1. 5/2/2007Cohen Group Meeting1 Luttinger Liquids Kevin Chan Cohen Group Meeting May 2, 2007

2. 5/2/2007Cohen Group Meeting2 Introduction Luttinger Liquid: system of 1D interacting fermions Reasons for interest: –Strong correlations –Real (quasi-)1D systems available experimentally –Exactly solvable

3. 5/2/2007Cohen Group Meeting3 Fermi vs. Luttinger Liquid Fermi Liquid –Landau –Elementary excitations are quasiparticles (fermions) –1-1 correspondence between quasiparticles and excitations of FEG –Weak correlations Luttinger Liquid –Tomonaga, Luttinger –Elementary excitations are collective excitations (bosons) –Strong correlations

4. 5/2/2007Cohen Group Meeting4 Luttinger Model Noninteracting Hamiltonian: Second quantization: Spinless fermions Two species Linear dispersion

5. 5/2/2007Cohen Group Meeting5 Bosonization Density operators: Commutation relations: Boson operators:

6. 5/2/2007Cohen Group Meeting6 Interaction Hamiltonian Scattering of same species: Different species: Note: scattering must conserve momentum

7. 5/2/2007Cohen Group Meeting7 Solution of Model with Spin Now include spin: Define operators that obey Boson commutation relations:

8. 5/2/2007Cohen Group Meeting8 Spin-Charge Separation Hamiltonian separates into charge and spin parts: Diagonalize charge density part: For delta function potential: Spin and charge have different speeds – spin-charge separation!

9. 5/2/2007Cohen Group Meeting9 1D versus 3D Luttinger Liquids: –Charge and spin density waves (bosons) –Can show that Fermi surface is unstable to perturbation Consider particle-hole excitation spectrum: 1D3D

10. 5/2/2007Cohen Group Meeting10 Experiment: Carbon Nanotubes (ropes) For LL, conductance should follow power law: Bockrath et al., Nature

11. 5/2/2007Cohen Group Meeting11 CNT Junctions Conductance measurements for two joined CNT segments I II Yao et al., Nature

12. 5/2/2007Cohen Group Meeting12 Spin-Charge Separation in Nanowires Observed in tunneling between parallel GaAs nanowires Results not all explained by theory

13. 5/2/2007Cohen Group Meeting13 Conclusion Luttinger model gives interesting phenomena Some experimental evidence in CNTs, nanowires (also polymers) Experimental situation not settled: –Spin-charge separation in CNTs? –Coulomb blockade?

14. 5/2/2007Cohen Group Meeting14 Instability of Fermi Surface