Luttinger Liquids - University of California, Berkeley
Luttinger Liquids Kevin Chan Cohen Group Meeting May 2, 2007 5/2/2007 Cohen Group Meeting 1 Introduction Luttinger Liquid: system of 1D interacting fermions Reasons for interest: Strong correlations
Real (quasi-)1D systems available experimentally Exactly solvable 5/2/2007 Cohen Group Meeting 2 Fermi vs. Luttinger Liquid Fermi Liquid Luttinger Liquid Landau
Elementary excitations are quasiparticles (fermions) 1-1 correspondence between quasiparticles and excitations of FEG Weak correlations 5/2/2007 Tomonaga, Luttinger Elementary excitations are collective excitations (bosons) Strong correlations
Cohen Group Meeting 3 Luttinger Model Noninteracting Hamiltonian: Second quantization: Spinless fermions Two species Linear dispersion 5/2/2007 Cohen Group Meeting
4 Bosonization Density operators: Commutation relations: 5/2/2007 Boson operators: Cohen Group Meeting 5
Interaction Hamiltonian Scattering of same species: Different species: 5/2/2007 Cohen Group Meeting Note: scattering must conserve momentum 6 Solution of Model with Spin
Now include spin: Define operators that obey Boson commutation relations: 5/2/2007 Cohen Group Meeting 7 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! 5/2/2007 Cohen Group Meeting 8
Luttinger Liquids: 1D versus 3D Charge and spin density waves (bosons) Can show that Fermi surface is unstable to perturbation Consider particle-hole excitation spectrum: 1D
5/2/2007 3D Cohen Group Meeting 9 Experiment: Carbon Nanotubes (ropes) For LL, conductance should follow power law: Bockrath et al., Nature 5/2/2007 Cohen Group Meeting
10 CNT Junctions Conductance measurements for two joined CNT segments I 5/2/2007 Yao et al., Nature II Cohen Group Meeting 11
Spin-Charge Separation in Nanowires Observed in tunneling between parallel GaAs nanowires Results not all explained by theory 5/2/2007 Cohen Group Meeting 12 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? 5/2/2007 Cohen Group Meeting 13 Instability of Fermi Surface 5/2/2007
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