Download - Fermi-Luttinger Liquid
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Fermi-Luttinger Liquid Fermi-Luttinger Liquid
Leonid Glazman, U of M Leonid Glazman, U of M Maxim Khodas, U of MMaxim Khodas, U of MMichael Pustilnik, Georgia Tech Michael Pustilnik, Georgia Tech
Alex KamenevAlex Kamenev
in collaboration with
PRL 96, 196405 (2006); arXiv:cond-mat/0702.505arXiv:cond-mat/0705.2015
RPMBT14, Jul., 2007
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One-dimensional …One-dimensional …
M. Chang, et al 1996 Dekker et al 1997 Bockrath, et al 1997
Auslaender et al 2004 I. Bloch 2004
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Spectral Function
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d>1: Fermi Liquid
Energy relaxation rate:
interaction potential
Spectral density:
The same for holes
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d=1d=1
Energy relaxation rate:
? ?
Spectral density:
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Luttinger modelLuttinger model
Dzaloshinskii, Larkin 1973
Energy relaxation rate:
Spectral density:
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Luttinger model (cont)Luttinger model (cont)
Haldane, 1983
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1D with non-linear dispersion: Holes 1D with non-linear dispersion: Holes
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1D with non-linear dispersion: Particles1D with non-linear dispersion: Particles
Energy relaxation rate:
interaction potential Does not work for integrable models
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Particles (cont)Particles (cont)
Fermi head with the Luttinger tail
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Spectral Edges Spectral Edges
Shake up or X-ray singularity
(cf. Mahan, Nozieres,…)
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Structure Factor Structure Factor
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Luttinger approximation Luttinger approximation
Linear dispersion
Exact result within the Luttinger approximation.
How does the dispersion curvature and interactions affect the structure factor ?
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Spectrum curvature Spectrum curvature ++ interactions interactions
interactions
Fourier components of the interaction potential V
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AFM spin chainAFM spin chain
N 200. For this case we have calculated2 200 000 form factors
S. Nagler, et al 2005
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1D Bose Liquid1D Bose Liquid
Constant-q scanCaux, Calabrese, 2006Lieb-Liniger model, 1963
Bosons with the strong repulsion =Fermions with the weak attraction – changes sign.
Bose-Fermi mapping (1D)
1D hard-core bosons = free fermions (Tonks-Girardeau) Divergence at the upper edge
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Structure factor: conclusionsStructure factor: conclusions
S 0
S q( , )Fermions
Power law singularities at the spectral edges (Lieb modes) with q-dependent exponents.
S q( , )Bosons
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Fermi-Luttinger Liquid Fermi-Luttinger Liquid
Hole’s mass-shell is described by the Luttinger liquid (with momentum-dependent exponent).
Particle’s mass-shell is described by the Fermi liquid (with smaller relaxation rate).
Spectral edges of the spectral function and the structure factor exhibit power-law singularities.
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Boson-Fermion mapping
Hydrodynamics
Summary of bosonic exponents Summary of bosonic exponents
?
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Numerics (preliminary) Numerics (preliminary)
Courtesy of J-S. Caux
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Numerics (preliminary) Numerics (preliminary)
Courtesy of J-S. Caux