1/N expansion for strongly correlated quantum Fermi gasand its application to quark matterHiroaki Abuki(Tokyo University of Science)Tomas Brauner (Frankfurt University)

Based on PRD78, 125010 (2008)

27 May 2009, @Komaba

OutlineIntroduction

Nonrelativistic Fermi gas

FormulationResults

Dense relativistic Fermi gas

Nambu-Jona Lasinio (NJL) descriptionHigh density approximationResults

Summary

IntroductionCold atom system in the Feshbach resonance attracts renewed interests on the BCS/BEC crossover: Leggett(80), Nozieres Schmitt-Rink(85)

Interaction tunable via Magnetic field!!

K40 , Li6 atomic system in the laser trap

Regal et al., Nature 424, 47 (2003): JILA gropStrecker et al., PRL91, (2003): Rice groupZwierlen et al., PRL91 (2003): MIT groupChin et al., Science 305, 1128 (2004): Austrian groupetc, etc

+1 0 -1 strong attraction weak attraction

BCS

1957Nave application of BCSleads power law blow up

BEC

Smooth crossover BCS/BEC:Eagles (1969), Leggett (1980) Nozieres & Schmitt-Rink (1985)brokensymmetryphase

Unitaryregime

no smallexpansionparameter

no reliabletheoreticalframework

IntroductionNonperturbative, but universal thermodynamics at the unitarity

Theoretical challenges to describe such strongly correlated Fermi gas

Gas in Unitary limit: nonperturbative but with universalityX At T=0, thermodynamic quantities would have the form:X Universal, does not depend on microscopic details of the 2-body force ex. Cold atoms, Neutron gas with n-1/3 |as(1S0)| =18 fmX Non-perturbative information condenses in the universal parameter x Greens function Monte Carlo simulation:

Extrapolation of infinite ladder sum in the NSR split:

e-expansion around 4-space dimension:

Experiment:

Nishida, Son, PRL97 (2006) 050403: Next-to-leading order, x =0.475H. Heiselberg, PRA 63, 043606 (01); T. Schafer et al, NPA762, 82 (05), x =0.32Carlson-Chang-Pandharipande-Schmidt, PRL91, 050401 (03), x =0.44(1)Astrakharchik-Boronat-Casulleras-Giorgini, PRL93, 200404 (04), x =0.42(1)Bourdel et al., PRL91, 020402 (03); x 0.7 but for T/TF > 0.5 and also in a finite trap

1/N expansion applied to Fermi gasfluctuation effects are important!

systematic, controlled expansion possible when spin SU(2) generalized to SP(2N)

X Nikolic, Sachidev, PRA75 (2007) 033608 (NS)1. TC at unitarity

X Veillette, Sheehy, Radzihovsky, PRA75 (2007) 043614 (VSR)1. TC at unitarity2. T=0, x parameter at and off the unitality

1/N expansionIn this work,

X Tc at and off the unitarityand analytic asymptoticbehavior in the BCS limit

X Apply 1/N spirit to the relativistic fermion system, Possible impacts on QCD?

1/N expansion, philosophy (1)Euclidian lagrangian

Extend SU(2) Sp(2N) by introducing N copies of spin doublet: flavor

1/N expansion, philosophy (2)SU(2) singlet Cooper pair Sp(2N) singlet pairing field

No additional symmetry breaking, no unwanted NG bosons other than the Anderson-Bogoliubov associated with correct U(1) (total number) breaking

Counting by factor of N (1)Bosonized action

Enables us to perform formal expansionin 1/N

Each boson f-propagator contributes 1/N and fermion loop counts N from the trace factor

Equivalent to expansion in # of bosonic loops

Counting by factor of N (2)LO in 1/N equivalent to MFA

NLO in 1/N one boson loop corrections

At the end, we set N=1:

1/1 is not really small, but at least gives a systematic orderingof corrections beyond MFA

Pressure up to NLO (VSR)Thermodynamic potential at NLO

At NLO, bosons contributeAnderson-Bogoliubov (phason), andSigma mode (ampliton), they are mixedFermion one loopBoson one loopD=f

Coupled equations to be solvedEquations that have to be solved:

For T=0

For Tc

Gapless-Conserving dichotomySelf-consistent solutions to these coupled equations? Dangerous!

Violation of Goldstone theorem

Universal artifact in common with conserving approximation (Luttinger-Ward, Kadanoff-BaymsF-derivable): Well-known longstanding problem:

Gapless-conserving dichotomyX Haussmann et al, PRA75 (2007) 023610X Strinati and Pieri, Europphys. Lett. 71 359 (2005)X T. Kita, J. Phys. Soc. Jpn. 75, 044603 (2006)

The way to bypass the problem:order by order expansionWhat to be solved is of type:

We also expand

to find solution order by order

O(1): (MFA)

O(1/N):

Order by order expansionDetailed form of NLO equations

for T=0:

for Tc:

Relation to other approaches (1)Nozieres-Schmitt-Rink theory

1/N correction to Thouless criterion missingNot really systematic expansion about MF:Solve the number equation in (m, T) non-perturbatively in 1/N1/N (NLO) term in # eq. dominates in the strong coupling and recovers the BEC limitThe phase diagram in (m, T)-plane unaffected: Only affects the equal density contours in the (m, T)-plane

Relation to other approaches (2)Haussmanns self-consistent theorybesed on Luttinger-Ward formalism

1/N correction to thouless criterion includedSolve the coupled equations self-consistently Leads several problems related to gapless-conserving dichotomy: LO pair propagator gets negative mass even above Tc Negative weight to partition function!X Haussmann et al, PRA75 (2007) 023610

The results: UnitarityNSVSR

1/N corrections to (TC, mC), formally equivalent,but they are large!Corrections are a bit smaller at T=0T=0

The results: Off the unitarity at T=0 from VSR

BECBCSMonte Calro results atunitarity are locatedbetween MF(LO) andthe NLO result

1/N corrections seemto work at leastin the correct direction

But the obtained valuex=0.28 not satisfactoryMonte Calro:Carlson et al, PRL91 (2003)x(MF)=0.5906 (Leggett)x(MC)=0.44(1) (Carlson)x(1/N)=0.28 (VSR)MF : 0.6864MC : 0.541/N : 0.49

Mid-SummaryExtrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion

TC useless at unitarity, even negative!Only qualitative conclusion, fluctuation lower TC

1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7): E.Burovski et al., PRL96 (2006) 160402

b is natural parameter? Needs convincing justification!

Expansion about MF fails in BEC

We may, however, expect that 1/N expansion still gives useful prediction in the BCS region

Result for TC : Off the unitarity1/N to bC (1/TC)1/N to TCNSRLO (MFA) TC reduced by a constant factor in the BCS limit! Chemical potential in the BCS limit governed by perturbative corrections: Reproduces second-order analytic formulac.f. Fetter, Waleckas textbook1st 2nd 0.218

Why 1/N reproduces perturbative m? g0, O(N)g2, O(1)g, O(1)g2, O(1/N) is LO in 1/N(c) included in RPA (NLO in 1/N)(d) is NNLO not included here, but this is zero

What is the origin of asymptotic offset in TC then?Weak coupling analytical evaluation possible in the deep BCSThe BCS limit:kFas -0

Pair (fluctuation)propagator extremelysensitive to variationof mSingularity in mD2W and slow convergence of mC to EF responsible!

1/N expansion in dense, relativistic Fermi system, Color superconductivityMotivation

What is the impact of pair fluctuation on (m, T)-phase diagram?

In the NSR scheme, only the (m, r)-relation gets modified: No change in (m, T)-phase diagram

see, Nishida-Abuki, PRD (05), Abuki, NPA (07)

Are fluctuation effects different for several pairing patterns?

1/N expansion in dense, relativistic Fermi system: Color superconductivitytake NJL (4-Fermi) model

Several species with equal mass, equal chemical potentialqq pairing in total spin zero,Arbitrary color-flavor structure:Different fluctuation channels

2SC:

CFL:3 diquark flavor9 diquark flavor

Economical way to introduce expansion parameter N possible? What about extending NC=3 to NC=N?

However, diquark is not color singlet Full RPA series not resummed at any finite order in 1/N unless coupling O(1)

If coupling scales as O(1), the expansion in 1/N will not be under controlThis type of planer (ladder) graphwill have growing power of NWith # of loops!

No way but to introduce new flavor, taste of quarks q qi (i=1,2,3,,N)

Lagrangian has SU(3)CSO(N)(flavor group)

Assume SO(N)-singlet Cooper pair, then

No unwanted NG bosons other than AB mode

We make a systematic expansion in 1/N and set N=1 at the end of calculation: Expansion in bosonic loops:Construction is general, can be applied to any pattern of Cooper pairing

1/N expansion to shift of TC Only interested in shift of TC in (m,T)-phase diagram

Not interested in (m, r)-relation here since the density can not be controlled: m is more fundamental quantity in equilibrium

Then consider Thouless criterion alonePair fluctuation becomes massless at TC

1/N expansion to inverse boson propagator, NLO Thouless criterion

Boson propagator at LO:

NLO correction to boson self energycpair : CooperonPm=0LO O(N) NLO O(1) O(1)vertex:

NLO correction to boson self energy

Information of color/flavor structure of pairing pattern condenses in simple algebraic factor NB/NFcdabflavor-structure of the graph gives

Pairing pattern dependent algebraic factorInformation of flavor structure of pairing pattern condenses in simple algebraic factor NB/NF

NLO fluctuation effect in CFL is twice as large as 2SCMean field Tcs split at NLO

pairingNBNFNB/NFBCS1112SC361/2CFL991

High density approximationNLO integral badly divergent

Then take advantage of HDET

In the far BCS region, the pairing and Fermi energy scales are well separated

Only degrees of freedom close to Fermi surface are relevant for pairing physics

We want to avoid interference with irrelevant scales, in particular all vacuum divergences

We can renormalize the bare coupling G in favor of mean field gap D0 or mean field Tc(0)

1/N correction to Tc, final resultIn this framework

In the weak coupling limit TC(0)=0.567D0Use TC(0)/m as parameter for coupling strength

gives

Numerical results for universal functionTC(0)/mfNLO Fluctuation suppressesTC significantly

Suppression of order of30% atphenomenologicallyinteresting couplingstrength weakstrong

Implication to QCD phase diagramSuppression of TC is phase dependent: CFL TC is more suppressed than 2SC one

Schematic phase diagram: There is quantum-fluctuationdriven 2SC window even ifMs=0 is assumed.

Suppression of Tc is orderof 10% : Non-negligible

SummaryGeneral remarks on 1/N expansionPerturbative extrapolation based on MF values of D, m, T, Avoids problems with self-consistency, technically very easyOnly reliable when the NLO corrections are small (in BCS, not in molecular BEC region)Efimov-like N-body (singlet) bound state can contribute? If yes, at which order of N?Color superconducting quark matterFluctuation corrections non-negligibleDifferent suppressions in TC according to pairing pattern competition of various phasesImprovement necessary: Fermi surface mismatch, Color neutrality, etc.Generalization below the critical temperatureApplication to pion superfluid, # of color is useful

NG theorem preserved order by order, Goldstone boson remains massless at any order of N