Dimensional crossover in ultracold Fermi gases from Functional - PowerPoint PPT Presentation

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 Renormalisation 3. Dimensional crossover 4. Conclusion 1

Physics of ultracold atoms

Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2

Ultracold: ℓ/𝜇 th ≲ 1 Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2

Ultracold: ℓ/𝜇 th ≲ 1 Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2

Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2 Ultracold: ℓ/𝜇 th ≲ 1

Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2 Ultracold: ℓ/𝜇 th ≲ 1

Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2 Ultracold: ℓ/𝜇 th ≲ 1 𝑊 ext = ℏ 𝜕 2 (𝑠/ℓ osc ) 2

Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2 Ultracold: ℓ/𝜇 th ≲ 1

Scales • interparticle spacing 𝑜 = ℓ −𝑒 • thermal wavelength 𝜇 th • van der Waals length 𝜇 vdW • oscillator length ℓ osc • scattering length 𝑏 2 Dilute: 𝑏 𝑜 1/𝑒 ≪ 1 Ultracold: ℓ/𝜇 th ≲ 1

Scale hierarchy & Hamiltonian ⃗ 𝑏 𝑁 ̂ 𝑜 = ̂ with 𝑦) 2 ] 𝑜( ⃗ 𝑏( ⃗ 𝑦 𝑏 † ( ⃗ 3 Effective Hamiltonian valid on scales ≫ ℓ vdW : ̂ adapted from Boettcher et al. Nuclear Physics B (2012) van der Waals interparticle thermal wavelength oscillator length interaction scale length of trap spacing 𝐼 = ∫ [ ̂ 𝑦) (−ℏ ∇ 2 2 𝑁 + 𝑊 ext ( ⃗ 𝑦)) ̂ 𝑦) + 𝑕 Λ ̂ 𝑏 and 𝑕 Λ = 4 𝜌 ℏ 2 𝑏 † ̂

Feshbach resonances 4 2-atom scattering 𝜉(𝐶) = Δ𝜈 (𝐶 − 𝐶 0 ) Δ𝐶 𝐶 − 𝐶 0 ) with 𝜉 → 0 @ resonance Regal et al. 2006 BEC 𝑏 = 𝑏 bg (1 − BCS

Feshbach resonances 4 ) 𝐶 − 𝐶 0 Δ𝐶 𝜉(𝐶) = Δ𝜈 (𝐶 − 𝐶 0 ) with 𝜉 → 0 @ resonance adapted from Boettcher et al. Nuclear Physics B (2012) van der Waals oscillator length interparticle thermal wavelength interaction scale length of trap spacing Feshbach resonance Regal et al. 2006 BEC 𝑏 = 𝑏 bg (1 − BCS

The BCS-BEC crossover in 3D 5 two-component fermionic atoms

The BCS-BEC crossover in 3D 5 two-component fermionic atoms fermions with attractive interactions

The BCS-BEC crossover in 3D 5 two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms

The BCS-BEC crossover in 3D 5 two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms low 𝑈 BCS-superfmuidity (condensation of Cooper pairs)

The BCS-BEC crossover in 3D 5 two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms low 𝑈 BCS-superfmuidity (condensation of Cooper pairs) low 𝑈 BEC (Bose-Einstein condensate)

The BCS-BEC crossover in 3D 5 two-component fermionic atoms fermions with attractive interactions bound molecules of two atoms low 𝑈 BCS-superfmuidity (condensation of Cooper pairs) low 𝑈 BEC (Bose-Einstein condensate) Crossover (magnetic fjeld)

The BCS-BEC crossover in 3D 5 Randeria Nature (2010) 𝑙 𝐺 = (3 𝜌 2 𝑜) 1/3

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

BCS-BEC physics from Functional Renormalisation

The functional RG Flow equation (Wetterich 1993) Exact 1-loop equation 7 full quantum effective action bosons fermions IR UV 𝜖 𝑙 Γ 𝑙 = 1 2 STr [(Γ (2) + 𝑆 𝑙 ) −1 𝜖 𝑙 𝑆 𝑙 ] ∂ t Γ k [ φ, ψ ] = 1 1 2

