Non-standard pairing in asymmetric trappedFermi gases

Michael Urban

IPN Orsay

Outline

1. Standard (BCS) pairing

2. Breached pairing (Sarma phase)

3. LOFF (FFLO) phase

4. BdG equations and results

5. Non-standard pairing in a rotating trap

Hamiltonian of a dilute trapped Fermi gas

! Consider a Fermi gas with two “spin” states σ =↑, ↓(in atomic gases these are usually two hyperfine states)

! Mean distance between the atoms ∼ 10−7 mrange of the interatomic potential ∼ 10−10 m→ potential may be replaced by a contact pseudopotential

! Hamiltonian:

H =

∫d3r

[∑

σ

ψ†σ

(−∇2

2m+ Vtrap(r)

)ψσ + gψ†

↑ψ†↓ψ↓ψ↑

]

! Vtrap(r) = (approximately) harmonic trap potential

! Coupling constant related to scattering length by g =4πa

m! Here: consider attractive interaction (a < 0)

BCS pairing (1)

! Consider a uniform gas (Vtrap = 0) with equal populations(µ↑ = µ↓), neglect Hartree potential VHartree σ = gn−σ

! BCS gap equation:

∆ = −g

∫ Λ d3p

(2π)3∆

2E(1 − 2f (E ))

where ξ =p2

2m− µ, E =

√ξ2 +∆2, f (E ) =

1

eE/T + 1! Divergence for Λ→ ∞! Solution: express g in terms of the scattering length

∆ = −4πa

m

∫ Λ d3p

(2π)3

(∆

2E(1 − 2f (E )) − ∆

2ξ

)

→ finite result for Λ→ ∞

BCS pairing (2)

! Occupation numbers:

n(p) =(1 − ξ

E

)(1 − f (E )) +

(1 +

ξ

E

)f (E )

= 1 − ξ

2E(1 − 2f (E ))

Sarma phase (breached pairing) (1)

! Consider now µ↑ > µ↓

! Define µ =µ↑ + µ↓

2, δµ = µ↑ − µ↓ > 0

! Gap equation:

∆ = −4πa

m

∫d3p

(2π)3

(∆

2E(1 − f (E+) − f (E−)) − ∆

2ξ

)

where ξ =p2

2m− µ, E =

√ξ2 +∆2, E± = E ± δµ

2

! If ∆ <δµ

2, there are p1, p2 such that E− < 0 for p1 < p < p2

→ gapless superfluid

! At T = 0, if ∆ >δµ

2, the solution is the standard BCS

solution (n↑ = n↓) with µ = µ

Sarma phase (breached pairing) (2)

! Occupation numbers:

n↑(p) =(1 − ξ

E

)(1 − f (E+)) +

(1 +

ξ

E

)f (E−)

n↓(p) =(1 − ξ

E

)(1 − f (E−)) +

(1 +

ξ

E

)f (E+)

Phase separation [Bedaque et al., PRL 91, 247002 (2003)]

! Thermodynamic potential Ω at T = 0 for different δµ (µ fixed)

! Solutions of gap equation= extrema of Ω

! Trivial solution ∆ = 0

! Minimum at ∆BCS

(for δµ not too large)

! Maximum at ∆Sarma < ∆BCS

! First-order phase transition between from BCS to normal phase(phase separation)

! Sarma phase is always a maximum → unstable

! However, the Sarma phase can exist at finite temperature

Sarma phase in a trap (1) [Gubbels et al., PRL 97, 210402 (2006)]

! Local-density approximation (LDA): µσ(r) = µσ − Vtrap(r)

! Consider unitary limit |a| → ∞ (Feshbach resonance)Assume that mean-field theory is qualitatively correct

! Depending on polarization and temperature, there are twopossibilities called “phase separation” and “Sarma phase”:

! The Sarma-normal transition is second order→ no discontinuity in ∆(r) and density profiles

! The BCS-normal transition is first order→ discontinuity; possibly surface tension

Sarma phase in a trap (2)

! Phase diagram:

