1. Fermi-Luttinger Liquid Michael Pustilnik, Georgia Tech
Alex Kamenev
in collaboration with
Leonid Glazman, U of M Maxim Khodas, U of M
Michael Pustilnik, Georgia Tech
PRL 96, (2006);
arXiv:cond-mat/
arXiv:cond-mat/
RPMBT14, Jul., 2007

2. One-dimensional … M. Chang, et al 1996 Dekker et al 1997
Bockrath, et al 1997
Auslaender et al 2004
I. Bloch 2004

3. Spectral Function

4. d>1: Fermi Liquid Energy relaxation rate: The same for holes
Spectral density:
Energy relaxation rate:
interaction potential
The same for holes

5. d=1
Spectral density: ? ?
Energy relaxation rate:

6. Luttinger model Energy relaxation rate: Spectral density:
Dzaloshinskii, Larkin 1973
Spectral density:
Energy relaxation rate:

7. Luttinger model (cont)
Haldane, 1983

9. 1D with non-linear dispersion: Holes

10. 1D with non-linear dispersion: Particles
Energy relaxation rate:
interaction potential
Does not work for integrable models

11. Particles (cont)
Fermi head with the Luttinger tail

12. Spectral Edges
Shake up or
X-ray singularity
(cf. Mahan, Nozieres,…)

13. Structure Factor

14. Luttinger approximation
Linear
dispersion
Exact result within the
Luttinger approximation.
How does the dispersion curvature and
interactions affect the structure factor ?

15. Spectrum curvature + interactions
Fourier components of the interaction potential V

16. AFM spin chain N 200. For this case we have calculated
form factors
S. Nagler, et al 2005

17. 1D Bose Liquid Bose-Fermi mapping (1D)
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
Caux, Calabrese, 2006
Lieb-Liniger model, 1963
Constant-q scan

18. Structure factor: conclusions
Power law singularities at the
spectral edges (Lieb modes)
with q-dependent exponents.
Bosons
Fermions

19. 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.

20. Summary of bosonic exponents
Boson-Fermion mapping
Hydrodynamics ?

21. Numerics (preliminary)
Courtesy of J-S. Caux

22. Numerics (preliminary)
Courtesy of J-S. Caux