One Dimensional Electronic Systems: Quantum Hall Edge States: Chiral Luttinger Liquid

1. One Dimensional Electronic Systems: • Quantum Hall Edge States: • Chiral Luttinger Liquid • Quantum Wires • Semiconductor wires • Polyacetalene • Carbon Nanotubes

2. Integer Quantum Hall Effect with edges (Halperin 1982) x Bulk Landau levels: L R y n=1 E EF Chiral Fermions yR Fermi Liquid exponents (g=1) even with interactions

3. Fractional Quantum Hall Effect: Chiral Luttinger Liquid Wen 1990 v • Assume a single edge mode: • Laughlin States n=1/m • “Sharp” Edge Incompressible Fluid n=1/m g is determinied by the Chiral Anomaly

4. Fractional Quantum Hall Effect: Chiral Luttinger Liquid Wen 1990 v • Single Mode approximation: • Valid for • Laughlin States n=1/m • “Sharp” Edge J=sxyE Incompressible Fluid n=1/m Equation of motion :

5. Fractional Quantum Hall Effect: Chiral Luttinger Liquid Wen 1990 v • Assume a single edge mode: • Laughlin States n=1/m • “Sharp” Edge Incompressible Fluid n=1/m Electron op. (charge e) Laughlin Q.P. op. (charge e/m)

6. Quantum Point Contact: Analog of single barrier in L.L. L Weak (electron) Tunneling e n=1/m R Weak (quasiparticle) Backscattering T quasiparticle e*=e/m n=1/m B

7. Early point contact experiments (Milliken, Umbach and Webb 1994) For weak tunneling theory predicts G ~ T4

8. Resonance Lineshapes (Milliken, Umbach and Webb 1994) For “perfect” resonances theory predicts: Difficulty for interpretation: Gates produce a smooth confining potential

9. Cleaved Edge Experiments Chang, Pfeiffer and West, 1996 Tunneling from metal to n=1/3 at an atomically sharp edge Predicted Scaling Form: c I(V,T)/G(T) Fit from Experiment: a ~ 2.7 V(mV) cV/T

10. Dependence of Tunneling Exponent on Filling Factor Theory: Kane, Fisher 95 Shytov, Levitov, Halperin 98 Exponent should depend on “neutral modes” predicted to be present for n≠1/m. Leads to plateau structure in a(n) Experiment: Grayson, et al, 1998 Exponent seems to depend only on the “charge mode”: a ~ 1/n. Experiments must not be in the low energy, long wavelength limit described by theory.

11. -e* Ib e* Shot Noise in a point contact: DePicciotto et. al.; Saminadayer et. al. 97 Direct measurement of the charge of the Laughlin Quasiparticle It Strong Pinch Off e tunneling e -e S = 2e It Weak Pinch Off Q.P. tunneling S = 2e* Ib e*=e/3

12. Quantum Wires: 1D for E < DE: the subband spacing 1. Split Gate Devices: (Tarucha et al. 95) DE ~ 5 meV 2. Cleaved Edge Overgrowth (Yacoby et al. 96) DE ~ 20 meV

13. Molecular Wire: Polyacetalene (See Heeger et al. Rev. Mod. Phys. 1988) E E 1 EF EF D k k Truly 1D .... but Peierl’s Instability: Any commensurate 1D metal is unstable Elastic energy cost :~ D2 Electronic energy gain: ~ lD2 log w0/D Peirels Energy Gap: D ~ w0 e-1/l ~ 1.8 eV

14. “Armchair” [N,N] Carbon Nanotubes E k • One Dimensional for • Peierls Instability suppressed for large N ~ R/a: Elastic energy cost :~ ND2 Electronic energy gain: ~ 1 lD2 log w0/D D ~ w0 e-N/l

15. Low Energy Theory L R 4 Channels:

16. Electron Interactions • “Backscattering Interactions” • Violate Chiral symmetry • Have form l cos(fa+fb-fc-fd) • Can lead to electronic Instabilities • Depend on the short range part of V(r) ( q ~ 1/a ) • Suppressed by 1/N • 2. “Forward Scattering Interactions” • Preserve Chiral symmetry • Have form • Lead to Luttinger Liquid • Depend on Long range part of V(r) • NOT suppressed by 1/N • Screened Coulomb Interaction :

17. Luttinger Liquid Theory Charge/Spin/Flavor variables: 1 Charge Mode 3 Neutral Modes Spin Charge Separation: w vrq vFq q

18. Tunneling Experiments Metal contact to rope of nanotubes G(T) ~ Ta Bockrath et al. 1999 Theory: abulk= (g+1/g-2)/8 ~.25 aend= (1/g-1)/4 ~ .65 End contacted Bulk contacted

19. Tunneling across a barrier in a single nanotube Yao et al. 99 Scaling Plot TaF(V/T)