5/2/2007 Cohen Group Meeting 1

Luttinger Liquids

Kevin Chan

Cohen Group Meeting

May 2, 2007

5/2/2007 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

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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

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Luttinger Model

• Noninteracting Hamiltonian:

• Second quantization:• Spinless fermions• Two species• Linear dispersion

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Bosonization• Density operators:

• Commutation relations:

• Boson operators:

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Interaction Hamiltonian

• Scattering of same species:

• Different species:

Note: scattering must conserve momentum

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Solution of Model with Spin

• Now include spin:

• Define operators that obey Boson commutation relations:

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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!

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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

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Experiment: Carbon Nanotubes (ropes)• For LL, conductance should follow power law:

Bockrath et al., Nature

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CNT Junctions

• Conductance measurements for two joined CNT segments

I II Yao et al., Nature

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Spin-Charge Separation in Nanowires

• Observed in tunneling between parallel GaAs nanowires

• Results not all explained by theory

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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?

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Instability of Fermi Surface