Giuseppe Morandi and me PhD In Bologna years 1994-96: very fruitful and formative period. Collaboration continued for some years after. What I have learned from him Follow the beauty (and don’t care too much about the experiments) Giuseppe had a great esthetic sense which guided him in deciding what was worth to be investigated and what was not. To some extent “beautiful” or elegant were synonyms of “interesting” or “important” to him. Timeo hominem unius libri Giuseppe was a man of many books (he was really fond of them) and a deeply learned scholar. At a time when people were still not so obsessed by citations (and thus did not demand them explicitly), he was extremely attentive to other people’s work. Do not oversell your work Giuseppe preferred understatements to emphatic claims.

Transition temperature of a superfluid Fermi gas throughout

the BCS-BEC crossover

Leonardo Pisani, Andrea Perali, Pierbiagio Pieri, Giancarlo C. Strinati

School of Science and Technology, Physics Division, University of Camerino, Italy

Physics & Geometry, Nov. 24, 2017, Bologna

The BCS-BEC crossover

Gas of fermions interacting via an attractive potential: Weak attraction: Cooper pairs form at low temperature according to BCS theory. Largely-overlapping pairs form and condense at the same temperature (Tc ). Strong attraction: the pair-size shrinks and pair-formation is no longer a cooperative phenomenon. Non-overlapping pairs (composite bosons) undergo Bose-Einstein condensation at low temperature. Pair-formation and condensation become unrelated.

How does the system evolves from one regime to the other one?

The standard BCS effective potential (constant attraction –V within a shell of width ωD about the Fermi energy EF) is not suitable for studying the BCS-BEC crossover (Fermi surface meaningless in the BEC limit). Two body attractive potential V(k-k’) is OK. Often approximated by a separable potential . For example NSR (or Yamaguchi) potential For (diluteness condition) the attractive potential is effectively short-ranged and can be parametrized completely in terms of the s-wave scattering length

1/ (kFa)-1 0 +1

BCS BEC

Diluteness condition normally satisfied in experiments with ultracold Fermi atoms for which the dimensionless effective coupling parameter can be tuned from BCS to BEC limit by using appropriate Fano-Feshbach resonances.

1/ (kFa)

The interaction potential

V (k,k ') = −V0 w(k)w(k ') w(k) =1/ 1+ (k / k0 )2

V (k,k ')

k0 >> kF ~ n1/3

a :1V0

=m4πa

−dk(2π )3|k|<k0

∫ mk2 [example for a sharp cutoff ] w(k) =Θ(k0 − k)

where for weak coupling.

BCS equation for Tc (for contact potential): (1) where with With weak-coupling assumption one gets For stronger coupling Tc becomes of order of (or larger than) TF : necessary to use number equation to adjust chemical potential. BCS number equation at Tc (and above) reduces to free Fermi gas relation (2) Eagles was first to notice (1969) that simultaneous solution of (1) and (2) yields in the BEC limit a “pairing” temperature T* rather than critical temperature for condensation.

where . Eq. (1) then yields:

BCS critical temperature throughout the BCS-BEC crossover

−m4πa

=Πpp (0)

Πpp (0) ≡dk(2π )3∫ tanh(ξk / 2Tc )

2ξk−mk2

⎛

⎝⎜

⎞

⎠⎟

Tc << µ

Πpp (0) =mkµ2π 2 ln

8eγ−2µπ Tc

⎛

⎝⎜

⎞

⎠⎟ kµ ≡ 2mµ

Tc =8eγ−2µπ

exp π2kµa

⎛

⎝⎜⎜

⎞

⎠⎟⎟=8eγ−2EF

πexp π

2kFa⎛

⎝⎜

⎞

⎠⎟

ξk ≡ εk −µ .

Interlude: a threefold role for ladder diagrams

+...+ +k+Q

−k

k+Q

−k

k’+Q

−k’

=Γ (Q)0

The sum of ladder diagrams has a threefold role: •  The BCS equation for Tc corresponds to the equation (Thouless

criterion): the two-particle Green’s function diverges at zero center-of-mass momentum and frequency.

•  In the BEC limit, becomes (except for numerical factors) the free propagator of molecules (composite bosons).

•  For a contact potential, the series of ladder diagrams replaces the bare interaction

in diagrammatic theory. This is because the “bare” potential strength V0 vanishes as the momentum cutoff . Every particle-particle rung in the ladder series is ultraviolet divergent and compensates a vanishing V0 , making finite for .

