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Dimensional crossover in ultracold Fermi gases from Functional Renormalisation Bruno Faigle-Cedzich Cold Quantum Coffee Heidelberg University 8 th May 2018 Table of contents 1. Physics of ultracold atoms 2. BCS-BEC physics from Functional

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SLIDE 1

Dimensional crossover in ultracold Fermi gases

from Functional Renormalisation

Bruno Faigle-Cedzich 8th May 2018

Cold Quantum Coffee

Heidelberg University

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Table of contents

  • 1. Physics of ultracold atoms
  • 2. BCS-BEC physics from Functional Renormalisation
  • 3. Dimensional crossover
  • 4. Conclusion

1

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Physics of ultracold atoms

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Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th
  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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SLIDE 7

Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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SLIDE 8

Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

π‘Šext = ℏ πœ•

2 (𝑠/β„“osc)2

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SLIDE 10

Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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SLIDE 11

Scales

  • interparticle spacing π‘œ = β„“βˆ’π‘’
  • thermal wavelength πœ‡th

Ultracold: β„“/πœ‡th ≲ 1 Dilute: 𝑏 π‘œ1/𝑒 β‰ͺ 1

  • van der Waals length πœ‡vdW
  • oscillator length β„“osc
  • scattering length 𝑏

2

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Scale hierarchy & Hamiltonian

van der Waals length interaction scale interparticle spacing

adapted from Boettcher et al. Nuclear Physics B (2012)

thermal wavelength

  • scillator length
  • f trap

Effective Hamiltonian valid on scales ≫ β„“vdW: Μ‚ 𝐼 = ∫

βƒ— 𝑦

[ Μ‚ 𝑏†( βƒ— 𝑦) (βˆ’β„ βˆ‡2 2 𝑁 + π‘Šext( βƒ— 𝑦)) Μ‚ 𝑏( βƒ— 𝑦) + 𝑕Λ Μ‚ π‘œ( βƒ— 𝑦)2] with Μ‚ π‘œ = Μ‚ 𝑏† Μ‚ 𝑏 and 𝑕Λ = 4 𝜌 ℏ2

𝑁

𝑏

3

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Feshbach resonances

4

2-atom scattering

BEC BCS

Regal et al. 2006

πœ‰(𝐢) = Ξ”πœˆ (𝐢 βˆ’ 𝐢0) 𝑏 = 𝑏bg (1 βˆ’ Δ𝐢 𝐢 βˆ’ 𝐢0 ) with πœ‰ β†’ 0 @ resonance

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Feshbach resonances

4 van der Waals length interaction scale interparticle spacing

adapted from Boettcher et al. Nuclear Physics B (2012)

Feshbach resonance thermal wavelength

  • scillator length
  • f trap

BEC BCS

Regal et al. 2006

πœ‰(𝐢) = Ξ”πœˆ (𝐢 βˆ’ 𝐢0) 𝑏 = 𝑏bg (1 βˆ’ Δ𝐢 𝐢 βˆ’ 𝐢0 ) with πœ‰ β†’ 0 @ resonance

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SLIDE 15

The BCS-BEC crossover in 3D

5

two-component fermionic atoms

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The BCS-BEC crossover in 3D

5

two-component fermionic atoms fermions with attractive interactions

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The BCS-BEC crossover in 3D

5

two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms

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The BCS-BEC crossover in 3D

5

two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms low π‘ˆ BCS-superfmuidity (condensation of Cooper pairs)

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The BCS-BEC crossover in 3D

5

two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms low π‘ˆ BCS-superfmuidity (condensation of Cooper pairs) low π‘ˆ BEC (Bose-Einstein condensate)

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The BCS-BEC crossover in 3D

5

two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms low π‘ˆ BCS-superfmuidity (condensation of Cooper pairs) low π‘ˆ BEC (Bose-Einstein condensate) Crossover (magnetic fjeld)

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The BCS-BEC crossover in 3D

5

𝑙𝐺 = (3 𝜌2 π‘œ)1/3

Randeria Nature (2010)

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Features

advantages

  • couplings are tunable
  • high precision experiments
  • microphysics known

challenges

  • from microphysic laws to macroscopic observation

including fmuctuations

  • large couplings
  • different effective degrees of freedom
  • microphysics: single atoms & molecules
  • macrophysics: bosonic collective degrees of freedom

6

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SLIDE 23

BCS-BEC physics from Functional Renormalisation

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The functional RG

UV IR

Flow equation (Wetterich 1993)

