Finite-size and Particle-number Effects in an
Ultracold Fermi Gas at Unitarity
Michael M. Scherer

Institute for Theoretical Solid State Physics, RWTH Aachen

presented at YITP, Kyoto, August 26th 2011

J. Braun, S. Diehl, M. M. Scherer, [arXiv:1108.xxxx]
1Outline

1. BCS-BEC Crossover: Basics and Phase diagram
2. Overview: FRG Studies of the BCS-BEC Crossover
3. Finite System Size Study
4. Conclusions & Outlook21. BCS-BEC Crossover: Basics and Phase diagram

3

BCS-BEC Crossover: Basics• Ultracold gases of fermionic atoms (6Li, 40K) → 2 accessible hyperfine spin states• Spin-balanced case (for strong spin-imbalance, cf. talk by R. Schmidt)
inverse s-wave scattering length (a kF)-1 in units of the Fermi momentum
weak attractive i.a.

0bound state disappears

bound state,
weak i.a. between molecules

Unitary regime

T < Tc: BEC of molecules

Cooper pairs

T < Tc: BCS-Superfluid• (a kF)-1 < −1: weakly attractive, Cooper pairing → below Tc : BCS superfluidity• (a kF)-1 > 1: two-body bound state, formation of molecules → below Tc : interacting BEC• |(a kF)-1| < 1: strongly correlated regime, Unitarity limit at (a kF)-1 → 04

BCS-BEC Crossover: Feshbach Resonance
6LiScattering length �1000 a0 �• Feshbach resonance: Vary effective i.a. strength or (a kF)-1 by external magnetic field

10(a kF)-150

BECBCS

�5
�10600 700 800 900 1000 1100 1200
Magnetic field �G�• Feshbach resonance allows to tune i.a. strength arbitrarily in an experiment• Challenge and testing ground for non-perturbative approaches to strongly interacting QFTs5BCS-BEC Crossover: Universality
inverse s-wave scattering length (a kF)-1 in units of the Fermi momentum
weak attractive i.a.

0bound state,
weak i.a. between molecules

Unitary regime

T < Tc: BEC of molecules

Cooper pairs

T < Tc: BCS-Superfluid• Limit of broad Feshbach resonances (e.g. experiments with 6Li and 40K)
• TD quantities independent of microscopic details → can be expressed by two parameters:(a kF)-1 & T/TF
• Units set by density n = (kF)3/(3π2)
6BCS-BEC Crossover: Model• Microscopic model:†S[ψ , ψ] =

� 1/T0dτ�

��
λ̄
ψ†
23†2
�d x ψ (∂τ − ∇ − µ)ψ +(ψ ψ)2• two-component Grassmann field: ψ = (ψ1 , ψ2 )• chemical potential: µ• natural units: � = kB = 2m = 1• bare four-fermion coupling: λ̄ψ = λ̄ψ (B)• Observable thermodynamics from grand canonical partition function:

ZG =�

†

Dψ Dψe−S[ψ † ,ψ]... to be evaluated non-perturbatively!
7

BCS-BEC Crossover: Phase Diagram• Critical temperature for the phase transition to superfluidity
Tc/TF
0.4
Interacting BEC

normal fluid

1[Baym et al. 1999]
2[Arnold & Moore 2001]

0.3

3[Blaizot et al. 2004]

BCS+Gorkov
QMC

0.2

4[Burovski et al. 2006]
5[Akkineni et al. 2007]
6[Bulgac et al. 2008]
7[Wingate et al. 2010]superfluid

0.1

0.0

�2

�101

2

3

4(akF)-18

2. FRG Studies of the BCS-BEC Crossoverwith Stefan Flörchinger, Sebastian Diehl, Holger Gies, Jan Pawlowski
and Christof Wetterich

9Functional RG and Theory Space• Continuum formulation in terms of effective action
Γ[χeq ] = ΦG /T,• Wetterich equation:whereΦG = −T ln Z����−11(2)∂k Γk [χ] = STr Γk + Rk
∂ k Rk2Theory SpaceΓ0 (T = 0, n = 0) → scattering physics
ΓΛ = S[MIT experiment]

Γ0 (T "= 0, n "= 0) → thermodynamics
10

Hubbard-Stratonovich Field and Yukawa Coupling†S[ψ , ψ] =� 1/T0dτ�

� †�
λ̄
ψ
2
†2
� − µ)ψ +d x ψ (∂τ − ∇(ψ ψ)
23• Introduce a complex scalar by Hubbard-Stratonovich transformation:φ̄λ̄ψh̄h̄Complex scalar: Molecule field, Cooper pairs,...