Γ0−1(Q = 0) = 0

Γ0 (Q)

Q ≡ (Q,Ων )

Γ0 (Q)

k0 →∞Γ0 k0 →∞

Nozières-Schmitt-Rink approach to the BCS-BEC crossover

Consider the simplest self-energy which can be constructed with ladder series:

In this way Tc approaches BE critical temperature in the BEC limit. True also when the Dyson’s equation is used to calculate G and then n (T-matrix self-energy approach).

k

Q−k

kΣ(k) = Γ (Q)0

Use number equation n=Tr G, where coupled with the standard BCS equation for critical temperature.

G =G0 +G0 ΣG0

G−1 =G0−1 −Σ

0

1

2

3

-2 -1 0 1 2

T c /

T BEC

(kF a)-1

TcBCS

TcT-matr

TcNSR

NSR theory in the weak-coupling limit

Common belief: The NSR curve for Tc interpolates between the BCS and BE critical temperatures. Not really true: in the BCS limit the NSR Tc approaches the BCS critical temperature divided by a factor e1/3. Reason? In weak coupling: The number equation in NSR approach then yields:

Tc =8eγµπe2

exp π2kµa

⎛

⎝⎜⎜

⎞

⎠⎟⎟

kµ ≡ 2mµ ≅ kF 1+23π

kFa⎛

⎝⎜

⎞

⎠⎟

µ = EF + 2πna /m

Tc =8eγEF

πe2exp π

2kFaF (1+ 2kFa / 3π )⎛

⎝⎜

⎞

⎠⎟

=8eγEF

πe2exp π

2kFaF(1− 2kFa / 3π )

⎛

⎝⎜

⎞

⎠⎟

=8eγEF

πe2exp π

2kFa⎛

⎝⎜

⎞

⎠⎟1e1/3

Σ(k)→ 2πnaF /m ≡ Σ0 .Γ0 →− 4πaF /m

0.01

0.1

1

10

-3 -2 -1 0 1 2

T c /

T BEC

(kF aF)-1

TcBCS

TcT-matr

TcNSR

Insert it in Tc equation:

(kFa)−1

Recovering BCS critical temperature in the weak-coupling limit

If the fermion propagators within the ladder series are dressed with first-order self-energy then and the BCS result for Tc is recovered. Diagrammatically, to lowest order in weak coupling, it corresponds to “dressing” the composite-boson propagator Γ with a “composite-boson self-energy” insertion ΣΒ .

Σ0 = 2πna /m µ→ µ ' = µ −Σ0 = EF

Γ−1(Q) = Γ0−1(Q)−ΣPop

B (Q) ΣPopB = 2×

Γ0

) ( where

yielding:

++= 0

0

0

Γ0 Γ0 ΓΓ ΓΓ

Γ

Γ−1(Q = 0) = 0

“Dressed” Thouless criterion

⇔m4πaF

+Πpp (0)+ΣPopB (0) = 0

m4πa

+mkµ2π 2 ln

8eγ−2µπ Tc

⎛

⎝⎜

⎞

⎠⎟+

mkµ6π 2 = 0

ΣPopB (0) = −2 G0 (k)

2

k,Q∑ G0 (−k)G0 (Q− k)Γ0 (Q)

≅ −4πam

n0m2

4π 2kµln 8e

γ−2µπ T

&

'(

)

*+

πkµa

+ ln 8eγ−2µπ Tc

⎛

⎝⎜

⎞

⎠⎟+13= 0

“Counter-term” 1/3 compensates -1/3 from expansion of kµ . BCS result for Tc is recovered.

For , ln(µ/Τ) originating from compensates the small parameter a !

Γ0k

Q-k

- k

k

Eq. for Tc becomes

T ≈ Tc G0 (k)k∑ 2G0 (−k)

≅ −4πam

kµ3

3π 2m2

4π 2kµ−

π2kµa

⎛

⎝⎜⎜

⎞

⎠⎟⎟ =

mkµ6π 2

The GMB correction in the weak-coupling limit (I)

But is BCS expression for Tc really correct in the weak-coupling limit? No! Gorkov and Melik-Barkhudarov (1961): For a low density attractive Fermi gas, the BCS expression is reduced by a factor (4e)1/3 ~ 2.2 in the weak-coupling limit. Reason? Same mechanism just discussed: compensation of small parameter a with large factor ln(µ/Τ) originating from particle-particle bubble . Dress Γ with following “self-energy” insertion: which contains two pp bubbles and two Γ0 ~ a in weak coupling.

