universality in ultra-cold fermionic atom gases. with s. diehl, h.gies, j.pawlowski s. diehl,...
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Universality in ultra-cold Universality in ultra-cold fermionic atom gasesfermionic atom gases
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Universality in ultra-cold Universality in ultra-cold fermionic atom gasesfermionic atom gases
withwith
S. Diehl , H.Gies , S. Diehl , H.Gies , J.PawlowskiJ.Pawlowski
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BEC – BCS crossoverBEC – BCS crossover
Bound molecules of two atoms Bound molecules of two atoms on microscopic scale:on microscopic scale:
Bose-Einstein condensate (BEC ) for low TBose-Einstein condensate (BEC ) for low T
Fermions with attractive interactionsFermions 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 pairsby condensation of Cooper pairs
CrossoverCrossover by by Feshbach resonanceFeshbach resonance as a transition in terms of external as a transition in terms of external magnetic fieldmagnetic field
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Feshbach resonanceFeshbach resonance
H.Stoof
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scattering lengthscattering length
BCS
BEC
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chemical potentialchemical potential
BEC
BCS
inverse scattering length
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BEC – BCS crossoverBEC – BCS crossover qualitative and partially quantitative qualitative and partially quantitative
theoretical understandingtheoretical understanding mean field theory (MFT ) and first mean field theory (MFT ) and first
attempts beyondattempts beyond
concentration : c = a kF
reduced chemical potential : σ˜ = μ/εF
Fermi momemtum : kF
Fermi energy : εF
binding energy :
T = 0
BCS
BEC
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concentrationconcentration c = a kF , a(B) : scattering length needs computation of density n=kF
3/(3π2)
BCS
BEC
dilute dilutedense
T = 0
non-interactingFermi gas
non-interactingBose gas
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universalityuniversality
same curve for Li and K atoms ?
BCS
BEC
dilute dilutedense
T = 0
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Quantum Monte Carlo
different methodsdifferent methods
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who cares about details ?who cares about details ?
a theorists game …?a theorists game …?
RGMFT
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precision many body theoryprecision many body theory- quantum field theory- quantum field theory - -
so far :so far : particle physics : particle physics : perturbative calculationsperturbative calculations
magnetic moment of electron : magnetic moment of electron :
g/2 = 1.001 159 652 180 85 ( 76 ) g/2 = 1.001 159 652 180 85 ( 76 ) ( Gabrielse et ( Gabrielse et al. )al. )
statistical physics : universal critical exponents statistical physics : universal critical exponents for second order phase transitions : for second order phase transitions : νν = 0.6308 = 0.6308 (10)(10)
renormalization grouprenormalization group lattice simulationslattice simulations for bosonic systems in particle for bosonic systems in particle
and statistical physics ( e.g. QCD )and statistical physics ( e.g. QCD )
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QFT with fermionsQFT with fermions
needed:needed:
universal theoretical tools for universal theoretical tools for complex fermionic systemscomplex 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 QFT for non-relativistic fermionsfermions
functional integral, actionfunctional integral, action
τ : euclidean time on torus with circumference 1/Tσ : effective chemical potential
perturbation theory:Feynman rules
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variablesvariables
ψψ : Grassmann variables : Grassmann variables φφ : bosonic field with atom number : bosonic field with atom number
twotwo
What is What is φφ ? ? microscopic molecule, microscopic molecule, macroscopic Cooper pair ?macroscopic Cooper pair ?
All !All !
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parametersparameters
detuning detuning νν(B)(B)
Yukawa or Feshbach coupling hYukawa or Feshbach coupling hφφ
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fermionic actionfermionic action
equivalent fermionic action , in equivalent fermionic action , in general not localgeneral not local
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scattering length ascattering length a
broad resonance : pointlike limitbroad resonance : pointlike limit large Feshbach couplinglarge Feshbach coupling
a= M λ/4π
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parametersparameters
Yukawa or Feshbach coupling hYukawa or Feshbach coupling hφφ
scattering length ascattering length a
broad resonance :broad resonance : h hφφ drops outdrops out
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concentration cconcentration c
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universalityuniversality Are these parameters enough for a Are these parameters enough for a
quantitatively precise description ?quantitatively precise description ?
Have Li and K the same crossover when Have Li and K the same crossover when described with these parameters ?described with these parameters ?
Long distance physics looses memory of detailed Long distance physics looses memory of detailed microscopic properties of atoms and molecules !microscopic properties of atoms and molecules !
universality for cuniversality for c-1-1 = 0 : Ho,…( valid for broad = 0 : Ho,…( valid for broad resonance)resonance)
here: whole crossover rangehere: whole crossover range
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analogy with particle analogy with particle physicsphysics
microscopic theory not known -microscopic theory not known - nevertheless “macroscopic theory” nevertheless “macroscopic theory”
characterized by a finite number of characterized by a finite number of ““renormalizable couplings”renormalizable couplings”
mmee , , αα ; g ; g ww , g , g ss , M , M w w , …, …
here : here : cc , h , hφφ ( only ( only cc for broad for broad resonance )resonance )
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analogy with analogy with universal critical exponentsuniversal critical exponents
only one relevant parameter :only one relevant parameter :
T - TT - Tcc
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units and dimensionsunits and dimensions
c = 1 ; ħ =1 ; k = 1c = 1 ; ħ =1 ; k = 1 momentum ~ lengthmomentum ~ length-1-1 ~ mass ~ eV ~ mass ~ eV energies : 2ME ~ (momentum)energies : 2ME ~ (momentum)22
( M : atom mass )( M : atom mass ) typical momentum unit : Fermi momentumtypical momentum unit : Fermi momentum typical energy and temperature unit : Fermi typical energy and temperature unit : Fermi
energyenergy time ~ (momentum) time ~ (momentum) -2-2
canonical dimensions different from canonical dimensions different from relativistic QFT !relativistic QFT !