πœ–π‘™ Γ𝑙 = 1 2 STr [(Ξ“(2) + 𝑆𝑙)βˆ’1 πœ–π‘™ 𝑆𝑙] Exact 1-loop equation

βˆ‚t Ξ“k[Ο†, ψ] = 1 2

1

7 full quantum effective action

bosons fermions

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SLIDE 25

Regulator and truncation dependence

8

  • fmow of Γ𝑙 regulator

dependent, yet Ξ“ is not

  • during fmow all possible

interactions may be produced

  • πœ–π‘’Ξ“(π‘œ) depends on

Ξ“(π‘œ+1) and Ξ“(π‘œ+2) β†’truncation needed

adapted from Gies Springer (2012)

Theory space

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SLIDE 26

Action and truncation

Microscopic action 𝑇 = ∫

π‘Œ

[πœ”βˆ— (πœ–πœ βˆ’ βˆ‡2 βˆ’ 𝜈) πœ” + πœšβˆ— (πœ–πœ βˆ’ βˆ‡2 2 + πœ‰ βˆ’ 2𝜈) 𝜚 βˆ’ β„Ž (πœšβˆ— πœ”1 πœ”2 βˆ’ 𝜚 πœ”βˆ—

1 πœ”βˆ— 2) ]

with

  • πœ”: Grassmann fjeld
  • 𝜚: bosonic fjeld consisting of two atoms
  • πœ‰: detuning
  • 𝜐: Euclidean time on torus with circumference 1/π‘ˆ
  • 𝜈: chemical potential

9

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Units

Chosen such that: ℏ = 𝑙𝐢 = 2 𝑁 = 1 Consequences:

  • ℏ = 1: [momentum]=[length]βˆ’1 with typical momentum

unit 𝑙𝐺

  • 𝑙𝐢 = 1: [temperature]=[energy]
  • 2 𝑁 = 1: [momentum]=[energy] (equiv. 𝑑 = 1)

i.e. [𝑒] = π‘š2, [ βƒ— π‘ž] = π‘šβˆ’1, [π‘ˆ] = π‘šβˆ’2, [𝜈] = π‘šβˆ’2, [π‘œ] = π‘šβˆ’3 . β†’ canonical dimensions differ from relativistic QFT!

10

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Truncation: derivative expansion

  • vertices are expanded in powers of the momenta
  • effective average potential 𝑉𝑙 @ least to 2nd order in

𝜍 = πœšβˆ—πœš (2nd order phase transition)

βˆ‚t ( )βˆ’1 = F + B

βˆ‚t ( )βˆ’1 = M

11

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Ansatz for effective action Γ𝑙 = Ξ“kin + Ξ“int Ξ“kin[πœ”, 𝜚] = ∫

π‘Œ

[ βˆ‘

𝜏={1,2}

πœ”βˆ—

𝜏 (π‘‡πœ”πœ–πœ βˆ’ βˆ‡2 βˆ’ 𝜈) πœ”πœ + πœšβˆ— (π‘‡πœšπœ–πœ βˆ’ 1

2βˆ‡2) 𝜚] Ξ“int[πœ”, 𝜚] = ∫

π‘Œ

[𝑉 (πœšβˆ— 𝜚) βˆ’ β„Ž (πœšβˆ— πœ”1 πœ”2 βˆ’ 𝜚 πœ”βˆ—

1 πœ”βˆ— 2) ]

with

  • renormalised fjelds: πœ” = 𝐡1/2

πœ”

Μ… πœ”, 𝜚 = 𝐡1/2

𝜚

Μ… 𝜚

  • π‘‡πœ”,𝜚 = π‘Žπœ”,𝜚/π΅πœ”,𝜚
  • anomalous dimensions: πœƒπœ”,𝜚 = βˆ’ πœ–π‘’ log π΅πœ”,𝜚

and 𝑉(𝜍) =

𝑂

βˆ‘

π‘œ=1

π‘£π‘œ π‘œ! (𝜍 βˆ’ 𝜍0)π‘œ βˆ’ π‘œπ‘™ (𝜈 βˆ’ 𝜈0) +

𝑁

βˆ‘

𝑛=1

𝛽𝑛 𝑛! (𝜈 βˆ’ 𝜈0) (𝜍 βˆ’ 𝜍0)𝑛 , 𝑣1 = 𝑛2

𝜚 12

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Regularisation scheme

IR-regularisation: lim

π‘ž2/𝑙2β†’0 𝑆𝑙(π‘ž2)/𝑙2 > 0,

lim

𝑙2/π‘ž2β†’0 𝑆𝑙(π‘ž2) β†’ 0

Litim-type regulator: π‘†πœš,𝑙(𝑅) = π‘†πœš,𝑙(π‘Ÿ2) = (𝑙2 βˆ’ π‘Ÿ2 2 ) πœ„ (𝑙2 βˆ’ π‘Ÿ2 2 ) π‘†πœ”,𝑙(𝑅) = π‘†πœ”,𝑙(π‘Ÿ2) = [𝑙2 sgn (π‘Ÿ2 βˆ’ 𝜈) βˆ’ (π‘Ÿ2 βˆ’ 𝜈)] πœ„ (𝑙2 βˆ’ |π‘Ÿ2 βˆ’ 𝜈|) β†’analytic evaluation of Matsubara sums

13

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UV renormalisation

Connecting to experiment

  • initial condition for 𝑉𝑙: 𝑉Λ(𝜍) = (πœ‰Ξ› βˆ’ 2 𝜈) 𝜍
  • @ π‘ˆ = 0: connect to correct vacuum physics at unitarity

(π‘βˆ’1 = 0), i.e. 𝑛2

𝜚,𝑙=0 = 0 = 𝜈

Thus: 𝑏 = 𝑏(𝐢) = β„Ž2

Ξ›

8 𝜌 πœ‰(𝐢) , πœ‰(𝐢) = πœ‰Ξ› βˆ’ πœ€πœ‰(Ξ›)

14

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Universality

  • existence of FPs in RG fmow: macrophysics independent of

the microphysics

  • loss of memory of microphysics: start at the FP values

initial values β„Ž2

Ξ› = 6 𝜌2 Ξ›,

πœ‡πœš,Ξ› = Μƒ πœ‡πœš,βˆ— Ξ› , 𝑛2

𝜚,Ξ› = πœ‰Ξ› βˆ’ 2 𝜈,

π‘‡πœš,Ξ› = 1, 𝛽Λ = βˆ’2, π‘œΞ› = 𝜈3/2 3 𝜌2 Θ(𝜈).

15

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3D BCS-BEC crossover

π‘ˆ = 0: (with 𝑑 = 𝑙𝐺 𝑏)

  • 4
  • 2

2 4 0.0 0.2 0.4 0.6 0.8 1.0

  • 4
  • 2

2 4 0.0 0.5 1.0 1.5 2.0 2.5

π‘ˆ > 0:

  • 4
  • 2

2 4 0.0 0.5 1.0 1.5 2.0

  • 4
  • 2

2 4 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

16

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Dimensional crossover

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Why are quasi-2D systems interesting?

  • promising materials: graphene, high π‘ˆπ‘‘-superconductors,

layered semiconductors

  • pronounced infmuence of quantum fmuctuations
  • experimental accessibility via highly anisotropic trapping

potentials

  • for insuffjcient anisotropy β†’dimensional crossover

17

β†’disentangle dimensionality from many-body physics

ZΓΌrn et al. PRL (2015)

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Boundary conditions

  • delimit 𝑨-direction by potential well of length 𝑀

π‘Šbox(𝑨) = ⎧ { ⎨ { ⎩ 0 ≀ 𝑨 ≀ 𝑀 ∞ else

  • impose periodic boundary conditions: Ξ¨ = {πœ”, 𝜚}

Ξ¨(𝜐, 𝑦, 𝑧, 𝑨 = 0) = Ξ¨(𝜐, 𝑦, 𝑧, 𝑨 = 𝑀) β†’quantisation of momentum in 𝑨-direction: π‘Ÿπ‘¨ β†’ π‘™π‘œ = 2 𝜌 π‘œ 𝑀 , π‘œ ∈ β„•

  • spatial Matsubara sum

∫ π‘’π‘’π‘Ÿ (2 𝜌)𝑒 = 1 𝑀 βˆ‘

π‘™π‘œ

∫ π‘’π‘’βˆ’1π‘Ÿ (2 𝜌)π‘’βˆ’1

18

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Regulator and method

Litim-type regulator (as before) with βƒ— π‘Ÿ = Μ‚βƒ— π‘Ÿ + π‘Ÿπ‘¨ β†’ Μ‚βƒ— π‘Ÿ + π‘™π‘œ Idea:

  • initialise RG fmow at UV scale 𝑙 = Ξ› where ΓΛ coincides

with 𝑇 of 3D gas

  • 𝑀 introduces new length scale to 3D system
  • following the RG fmow we successively integrate out the 3rd

dimension β†’@ 𝑙 β†’ 0: system @ given confjnement length scale 𝑀 Note

  • UV scale is always chosen such that Ξ› ≫ (π‘€βˆ’1, 𝜈1/2, π‘ˆ 1/2)
  • system is effectively 2D if π‘€βˆ’1 ≫ all other many-body

scales

19

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Regulator and method

Litim-type regulator (as before) with βƒ— π‘Ÿ = Μ‚βƒ— π‘Ÿ + π‘Ÿπ‘¨ β†’ Μ‚βƒ— π‘Ÿ + π‘™π‘œ Idea:

  • initialise RG fmow at UV scale 𝑙 = Ξ› where ΓΛ coincides

with 𝑇 of 3D gas

  • 𝑀 introduces new length scale to 3D system
  • following the RG fmow we successively integrate out the 3rd

dimension β†’@ 𝑙 β†’ 0: system @ given confjnement length scale 𝑀 Note

  • UV scale is always chosen such that Ξ› ≫ (π‘€βˆ’1, 𝜈1/2, π‘ˆ 1/2)
  • system is effectively 2D if π‘€βˆ’1 ≫ all other many-body

scales

19

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SLIDE 39

Regulator and method

Litim-type regulator (as before) with βƒ— π‘Ÿ = Μ‚βƒ— π‘Ÿ + π‘Ÿπ‘¨ β†’ Μ‚βƒ— π‘Ÿ + π‘™π‘œ Idea:

  • initialise RG fmow at UV scale 𝑙 = Ξ› where ΓΛ coincides

with 𝑇 of 3D gas

  • 𝑀 introduces new length scale to 3D system
  • following the RG fmow we successively integrate out the 3rd

dimension β†’@ 𝑙 β†’ 0: system @ given confjnement length scale 𝑀 Note

  • UV scale is always chosen such that Ξ› ≫ (π‘€βˆ’1, 𝜈1/2, π‘ˆ 1/2)
  • system is effectively 2D if π‘€βˆ’1 ≫ all other many-body

scales

19

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Zero temperature

20

  • correct 3D-limit reached
  • qualitatively correct

behaviour of equation

  • f state (except very

small confjnements)

3D limit

  • 4
  • 2

2 4 0.0 0.2 0.4 0.6 0.8 1.0 3D limit

  • 4
  • 2

2 4 0.0 0.5 1.0 1.5 2.0 2.5

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SLIDE 41

Connecting to experiment

quasi-2d scattering length: 𝑏(pbc)

2D

= 𝑀 exp {βˆ’1 2 𝑀 𝑏3D } , 𝑏(trap)

2D

= ℓ𝑨 √ 𝜈 𝐡 exp {βˆ’βˆšπœˆ 2 ℓ𝑨 𝑏3D } quasi-2d crossover parameter: ln(𝑙𝐺 𝑏2D)

  • 4
  • 2

2 4 0.0 0.2 0.4 0.6 0.8 1.0

21

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SLIDE 42

Finite temperature

0.01 0.10 1 10 100 0.10 0.15 0.20 0.25 0.01 0.10 1 10 100 1000 0.10 0.15 0.20 0.25 0.30 0.1 1 10 100 1000 0.10 0.15 0.20 0.25 0.30

22

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SLIDE 43

Phase diagram

  • 4
  • 2

2 4 0.00 0.05 0.10 0.15 0.20 0.25 0.30

  • 4
  • 2

2 4 0.00 0.05 0.10 0.15 0.20 0.25

  • 3D phase diagram reproduced for large 𝑀
  • increased π‘ˆπ‘‘/π‘ˆπΊ around ln(𝑙𝐺 𝑏2D) ∼ 1

23

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SLIDE 44

Comparison to experiment

  • 4
  • 2

2 4 0.00 0.05 0.10 0.15 0.20

experimental data from Ries et al. PRL (2015)

24

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SLIDE 45

Conclusion

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SLIDE 46

Summary and Outlook

Conclusion

  • dimensional crossover in Fermi gas with FRG
  • qualitatively comparable to experiments

Outlook

  • employ harmonic trapping potential
  • quantitative precision
  • particle-hole fmuctuations
  • frequency dependent regulator
  • explore thermodynamics

25

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SLIDE 47

Thank you for your attention!

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SLIDE 48

Phase diagram from experiment

26

  • 8
  • 6
  • 4
  • 2

2 4 0.0 0.1 0.2 0.3

BEC T/TF ln(kFa2D)

0.00 0.10 0.20 0.30 0.40 0.50

Nq/N BCS

Ries et al. PRL (2015)