Regulator and truncation dependence 8 dependent, yet Γ is not • during fmow all possible interactions may be produced →truncation needed adapted from Gies Springer (2012) • fmow of Γ 𝑙 regulator Theory space • 𝜖 𝑢 Γ (𝑜) depends on Γ (𝑜+1) and Γ (𝑜+2)

Action and truncation Microscopic action 𝑇 = ∫ 𝑌 2 ) ] with • 𝜔 : Grassmann fjeld • 𝜚 : bosonic fjeld consisting of two atoms • 𝜉 : detuning • 𝜐 : Euclidean time on torus with circumference 1/𝑈 • 𝜈 : chemical potential 9 [𝜔 ∗ (𝜖 𝜐 − ∇ 2 − 𝜈) 𝜔 + 𝜚 ∗ (𝜖 𝜐 − ∇ 2 2 + 𝜉 − 2𝜈) 𝜚 − ℎ (𝜚 ∗ 𝜔 1 𝜔 2 − 𝜚 𝜔 ∗ 1 𝜔 ∗

Units Chosen such that: Consequences: unit 𝑙 𝐺 • 2 𝑁 = 1 : [ momentum ] = [ energy ] (equiv. 𝑑 = 1 ) i.e. [𝑢] = 𝑚 2 , [ ⃗ → canonical dimensions differ from relativistic QFT! 10 ℏ = 𝑙 𝐶 = 2 𝑁 = 1 • ℏ = 1 : [ momentum ] = [ length ] −1 with typical momentum • 𝑙 𝐶 = 1 : [ temperature ] = [ energy ] 𝑞] = 𝑚 −1 , [𝑈] = 𝑚 −2 , [𝜈] = 𝑚 −2 , [𝑜] = 𝑚 −3 .

Truncation : derivative expansion • vertices are expanded in powers of the momenta 𝜍 = 𝜚 ∗ 𝜚 (2nd order phase transition) 11 • effective average potential 𝑉 𝑙 @ least to 2nd order in F B ) − 1 = ∂ t ( + M ) − 1 = ∂ t (

12 𝜔 , 𝜚 = 𝐵 1/2 𝜚 𝛽 𝑛 𝑛=1 ∑ 𝑁 𝑣 𝑜 𝑜=1 ∑ 𝑂 𝑉(𝜍) = and 𝜚 Γ kin [𝜔, 𝜚] = ∫ 𝜚 ̅ ̅ 𝑌 𝑌 [ ∑ 𝜏={1,2} 𝜔 ∗ 2∇ 2 ) 𝜚] 𝜔 Γ int [𝜔, 𝜚] = ∫ 2 ) ] with • renormalised fjelds: 𝜔 = 𝐵 1/2 Ansatz for effective action Γ 𝑙 = Γ kin + Γ int 𝜏 (𝑇 𝜔 𝜖 𝜐 − ∇ 2 − 𝜈) 𝜔 𝜏 + 𝜚 ∗ (𝑇 𝜚 𝜖 𝜐 − 1 [𝑉 (𝜚 ∗ 𝜚) − ℎ (𝜚 ∗ 𝜔 1 𝜔 2 − 𝜚 𝜔 ∗ 1 𝜔 ∗ • 𝑇 𝜔,𝜚 = 𝑎 𝜔,𝜚 /𝐵 𝜔,𝜚 • anomalous dimensions: 𝜃 𝜔,𝜚 = − 𝜖 𝑢 log 𝐵 𝜔,𝜚 𝑜! (𝜍 − 𝜍 0 ) 𝑜 − 𝑜 𝑙 (𝜈 − 𝜈 0 ) + 𝑛! (𝜈 − 𝜈 0 ) (𝜍 − 𝜍 0 ) 𝑛 , 𝑣 1 = 𝑛 2