0.5

P

Sarma Phase

NormalPhase

a b c

T / T

c

0 10

0.5

1

Phase Separation

0.25

0.75

0.75

0.25

Experimental evidence [Partridge et al., PRL 97, 190407 (2006)]

! Density profiles of imbalanced 6Li gas at Rice University:

(d)(c)

(b)(a)

P=0.45T=0.2 TF

Sarma phase

T=0.05 T!n

"n

"!n!n

!n

"n

"!n!n

P=0.5phase separation(surface tension)

F

Fulde-Ferrel-Larkin-Ovchinnikov (FFLO/LOFF) phase

! In the Sarma phase the atoms around pF are unpaired

! Improve this situation by pairing to total momentum q (= 0→ oscillating order parameter

! With the simplest possible ansatz

∆(r) = |∆|e iq·r

the gap equation and occupation numbers can be writtenexactly as in the breached-pairing (Sarma) phase but with

ξ =p2

2m+

q2

8m− µ, E± = E ±

(δµ − p · q

2m

)

! Determine q by minimizing the energy(more precisely, by maximizing the pressure for fixed µ↑, µ↓)

! More complicated ansatze (superposition of several planewaves, e.g. ∆ ∝ cos q · r) result in lower energy→ necessary to solve Bogoliubov-de-Gennes (BdG) equations

FFLO phase in the unitary limit[Bulgac and Forbes, PRL 101, 215301 (2008)]

! Mean-field theory not valid in the unitary limit (|a| → ∞)

! Use density-functional theory: “ASLDA” energy functionalfitted to quantum Monte-Carlo results

! The equations to be solved are similar to the BdG equations

! Spatial dependence of order parameter and densities fordifferent polarizations:

—— ∆(r)

······ n↑(r)—— n↓(r)

Bogoliubov-de Gennes (BdG) equations (1)

! In the case of a trapped system, there are two possibilities:! Local-density approximation (LDA): µσ(r) = µσ − Vtrap(r)! BdG equations (= HFB in nuclear physics)

! BdG equations can describe BCS, Sarma, and LOFF phases

! For ∆(r) = ∆∗(r) the BdG equations read

(−∇2

2m+ V↑(r) − µ↑

)u↑η(r) +∆(r)v↓η = E↑ηu↑η(r)

∆(r)u↑η −(−∇2

2m+ V↓(r) − µ↓

)v↓η(r) = E↑ηv↓η(r)

here we keep the Hartree field: Vσ(r) = Vtrap(r) + gn−σ(r)! Diagonalize BdG equations in a harmonic oscillator basis

! BCS: diagonal elements of ∆ij dominant! LOFF: important non-diagonal elements of ∆ij

Bogoliubov-de Gennes (BdG) equations (2)

! In practice, the basis has to be truncated at some energy Λ

! Regularization yields an effective coupling constant gΛ(r) suchthat ∆ is independent of Λ

∆(r) = gΛ(r)∑

η

v↓η(r)u↑η(r)f (E↑η)

(sum over positive and negative eigenvalues E↑η)

! Expressions for the densities:

n↑(r) =∑

η

|u↑η(r)|2f (E↑η), n↓(r) =∑

η

|v↓η(r)|2(1−f (E↑η))

! For simplicity, consider spherical symmetry:! Isotropic trap: Vtrap(r) = 1

2mω2r2

! In the case of LOFF phase: only oscillations in radial direction(pairing between different radial quantum numbers n)

BdG results (1) [Castorina et al., PRA 72, 025601 (2005)]

! Trap units: [E ] = !ω, [l ] =√

!/mω, etc.! Here: µ = 32, g = −1 → N = 17000, kF a = −0.7! Gap in the trap center as function of δµ = µ↑ − µ↓

0 2 4 6 8 10 12 14 16#µ

0

1

2

3

4

5

6

7

$ (

0)

P=0.3

µ = 32N = 17000

P=0.15

BdG(BCS)

LDA

(LOFF)

BdG results (2)

! r dependence of the gap and density profiles for P = 0.15(P = polarization = (N↑ − N↓)/(N↑ + N↓))

0 2 4 6 8 10r

0

4

8

12

16

%

−1

1

3

5

7

9

$

4 6 8 10−0.5

0

0.5

n

n"

!

