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Luttinger Liquids

Luttinger Liquids. Kevin Chan Cohen Group Meeting May 2, 2007. Introduction. Luttinger Liquid: system of 1D interacting fermions Reasons for interest: Strong correlations Real (quasi-)1D systems available experimentally Exactly solvable. Fermi Liquid Landau

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Luttinger Liquids

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

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

  3. 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 Fermi vs. Luttinger Liquid Cohen Group Meeting

  4. Noninteracting Hamiltonian: Second quantization: Spinless fermions Two species Linear dispersion Luttinger Model Cohen Group Meeting

  5. Density operators: Commutation relations: Boson operators: Bosonization Cohen Group Meeting

  6. Interaction Hamiltonian • Scattering of same species: • Different species: Note: scattering must conserve momentum Cohen Group Meeting

  7. Solution of Model with Spin • Now include spin: • Define operators that obey Boson commutation relations: Cohen Group Meeting

  8. 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! Cohen Group Meeting

  9. 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: 1D 3D Cohen Group Meeting

  10. Experiment: Carbon Nanotubes (ropes) • For LL, conductance should follow power law: Bockrath et al., Nature Cohen Group Meeting

  11. CNT Junctions • Conductance measurements for two joined CNT segments I Yao et al., Nature II Cohen Group Meeting

  12. Spin-Charge Separation in Nanowires • Observed in tunneling between parallel GaAs nanowires • Results not all explained by theory Cohen Group Meeting

  13. 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? Cohen Group Meeting

  14. Instability of Fermi Surface Cohen Group Meeting

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