Regularisation scheme IR-regularisation: lim lim Litim-type regulator: →analytic evaluation of Matsubara sums 13 𝑞 2 /𝑙 2 →0 𝑆 𝑙 (𝑞 2 )/𝑙 2 > 0, 𝑙 2 /𝑞 2 →0 𝑆 𝑙 (𝑞 2 ) → 0 𝑆 𝜚,𝑙 (𝑅) = 𝑆 𝜚,𝑙 (𝑟 2 ) = (𝑙 2 − 𝑟 2 2 ) 𝜄 (𝑙 2 − 𝑟 2 2 ) 𝑆 𝜔,𝑙 (𝑅) = 𝑆 𝜔,𝑙 (𝑟 2 ) = [𝑙 2 sgn (𝑟 2 − 𝜈) − (𝑟 2 − 𝜈)] 𝜄 (𝑙 2 − |𝑟 2 − 𝜈|)

UV renormalisation Connecting to experiment • @ 𝑈 = 0 : connect to correct vacuum physics at unitarity Thus: 𝑏 = 𝑏(𝐶) = ℎ 2 Λ 14 • initial condition for 𝑉 𝑙 : 𝑉 Λ (𝜍) = (𝜉 Λ − 2 𝜈) 𝜍 ( 𝑏 −1 = 0 ), i.e. 𝑛 2 𝜚,𝑙=0 = 0 = 𝜈 8 𝜌 𝜉(𝐶) , 𝜉(𝐶) = 𝜉 Λ − 𝜀𝜉(Λ)

Universality ̃ 𝑛 2 • existence of FPs in RG fmow: macrophysics independent of 𝜇 𝜚,∗ ℎ 2 initial values • loss of memory of microphysics: start at the FP values the microphysics 15 Λ = 6 𝜌 2 Λ, 𝜇 𝜚,Λ = Λ , 𝜚,Λ = 𝜉 Λ − 2 𝜈, 𝑜 Λ = 𝜈 3/2 𝑇 𝜚,Λ = 1, 𝛽 Λ = −2, 3 𝜌 2 Θ(𝜈).

3D BCS-BEC crossover 𝑈 > 0 : 16 𝑈 = 0 : (with 𝑑 = 𝑙 𝐺 𝑏 ) 1.0 2.5 0.8 2.0 0.6 1.5 0.4 1.0 0.2 0.5 0.0 0.0 - 4 - 2 0 2 4 - 4 - 2 0 2 4 0.35 2.0 0.30 0.25 1.5 0.20 1.0 0.15 0.10 0.5 0.05 0.0 0.00 - 4 - 2 0 2 4 - 4 - 2 0 2 4

Dimensional crossover

Why are quasi-2D systems interesting? • promising materials: graphene, high 𝑈 𝑑 -superconductors, layered semiconductors • pronounced infmuence of quantum fmuctuations • experimental accessibility via highly anisotropic trapping potentials • for insuffjcient anisotropy →dimensional crossover 17 →disentangle dimensionality from many-body physics Zürn et al. PRL (2015)

Boundary conditions Ψ(𝜐, 𝑦, 𝑧, 𝑨 = 0) = Ψ(𝜐, 𝑦, 𝑧, 𝑨 = 𝑀) (2 𝜌) 𝑒−1 𝑒 𝑒−1 𝑟 ∫ 𝑙 𝑜 𝑒 𝑒 𝑟 ∫ • spatial Matsubara sum 𝑜 ∈ ℕ • delimit 𝑨 -direction by potential well of length 𝑀 →quantisation of momentum in 𝑨 -direction: • impose periodic boundary conditions: Ψ = {𝜔, 𝜚} else ∞ 0 ≤ 𝑨 ≤ 𝑀 0 ⎩ { ⎨ { ⎧ 𝑊 box (𝑨) = 18 𝑟 𝑨 → 𝑙 𝑜 = 2 𝜌 𝑜 𝑀 , (2 𝜌) 𝑒 = 1 𝑀 ∑