• s-wave scattering length:h̄2a=−8πµM (B − B0 )11Truncation and Thermodynamic PhasesΓk [Φ] =�τ,�x��2�∇†2∗�ψ (∂τ − ∇ − µ)ψ + φ (Zφ ∂τ −)φ + U (ρ, µ) − h (φ∗ ψ1 ψ2 + φψ2∗ ψ1∗ )2• Effective potential: Expansion around the scale-dependent location of its minimum
1Uk (ρ, µ) = m (ρ − ρ0 ) + λ(ρ − ρ0 )22+ U (ρ0 , µ0 ) − n(µ − µ0 ) + α(µ − µ0 )(ρ − ρ0 )2

• Classification of the thermodynamic phases of the system:
Symmetric regime :ρ0 = 0,m2 > 0

Symmetry broken regime :ρ0 > 0,m2 = 0

Phase transition :ρ0 = 0,m2 = 0

12

BCS-BEC Crossover: Phase Diagram• Critical temperature for the phase transition to superfluidity:
Tc/TF
0.4

Interacting BEC
normal fluid

1[Baym et al. 1999]
2[Arnold & Moore 2001]
3[Blaizot et al. 2004]

0.3

BCS+Gorkov
QMC
0.2

4[Burovski et al. 2006]
5[Akkineni et al. 2007]
6[Bulgac et al. 2008]
7[Wingate et al. 2010]superfluid

0.1

FRG approach
8[Floerchinger, MS, Wetterich

2010]

0.0

�2�10124 (akF)-1• Scattering physics in BEC limit:• At Unitarity and T=0, n≠0:
Carlson et al. (2003) (QMC)
Perali et al. (2004) (t-matrix approach)
Floerchinger, MS, Wetterich (2010) (FRG)

3µ/EF
0.43
0.460.51

∆/EF
0.54
0.53
0.46 *

aM/a = 0.59

exact result: aM/a = 0.60 [Petrov et al. 2004]
9* more RG results are available:

10[Krippa 2008]
11[Bartosch et al. 2009]13

3. Finite System Size Studywith Jens Braun and Sebastian Diehl

14

Motivation• Analysis of data from lattice simulations (performed in a finite volume)• For unitary Fermi gas: Studies by MC community:[Wingate et al. 2009]
1

2[Kaplan et al. 2010]
3[Forbes et al. 2011]• Lattice studies:• Limited range of system sizes• Numerically expensive• Cannot investigate transition between finite system and continuum limit• FRG can! (recall talk by B. Klein)

Unitary regime
15

Setup for Finite-size Study• Finite cubic volume V with spatial extent L• Boundary conditions of fermions in spatial directions are periodic, cf. lattice [Bulgac et al. 2006][Kaplan et al. 2003]• Finite external pairing source J which couples to order-parameter field[Wingate et al. 2009]• Grand canonical ensemble: Average particle number fixed by chemical potential µ12
3

Universal Quantities• Investigate unitary regime (a → ∞) at T=0 and n≠0• In continuum limit we have the universal quantities:
Bertsch parameter:

Fermion gap:

µ
ξ=EF
∆EF

• In finite volume: Bertsch parameter and fermion gap will depend on L and J• Study deviation from TD limit as a function of L and J16

Truncation with External Source• Truncation:Γk [Φ] =�τ,�x�

∆2∗�ψ (∂τ − ∇ − µ)ψ + φ (Zφ ∂τ − )φ + U (ρ, µ) − h (φ∗ ψ1 ψ2 + φψ2∗ ψ1∗ )2
�
1+ √ J(φ + φ∗ )2†

• Source J allows to control symmetry breaking in a finite volume, cf. [Kaplan et al. 2003]
1

2[Wingate et al. 2009]

Finite Volume V=L3(φ/ψ)(τ,�x)→(ω, �q )Gotofrequency/momentumspacebyFT:•• In a finite volume, we obtain summation over discrete momenta:
ω(φ)= 2mπT,ω(ψ)= (2m + 1)πT,2π
�q = �n ,L

�n = (n1 , n2 , n3 ),ni , m ∈ Z• Flow equations can be evaluated for different system sizes L• In the limit (L → ∞) we recover the well known flow equations for infinite volume17

Truncation with External Source• Truncation:Γk [Φ] =�τ,�x�∆2∗�ψ (∂τ − ∇ − µ)ψ + φ (Zφ ∂τ − )φ + U (ρ, µ) − h (φ∗ ψ1 ψ2 + φψ2∗ ψ1∗ )2
�
1+ √ J(φ + φ∗ )2†

• Study flow of potential at T=0 to extract Bertsch parameter and fermion gap• In a first step we take into account the fermion fluctuations only∂t U (ρ̄, J, L, µ) = −2k 5 (BF> + BF< )sF

sF = �k2
k 4 + h̄2ϕ ρ̄� ��
1 � �
22 2
22 22
θ (kL) − (2π) �n + µL θ (2π) �n − µL ,3(kL)�n� � 2�1 � �<22 222 2
θ (kL) + (2π) �n − µL θ µL − (2π) �n .
BF =3(kL)
BF> =