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rescaled actionrescaled action
M drops outM drops out all quantities in units of kall quantities in units of kF F ifif
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what is to be computed ?what is to be computed ?
Inclusion of fluctuation effectsInclusion of fluctuation effectsvia via functional integralfunctional integralleads to leads to effective action.effective action.
This contains all relevant This contains all relevant information for arbitrary T information for arbitrary T and n !and n !
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effective actioneffective action
integrate out all quantum and integrate out all quantum and thermal fluctuationsthermal fluctuations
quantum effective actionquantum effective action generates full propagators and generates full propagators and
verticesvertices richer structure than classical actionricher structure than classical action
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effective actioneffective action
includes all quantum and thermal fluctuationsincludes all quantum and thermal fluctuations formulated here in terms of renormalized formulated here in terms of renormalized
fieldsfields involves renormalized couplingsinvolves renormalized couplings
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effective potentialeffective potential
minimum determines order parameterminimum determines order parameter
condensate fractioncondensate fraction
Ωc = 2 ρ0/n
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effective potentialeffective potential value of value of φφ at potential minimum : at potential minimum : order parameter , determines order parameter , determines
condensate fractioncondensate fraction second derivative of U with respect to second derivative of U with respect to
φφ yields correlation length yields correlation length derivative with respect to derivative with respect to σσ yields yields
densitydensity fourth derivative of U with respect to fourth derivative of U with respect to φφ
yields molecular scattering lengthyields molecular scattering length
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renormalized fields and renormalized fields and couplingscouplings
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challenge for ultra-challenge for ultra-cold atoms :cold atoms :
Non-relativistic fermion systems Non-relativistic fermion systems with precisionwith precision
similar to particle physics !similar to particle physics !
( QCD with quarks )( QCD with quarks )
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resultsresults
fromfrom
functional functional renormalization grouprenormalization group
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physics at different physics at different length scaleslength scales
microscopic theories : where the laws are microscopic theories : where the laws are formulatedformulated
effective theories : where observations are effective theories : where observations are mademade
effective theory may involve different degrees effective theory may involve different degrees of freedom as compared to microscopic theoryof freedom as compared to microscopic theory
example: microscopic theory only for example: microscopic theory only for fermionic atoms , macroscopic theory involves fermionic atoms , macroscopic theory involves bosonic collective degrees of freedom ( bosonic collective degrees of freedom ( φφ ) )
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T = 0
gap parametergap parameter
BCS BEC
Δ
BCSfor gap
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limitslimits
BCSfor gap
condensate fractionforbosons with scattering length0.9 a
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temperature dependence of temperature dependence of condensatecondensate
second order phase transition
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c -1 =1
c -1 =0
T/Tc
free BEC
universalcriticalbehavior
condensate fraction :condensate fraction :second order phase second order phase
transitiontransition
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crossover phase diagramcrossover phase diagram
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shift of BEC critical shift of BEC critical temperaturetemperature
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running couplings :running couplings :crucial for universality crucial for universality
for large Yukawa couplings hfor large Yukawa couplings hφφ : : only one relevant parameter conly one relevant parameter c all other couplings are strongly all other couplings are strongly
attracted to partial fixed pointsattracted to partial fixed points macroscopic quantities can be macroscopic quantities can be
predicted predicted
in terms of c and T/in terms of c and T/εεFF
( in suitable range for c( in suitable range for c-1 -1 ))
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Flow of Yukawa couplingFlow of Yukawa coupling
T=0.5 , c=1k2k2
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universality for broad universality for broad resonancesresonances
for large Yukawa couplings hfor large Yukawa couplings hφφ : : only one relevant parameter conly one relevant parameter c all other couplings are strongly all other couplings are strongly
attracted to partial fixed pointsattracted to partial fixed points macroscopic quantities can be predicted macroscopic quantities can be predicted
in terms of c and T/in terms of c and T/εεFF
( in suitable range for c( in suitable range for c-1 -1 ; density sets ; density sets scale )scale )
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universality for narrow universality for narrow resonancesresonances
Yukawa coupling becomes additional Yukawa coupling becomes additional parameterparameter
( marginal coupling )( marginal coupling ) also background scattering importantalso background scattering important
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Flow of Yukawa and four fermion Flow of Yukawa and four fermion couplingcoupling
h2/32π
λ ψ /8π
(A ) broad Feshbach resonance(C) narrow Feshbach resonance
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Universality is due Universality is due to to
fixed points !fixed points !
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not not all all quantities are quantities are universal !universal !
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bare molecule fractionbare molecule fraction(fraction of microscopic closed channel (fraction of microscopic closed channel
molecules )molecules ) not all quantities are universalnot all quantities are universal bare molecule fraction involves wave bare molecule fraction involves wave
function renormalization that depends on function renormalization that depends on value of Yukawa couplingvalue of Yukawa coupling
B[G]
Experimental points byPartridge et al.
6Li
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conclusionsconclusions
the challenge of precision :the challenge of precision :
substantial theoretical progress substantial theoretical progress neededneeded
““phenomenology” has to identify phenomenology” has to identify quantities that are accessible to quantities that are accessible to precision both for experiment and precision both for experiment and theorytheory
dedicated experimental effort neededdedicated experimental effort needed
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challenges for challenges for experimentexperiment
study the simplest systemstudy the simplest system identify quantities that can be identify quantities that can be
measured with precision of a few measured with precision of a few percent and have clear theoretical percent and have clear theoretical interpretationinterpretation
precise thermometer that does not precise thermometer that does not destroy probedestroy probe
same for densitysame for density
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