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:

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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

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Numerics (preliminary) Numerics (preliminary)

Courtesy of J-S. Caux

Numerics (preliminary) Numerics (preliminary)

Courtesy of J-S. Caux