Transcript Luttinger Liquids - University of California, Berkeley

Luttinger Liquids
Kevin Chan
Cohen Group Meeting
May 2, 2007
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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
• Luttinger Liquid
– Landau
– Elementary excitations
are quasiparticles
(fermions)
– 1-1 correspondence
between quasiparticles
and excitations of FEG
– Weak correlations
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– 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:
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• Boson operators:
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Interaction Hamiltonian
• Scattering of same
species:
• Different species:
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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
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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
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Yao et al., Nature
II
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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
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