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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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luttinger liquids

Luttinger Liquids

Kevin Chan

Cohen Group Meeting

May 2, 2007

Cohen Group Meeting

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

Cohen Group Meeting

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

Fermi vs. Luttinger Liquid

Cohen Group Meeting

luttinger model
Noninteracting Hamiltonian:

Second quantization:

Spinless fermions

Two species

Linear dispersion

Luttinger Model

Cohen Group Meeting

bosonization
Density operators:

Commutation relations:

Boson operators:

Bosonization

Cohen Group Meeting

interaction hamiltonian
Interaction Hamiltonian
  • Scattering of same species:
  • Different species:

Note: scattering must conserve momentum

Cohen Group Meeting

solution of model with spin
Solution of Model with Spin
  • Now include spin:
  • Define operators that obey Boson commutation relations:

Cohen Group Meeting

spin charge separation
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

1d versus 3d
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

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

Bockrath et al., Nature

Cohen Group Meeting

cnt junctions
CNT Junctions
  • Conductance measurements for two joined CNT segments

I

Yao et al., Nature

II

Cohen Group Meeting

spin charge separation in nanowires
Spin-Charge Separation in Nanowires
  • Observed in tunneling between parallel GaAs nanowires
  • Results not all explained by theory

Cohen Group Meeting

conclusion
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

instability of fermi surface
Instability of Fermi Surface

Cohen Group Meeting