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Universality in ultra-cold fermionic atom gases

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with S. Diehl, H.Gies, J.Pawlowski S. Diehl, H.Gies, J.Pawlowski

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BEC – BCS crossover Bound molecules of two atoms Bound molecules of two atoms on microscopic scale: on microscopic scale: Bose-Einstein condensate (BEC ) for low T Bose-Einstein condensate (BEC ) for low T Fermions with attractive interactions Fermions with attractive interactions (molecules play no role ) : (molecules play no role ) : BCS – superfluidity at low T BCS – superfluidity at low T by condensation of Cooper pairs by condensation of Cooper pairs Crossover by Feshbach resonance Crossover by Feshbach resonance as a transition in terms of external magnetic field as a transition in terms of external magnetic field

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Feshbach resonance H.Stoof

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scattering length BCSBEC

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chemical potential BEC BCS inverse scattering length

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BEC – BCS crossover qualitative and partially quantitative theoretical understanding qualitative and partially quantitative theoretical understanding mean field theory (MFT ) and first attempts beyond mean field theory (MFT ) and first attempts beyond concentration : c = a k F reduced chemical potential : σ˜ = μ/ε F Fermi momemtum : k F Fermi energy : ε F binding energy : T = 0 BCSBEC

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concentration c = a k F, a(B) : scattering length needs computation of density n=k F 3 /(3π 2 ) BCSBEC dilute dense T = 0 non- interacting Fermi gas non- interacting Bose gas

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universality same curve for Li and K atoms ? BCSBEC dilute dense T = 0

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Quantum Monte Carlo different methods

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who cares about details ? a theorists game …? a theorists game …? RG MFT

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precision many body theory - quantum field theory - so far : particle physics : perturbative calculations particle physics : perturbative calculations magnetic moment of electron : magnetic moment of electron : g/2 = 1.001 159 652 180 85 ( 76 ) ( Gabrielse et al. ) g/2 = 1.001 159 652 180 85 ( 76 ) ( Gabrielse et al. ) statistical physics : universal critical exponents for second order phase transitions : ν = 0.6308 (10) statistical physics : universal critical exponents for second order phase transitions : ν = 0.6308 (10) renormalization group renormalization group lattice simulations for bosonic systems in particle and statistical physics ( e.g. QCD ) lattice simulations for bosonic systems in particle and statistical physics ( e.g. QCD )

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QFT with fermions needed: needed: universal theoretical tools for complex fermionic systems universal theoretical tools for complex fermionic systems wide applications : wide applications : electrons in solids, electrons in solids, nuclear matter in neutron stars, …. nuclear matter in neutron stars, ….

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QFT for non-relativistic fermions functional integral, action functional integral, action τ : euclidean time on torus with circumference 1/T σ : effective chemical potential perturbation theory: Feynman rules

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variables ψ : Grassmann variables ψ : Grassmann variables φ : bosonic field with atom number two φ : bosonic field with atom number two What is φ ? What is φ ? microscopic molecule, microscopic molecule, macroscopic Cooper pair ? macroscopic Cooper pair ? All ! All !

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parameters detuning ν(B) detuning ν(B) Yukawa or Feshbach coupling h φ Yukawa or Feshbach coupling h φ

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fermionic action equivalent fermionic action, in general not local

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scattering length a broad resonance : pointlike limit broad resonance : pointlike limit large Feshbach coupling large Feshbach coupling a= M λ/4π

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parameters Yukawa or Feshbach coupling h φ Yukawa or Feshbach coupling h φ scattering length a scattering length a broad resonance : h φ drops out broad resonance : h φ drops out

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concentration c

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universality Are these parameters enough for a quantitatively precise description ? Are these parameters enough for a quantitatively precise description ? Have Li and K the same crossover when described with these parameters ? Have Li and K the same crossover when described with these parameters ? Long distance physics looses memory of detailed microscopic properties of atoms and molecules ! Long distance physics looses memory of detailed microscopic properties of atoms and molecules ! universality for c -1 = 0 : Ho,…( valid for broad resonance) universality for c -1 = 0 : Ho,…( valid for broad resonance) here: whole crossover range here: whole crossover range

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analogy with particle physics microscopic theory not known - microscopic theory not known - nevertheless “macroscopic theory” characterized by a finite number of nevertheless “macroscopic theory” characterized by a finite number of “renormalizable couplings” “renormalizable couplings” m e, α ; g w, g s, M w, … m e, α ; g w, g s, M w, … here : c, h φ ( only c for broad resonance ) here : c, h φ ( only c for broad resonance )

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analogy with universal critical exponents only one relevant parameter : T - T c T - T c

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units and dimensions c = 1 ; ħ =1 ; k = 1 c = 1 ; ħ =1 ; k = 1 momentum ~ length -1 ~ mass ~ eV momentum ~ length -1 ~ mass ~ eV energies : 2ME ~ (momentum) 2 energies : 2ME ~ (momentum) 2 ( M : atom mass ) ( M : atom mass ) typical momentum unit : Fermi momentum typical momentum unit : Fermi momentum typical energy and temperature unit : Fermi energy typical energy and temperature unit : Fermi energy time ~ (momentum) -2 time ~ (momentum) -2 canonical dimensions different from relativistic QFT ! canonical dimensions different from relativistic QFT !

