1. Density imbalanced mass asymmetric mixtures in one dimension
Evgeni Burovski
Giuliano Orso
Thierry Jolicoeur
LPTMS, Orsay
FERMIX-09, Trento

2. Effective low-energy theory,
a.k.a. ``bosonization’’
Two-component mixtures: use pseudo-spin notation σ=, 
(Haldane, 81)

3. Effective low-energy theory, cont’d
Non-interacting fermions:
Effect of interactions:
higher harmonics

4. The effect of higher harmonics
( p and q are integers )
p =q = 1  spin gap (attractive interactions)

5. Is this cos(…) operator relevant?
Renormalization group analysis ( Penc and Sólyom, 1990 ; Mathey, 2007) :
cos(…) is either relevant or irrelevant in the RG sence.
cos(…) is irrelevant  1D FFLO phase : gapless,
all correlations are algebraic,
cos(…) is relevant  ‘massive’ phase
massive
massless
A sufficient condition:
Notice the strong asymmetry between  and 

6. Quasi long range order 
In 1D no true long-range order is possible  algebraic correlations at most: 
i.e. the slowest decay  the dominant instability.
Equal densities ( p = q = 1 ), attractive interactions :
Unequal densities ( e.g. p = 2, q = 1 ) :
CDW/ SDW-z correlations are algebraic
SS correlations are destroyed (i.e. decay exponentially)
“trimer’’ ordering

7. A microscopic example:
-species: free fermions:
-species: dipolar bosons, a Luttinger liquid with as
( Citro et al., 2007 )
Take a majority of light non-interacting fermions and
a minority of heavy dipolar bosons:
Switch on the coupling:
I. e.: (an infinitesimal attraction) opens the gap.

8. The Hubbard model
spin-independent hopping: Bethe-Ansatz solvable ( Orso, 2007; Hu et al., 2007)
two phases: fully paired (“BCS”) and partially polarized (“FFLO”)
“FFLO”
“BCS”
( cf. B. Wang et al., 2009 )
1 component gas

9. The asymmetric Hubbard: few-body
unequal hoppings: three-body bound states exist in vacuum (e.g., Mattis, 1986)
pair energy
What about many-body physics?

10. The asymmetric Hubbard model, correlations
unequal hoppings: the model is no longer integrable, hence use DMRG
superconducting correlations
‘commensurate’ densities
Majority of the heavy species: YES
Majority of the light species: NO

11. The asymmetric Hubbard model, correlations
unequal hoppings: the model is no longer integrable, hence use DMRG
superconducting correlations
‘commensurate’ densities
‘incommensurate’ densities
Majority of the heavy species: YES
Majority of the light species: NO

12. The asymmetric Hubbard model, cont’d
Broadening of the momentum distribution
is insensitive to the commensurability
long-range behavior is the same for
equal masses
unequal masses, incommensurate densities

13. The asym. Hubbard model, phase diagram
Multiple commensurate phases at low density

14. Conclusions and outlook
Multiple partially gapped phases possible in density- and mass-imbalanced
mixtures.
(Quasi-)long-range ordering of several-particle composites
D > 1 ?
Li-K mixtures ?
Mo’ info: EB, GO, and TJ, arXiv: