Density imbalanced mass asymmetric mixtures in one dimension

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