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rescaled action M drops out M drops out all quantities in units of k F if all quantities in units of k F if

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what is to be computed ? Inclusion of fluctuation effects via functional integral leads to effective action. This contains all relevant information for arbitrary T and n !

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effective action integrate out all quantum and thermal fluctuations integrate out all quantum and thermal fluctuations quantum effective action quantum effective action generates full propagators and vertices generates full propagators and vertices richer structure than classical action richer structure than classical action

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effective action includes all quantum and thermal fluctuations includes all quantum and thermal fluctuations formulated here in terms of renormalized fields formulated here in terms of renormalized fields involves renormalized couplings involves renormalized couplings

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effective potential minimum determines order parameter condensate fraction Ω c = 2 ρ 0 /n

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effective potential value of φ at potential minimum : value of φ at potential minimum : order parameter, determines condensate fraction order parameter, determines condensate fraction second derivative of U with respect to φ yields correlation length second derivative of U with respect to φ yields correlation length derivative with respect to σ yields density derivative with respect to σ yields density fourth derivative of U with respect to φ yields molecular scattering length fourth derivative of U with respect to φ yields molecular scattering length

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renormalized fields and couplings

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challenge for ultra-cold atoms : Non-relativistic fermion systems with precision Non-relativistic fermion systems with precision similar to particle physics ! similar to particle physics ! ( QCD with quarks ) ( QCD with quarks )

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results from functional renormalization group

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physics at different length scales microscopic theories : where the laws are formulated microscopic theories : where the laws are formulated effective theories : where observations are made effective theories : where observations are made effective theory may involve different degrees of freedom as compared to microscopic theory effective theory may involve different degrees of freedom as compared to microscopic theory example: microscopic theory only for fermionic atoms, macroscopic theory involves bosonic collective degrees of freedom ( φ ) example: microscopic theory only for fermionic atoms, macroscopic theory involves bosonic collective degrees of freedom ( φ )

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T = 0 gap parameter BCSBEC Δ BCS for gap

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limits BCS for gap condensate fraction for bosons with scattering length 0.9 a

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temperature dependence of condensate second order phase transition

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c -1 =1 c -1 =0 T/T c free BEC universal critical behavior condensate fraction : second order phase transition

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crossover phase diagram

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shift of BEC critical temperature

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running couplings : crucial for universality for large Yukawa couplings h φ : only one relevant parameter c only one relevant parameter c all other couplings are strongly attracted to partial fixed points all other couplings are strongly attracted to partial fixed points macroscopic quantities can be predicted macroscopic quantities can be predicted in terms of c and T/ε F in terms of c and T/ε F ( in suitable range for c -1 ) ( in suitable range for c -1 )

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Flow of Yukawa coupling T=0.5, c=1 k2k2 k2k2

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universality for broad resonances for large Yukawa couplings h φ : only one relevant parameter c only one relevant parameter c all other couplings are strongly attracted to partial fixed points all other couplings are strongly attracted to partial fixed points macroscopic quantities can be predicted macroscopic quantities can be predicted in terms of c and T/ε F in terms of c and T/ε F ( in suitable range for c -1 ; density sets scale ) ( in suitable range for c -1 ; density sets scale )

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universality for narrow resonances Yukawa coupling becomes additional parameter Yukawa coupling becomes additional parameter ( marginal coupling ) ( marginal coupling ) also background scattering important also background scattering important

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Flow of Yukawa and four fermion coupling h 2 /32π λ ψ /8π (A ) broad Feshbach resonance (C) narrow Feshbach resonance

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Universality is due to fixed points !

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not all quantities are universal !

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bare molecule fraction (fraction of microscopic closed channel molecules ) not all quantities are universal not all quantities are universal bare molecule fraction involves wave function renormalization that depends on value of Yukawa coupling bare molecule fraction involves wave function renormalization that depends on value of Yukawa coupling B[G] Experimental points by Partridge et al. 6 Li

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conclusions the challenge of precision : substantial theoretical progress needed substantial theoretical progress needed “phenomenology” has to identify quantities that are accessible to precision both for experiment and theory “phenomenology” has to identify quantities that are accessible to precision both for experiment and theory dedicated experimental effort needed dedicated experimental effort needed

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challenges for experiment study the simplest system study the simplest system identify quantities that can be measured with precision of a few percent and have clear theoretical interpretation identify quantities that can be measured with precision of a few percent and have clear theoretical interpretation precise thermometer that does not destroy probe precise thermometer that does not destroy probe same for density same for density

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end

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