k’’−k’

k k’’ k’

−k’k’’−k’−k−k

k’’−k

G0 (k)k∑ G0 (−k)

ΣGMBB (0) =

Tc =8eγEF

πe2exp π

2kFa⎛

⎝⎜

⎞

⎠⎟

1(4e)1/3

The GMB correction in the weak-coupling limit (II)

ΣGMBΓ (0) = G0 (k)

k,k ',k ''∑ G0 (−k)G0 (k ')G0 (−k ')G0 (k '')G0 (k ''− k − k ')Γ0 (k ''− k)Γ0 (k ''− k ')

~ −4πam

G0 (k)G0 (−k)k∑

⎛

⎝⎜

⎞

⎠⎟ −

4πam

G0 (k ')G0 (−k ')k '∑

⎛

⎝⎜

⎞

⎠⎟ G0 (k '')G0 (k ''− k − k ')k ''∑

k ≅ (kF, 0)

=1+O(kFa)

and select k ' ≅ (k 'F, 0)

=1+O(kFa)

and select

=12

d(k̂F ⋅∫ k̂ 'F )χph (kF +k 'F, 0)

= −mkF2π 2 ln(4e)

1/3

m4πa

+mkF2π 2 ln

8eγ−2EFπ Tc

⎛

⎝⎜

⎞

⎠⎟−

mkF2π 2 ln(4e)

1/3 = 0Equation for Tc :

Note: for standard BCS potential, the GMB reduction is instead exp(ωD/EF) ~ 1 since ωD << EF . GMB correction irrelevant in this case.

GMB (and Popov) correction throughout the BCS-BEC crossover

Σ(k) =

Q−k

k k

(Q)Γ

= + +Γ0 Γ

Γ

Γ Γ0Γ0 ΓΓ2×

Γ−1(Q) = Γ0−1(Q)−ΣPop

B (0)−ΣGMBB (0)

Γ−1(Q = 0) = 0 ⇔m4πaF

+Πpp (0)+ΣPopB (0)+ΣGMB

B (0) = 0

n = Tr G= 2T dk(2π )3

G(k,ωn )eiωn 0

+

∫n∑

Coupled equations:

G−1(k) =G0−1(k)−Σ(k)

ΣGMBB (0) = G0 (k)

k,k ',k ''∑ G0 (−k)G0 (k ')G0 (−k ')G0 (k '')G0 (k ''− k − k ')Γ(k ''− k)Γ(k ''− k ')

ΣPopB (0) = −2 G0 (k)

2

k,Q∑ G0 (−k)G0 (Q− k)Γ(Q)

Retain full momentum and frequency dependence of Γ and avoid all weak-coupling approximations when calculating and ΣPop

B ΣGMBB :

Calculation quite non-trivial!

-0.03-0.025

-0.02-0.015

-0.01-0.005

0 0.005

0.01 0.015

-5 -4 -3 -2 -1 0 1 2 3

ΣB/(

2mk F

)

(kFaF)-1

POPOVGMB

Numerical results: GMB and Popov self-energies at Tc

Inclusion of both self-energies is important from weak coupling to past unitarity. To some extent, they compensate each other.

A. Bulgac et al., PRA 78, 023625 (2008)

E. Burovski et al., PRL 101, 090402 (2008)

Numerical results for the critical temperature

The resulting curve for Tc converges to the GMB result for weak coupling and to the BE critical temperature for strong coupling. In the intermediate coupling region it agrees very well with diag-QMC and lattice QMC.

0

0.05

0.1

0.15

0.2

0.25

0.3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

T c/T

F

(kFa)-1

GMB+POPOVQMC Bulgac

QMC BurovskiT-matrix

TcBCS-GMB

Summary and conclusions

•  GMB found that for a dilute attractive Fermi gas the BCS critical temperature is

reduced by a numerical factor.

•  Their work was formulated in the weak coupling regime of the attraction. •  Standard diagrammatic approximations for the BCS-BEC crossover (NSR, T-

matrix) fail to recover GMB result in weak coupling.

•  We have reformulated GMB’s work in a way which is amenable to extension to the whole BCS-BEC crossover.

•  Consistency in the weak-coupling limit required us to also include “Popov” two-particle self-energy.

•  The resulting curve for Tc agrees very well with available QMC data.

Thank you!