap