Density imbalanced mass asymmetric mixtures in one dimension
Evgeni Burovski
LPTMS, Orsay
Giuliano Orso Thierry Jolicoeur
FERMIX-09, Trento
Effective low-energy theory,
a.k.a. ``bosonization’’
Two-component mixtures: use pseudo-spin notation σ=,
(Haldane, 81)
mm
nn
Effective low-energy theory, cont’d
Non-interacting fermions:
Effect of interactions:
higher harmonics
The effect of higher harmonics
( p and q are integers )
p =q = 1 spin gap (attractive interactions)
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
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
A microscopic example:
-species: free fermions:
-species: dipolar bosons, a Luttinger liquid with
( Citro et al., 2007 )as
Take a majority of light non-interacting fermions and a minority of heavy dipolar bosons:
I. e.: (an infinitesimal attraction) opens the gap.
Switch on the coupling:
The Hubbard model
• spin-independent hopping: Bethe-Ansatz solvable ( Orso, 2007; Hu et al., 2007) two phases: fully paired (“BCS”) and partially polarized (“FFLO”)
“BCS”
“FFLO”
( cf. B. Wang et al., 2009 )
1 component gas
The asymmetric Hubbard: few-body
unequal hoppings: three-body bound states exist in vacuum (e.g., Mattis, 1986)
0.0 0.5 1.0
-0.4
-0.2
0.0
U = -10
U = -5
U = -1
pair energy
What about many-body physics?
The asymmetric Hubbard model, correlations
0.0 0.1 0.2 0.3 0.4
10-6
10-5
10-4
10-3
10-2
x/L
x/L
|(x)|
unequal hoppings: the model is no longer integrable, hence use DMRG
superconducting correlations
‘commensurate’ densities
Majority of the heavy species: YESMajority of the light species: NO
The asymmetric Hubbard model, correlations
0.0 0.1 0.2 0.3 0.4
10-6
10-5
10-4
10-3
10-2
x/L
x/L
|(x)|
superconducting correlations
‘incommensurate’ densities
Majority of the heavy species: YESMajority of the light species: NO
unequal hoppings: the model is no longer integrable, hence use DMRG
‘commensurate’ densities
The asymmetric Hubbard model, cont’d
0 1 2 3
|k|
k
0.0 0.1 0.2 0.3 0.4
10-6
10-5
10-4
10-3
10-2
x/Lx/L
|(x)|
long-range behavior is the same for • equal masses• unequal masses, incommensurate densities
Broadening of the momentum distribution is insensitive to the commensurability
The asym. Hubbard model, phase diagram
-2.0 -1.8 -1.6 -1.4
-2.5
-2.4incommens.
2+31+1
1+2
1+3
0+1
vacuum
h
|U|/2
Multiple commensurate phases at low density
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:0904.0569
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