Universality in ultra-cold Universality in ultra-cold fermionic atom gasesfermionic atom gases

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

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

Feshbach resonanceFeshbach resonance

H.Stoof

scattering lengthscattering length

BCS

BEC

chemical potentialchemical potential

BEC

BCS

inverse scattering length

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

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

universalityuniversality

same curve for Li and K atoms ?

BCS

BEC

dilute dilutedense

T = 0

Quantum Monte Carlo

different methodsdifferent methods

who cares about details ?who cares about details ?

a theorists game …?a theorists game …?

RGMFT

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 )

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 , ….….

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

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 !

parametersparameters

detuning detuning νν(B)(B)

Yukawa or Feshbach coupling hYukawa or Feshbach coupling hφφ

fermionic actionfermionic action

equivalent fermionic action , in equivalent fermionic action , in general not localgeneral not local

scattering length ascattering length a

broad resonance : pointlike limitbroad resonance : pointlike limit large Feshbach couplinglarge Feshbach coupling

a= M λ/4π

parametersparameters

Yukawa or Feshbach coupling hYukawa or Feshbach coupling hφφ

scattering length ascattering length a

broad resonance :broad resonance : h hφφ drops outdrops out

concentration cconcentration c

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

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 )

analogy with analogy with universal critical exponentsuniversal critical exponents

only one relevant parameter :only one relevant parameter :

T - TT - Tcc

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 !

rescaled actionrescaled action

M drops outM drops out all quantities in units of kall quantities in units of kF F ifif

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 !

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

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

effective potentialeffective potential

minimum determines order parameterminimum determines order parameter

condensate fractioncondensate fraction

Ωc = 2 ρ0/n

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

renormalized fields and renormalized fields and couplingscouplings

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 )

resultsresults

fromfrom

functional functional renormalization grouprenormalization group

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 ) )

T = 0

gap parametergap parameter

BCS BEC

Δ

BCSfor gap

limitslimits

BCSfor gap

condensate fractionforbosons with scattering length0.9 a

temperature dependence of temperature dependence of condensatecondensate

second order phase transition

c -1 =1

c -1 =0

T/Tc

free BEC

universalcriticalbehavior

condensate fraction :condensate fraction :second order phase second order phase

transitiontransition

crossover phase diagramcrossover phase diagram

shift of BEC critical shift of BEC critical temperaturetemperature

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 ))

Flow of Yukawa couplingFlow of Yukawa coupling

T=0.5 , c=1k2k2

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 )

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

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

Universality is due Universality is due to to

fixed points !fixed points !

not not all all quantities are quantities are universal !universal !

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

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

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

end