�n

• This corresponds to an approximation on a meanfield level• Effects from boson fluctuations will be included later in the talk18

Investigation of the Source J• Truncation:Γk [Φ] =

�

τ,�x�

∆
2
∗
�ψ (∂τ − ∇ − µ)ψ + φ (Zφ ∂τ − )φ + U (ρ, µ) − h (φ∗ ψ1 ψ2 + φψ2∗ ψ1∗ )2
�
1+ √ J(φ + φ∗ )2†

• Study behaviour of TD observables with source J for infinite L• Generally, we can expand Bertsch parameter for small but finite J:
0-0.02-0.04-0.06
bjξJ − ξ0δξ =ξ0��252
3= − (δnJ ) + (δnJ ) + O (δnJ )3
9nJ − nJ=0
δnJ =nJ=0

-0.08
-0.1

• In lattice simulations only data for finite J isavailable → J=0 has to be extraced from fit(linear, higher order,...)• Linear fit justified for δnJ � 0.05

-0.12
-0.14
00.05

0.1
bnJ

0.15

0.2

19

Investigation of the Source J• Truncation:Γk [Φ] =�τ,�x�∆2∗�ψ (∂τ − ∇ − µ)ψ + φ (Zφ ∂τ − )φ + U (ρ, µ) − h (φ∗ ψ1 ψ2 + φψ2∗ ψ1∗ )2
�
1+ √ J(φ + φ∗ )2†

• Study behaviour of TD observables with source J for infinite L• Similarly, we can expand fermion gap for small but finite J:
0.2

ˆ (1) ≈ 0.749∆

0.15• fit from MF data
• In lattice simulations only data for finite J isavailable → J=0 has to be extraced from fit(linear, higher order,...)• Linear fit justified for δnJ � 0.05

b6

∆J −∆0ˆ (1) (δnJ ) + O((δnJ )2 )
δ∆ =
=∆
∆00.1
0.05
000.05

0.1

0.15

0.2

0.25

bnJ
20

Finite Volume V=L3
• The (average) particle number N is given by N = n L3
• Initial condition for RG flow of density is given by free Fermi gas nfree• nfree is determined by our choice for the chemical potential.
snfree = 3L�

�n,ni ∈Z32
�(µL →∞) µ 222 2
θ µL −(2π) �n−→
3π 2�Free Fermi Gas200

N150
100
50
001234522µL /(2/)

67821

Bertsch parameter at MF Levelj(j,N)/j'• Study Bertsch parameter as function of N (or µ) for various values of J at fixed L (fixes scale)• For large values of µ at fixed L and J: dimensionless Jµ-7/4 → 0, so ξ → ξ∞ for large L• For small N we observe shell effects (discontinuities)

1.05
10.95
0.9
0.85
0.8

decreasing source J

050

100
N150200� 2 �−1
1.05−N > 50 : δξ ∼ cN N
∼ cµ µL01• Large-N behaviour follows from definition of Bertsch parameterc only depend on JL7/2• Constants 0.95
0.9 our choice of L) Bertsch parameter is already close to cont. limit (>98%)• For N>200 (andbehaviour depends on J (very clearly for N<200)• Convergence
0.85

2j(j,!L0 )/j'2
322

Bertsch parameter at fixed density• Behavior of Bertsch parameter as a function of dimensionless quantity kFL for fixed density:
MFBMF

10

15

1.1

• Small kFL: shell effects(washed out for larger source)
approached(as it should for fixed density)• Finite source J: Does not

approach J →0 limit(due to constant fraction of nJ)

j(j,kFL)/j'• Large kFL: TD limit is

1.05

10.95

0.9
5202530kFL• BMF: Include boson loops in flow of effective potential, running wave-function renormalization

23Fermion Gap beyond MF• Fermion gap is more sensitive to the inclusion of order- parameter fluctuations

1015

1.15
1.1
6(j,kFL)/6'• For large source J: Gap almost
independent from kFL(wavelength of Goldstone smaller
than spatial extent L)

BMF

1.2

• Large kFL: TD limit is approached• Different values for different
densities and sources JMF

1.05
10.95
0.9→0.85

Order parameter fluctuations are
not affected by boundary

0.8
5202530kFL• Small source J: large deviations
J2mG = √2ρ0,m2R = m2G + λϕ ρ024

Conclusions• FRG connects BCS-/BEC-limits continuously with unitary regime and gives results with areasonable accuracy throughout the whole crossover• Using the FRG we have access to the shape of the volume and the particle number
dependence of observables over a wide range of system sizes• Volume-effects depend strongly on observable• Improves understanding of convergence of finite volume systems, useful for MC
simulations& Outlook• Finite size study of Tc(J,L)

25