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Fermi-Luttinger Liquid. Alex Kamenev. in collaboration with. Leonid Glazman, U of M Maxim Khodas, U of M. Michael Pustilnik, Georgia Tech. PRL 96 , 196405 (2006); arXiv:cond-mat/0702.505 arXiv:cond-mat/0705.2015. RPMBT14, Jul., 2007. One-dimensional …. Dekker et al 1997. - PowerPoint PPT Presentation
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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
One-dimensional …One-dimensional …
M. Chang, et al 1996 Dekker et al 1997 Bockrath, et al 1997
Auslaender et al 2004 I. Bloch 2004
Spectral Function
d>1: Fermi Liquid
Energy relaxation rate:
interaction potential
Spectral density:
The same for holes
d=1d=1
Energy relaxation rate:
? ?
Spectral density:
Luttinger modelLuttinger model
Dzaloshinskii, Larkin 1973
Energy relaxation rate:
Spectral density:
Luttinger model (cont)Luttinger model (cont)
Haldane, 1983
1D with non-linear dispersion: Holes 1D with non-linear dispersion: Holes
1D with non-linear dispersion: Particles1D with non-linear dispersion: Particles
Energy relaxation rate:
interaction potential Does not work for integrable models
Particles (cont)Particles (cont)
Fermi head with the Luttinger tail
Spectral Edges Spectral Edges
Shake up or X-ray singularity
(cf. Mahan, Nozieres,…)
Structure Factor Structure Factor
Luttinger approximation Luttinger approximation
Linear dispersion
Exact result within the Luttinger approximation.
How does the dispersion curvature and interactions affect the structure factor ?
Spectrum curvature Spectrum curvature ++ interactions interactions
interactions
Fourier components of the interaction potential V
AFM spin chainAFM spin chain
N 200. For this case we have calculated2 200 000 form factors
S. Nagler, et al 2005
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
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
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.
Boson-Fermion mapping
Hydrodynamics
Summary of bosonic exponents Summary of bosonic exponents
?
Numerics (preliminary) Numerics (preliminary)
Courtesy of J-S. Caux
Numerics (preliminary) Numerics (preliminary)
Courtesy of J-S. Caux