N=17000P=0.15

BdG

LDA(BCS)

(LOFF)

BdG results (3)

! Temperature dependence of the oscillation (P = 0.15)

4 5 6 7 8 9 10r

−0.3

−0.1

0.1

0.3$

T=0T=0.5T=1T=2

1

2

0.5T=0

N=17000P=0.15

BdG results (4)

! r dependence of the gap for P = 0.3

0 1 2 3 4 5 6 7 8 9 10r

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5$

N=17000P=0.3

LDABdG

(LOFF)

Angular FFLO phase (1) [Yanase, PRB 80, 220510(R) (2009)]

! So far: radial oscillations of ∆(r)Pairing between different radial quantum numbers n

! What about pairing between different angular momenta l ?Angular oscillations of ∆(r)Spontaneous breaking of rotational symmetry

! Consider quasi-2D trap (ωz * ωr )

! Beyond-mean-field effects treated within “Real-spaceself-consistent T-matrix approximation (RSTA)”

! Angular FFLO phase requires repulsive potential in the trapcenter (toroidal trap)

Angular FFLO phase (2)

! Order parameter at different polarizations:

(a) P=0 (b) P=0.21 (c) P=0.39

-0.4 0 0.4 0.8 1.2 1.6 -0.4 0 0.4 0.8 1.2 1.6 -0.4 0 0.4 0.8 1.2 1.6

(d) P=0.44 (e) P=0.49 (f) P=0.69

-1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5

Non-standard pairing in a rotating trap (1)

Bausmerth, Recati, and Stringari, PRL 100, 070401 (2008):

! Consider a balanced gas (kF a → ∞, T = 0) in a rotating trap

! Increase angular velocity Ω adiabatically → no vortices

! Simple energetic arguments (pairing vs.centrifugal energy) → phase separation:

! Non-rotating superfluid core! Rotating normal fluid

! Discontinuity of the density profile!

Urban and Schuck, PRA 78, 011601(R) (2008):

! BCS theory + LDA: gap equation analogous to that in theSarma phase but with E± = E ± (Ω × r) · p

! Rotating intermediate phase where ∆ (= 0but some pairs (E± < 0) are broken

! Densities, ∆, and current are continuous

Non-standard pairing in a rotating trap (2)

! Spherical trap (ωz = ωr ), plotresults as functions of r⊥ in theplane z = 0

! r⊥ ≤ r⊥1: fully paired! r⊥1 < r⊥ < r⊥2: partially paired! r⊥2 < r⊥: unpaired

! Partially paired phase energeticallyfavored

-550

-450

-350

-250

9.8 10 10.2 10.4 10.6

&'µ%

(h_ (

l-3 ho)

r) (lho)

1/(kF a) = 0N = 400 000* = 0.45 (

full calculationpaired

unpaired

0 40 80

120 160

% (l

-3 ho)

1/(kFa) = 0N = 400 000

*/( = 0*/( = 0.45

0 20 40 60 80

100

$ (

h_ (

)

0

40

80

120

0 4 8 12 16

|j| ((

l-2 ho)

r) (lho)

r)1 r)2

Summary and conclusions

! Non-standard pairing in imbalanced trapped Fermi gases:Sarma and LOFF phases

! Sarma phase (breached pairing): can exist at higher T(at low T : phase separation between BCS and normal state)

! Experimental evidence for Sarma phase: discontinuity indensity profiles disappears at some temperature

! LOFF phase: predicted by several theories,could exist between BCS and normal state at low T

! So far, no experimental evidence for LOFF phase

! BdG equations for imbalanced trapped Fermi gas giveoscillating order parameter near the BCS-normal phaseboundary – LOFF phase or surface effect?

! Phenomenon similar to Sarma phase in rotating trap