27 May 1/N expansion for strongly correlated quantum Fermi gas and its application to quark matter Hiroaki Abuki (Tokyo University of Science) Tomas Brauner (Frankfurt University) Based on PRD78, (2008)

Outline  Introduction  Nonrelativistic Fermi gas Formulation Results  Dense relativistic Fermi gas Nambu-Jona Lasinio (NJL) description High density approximation Results  Summary

Introduction  Cold 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!! K 40, Li 6 atomic system in the laser trap Regal et al., Nature 424, 47 (2003): JILA grop Strecker et al., PRL91, (2003): Rice group Zwierlen et al., PRL91 (2003): MIT group Chin et al., Science 305, 1128 (2004): Austrian group …etc, etc…

 strong attraction weak attraction  BCS Naïve application of BCS leads power law blow up BEC Smooth crossover BCS/BEC: Eagles (1969), Leggett (1980) Nozieres & Schmitt-Rink (1985) Unitarity limit broken symmetry phase Unitary regime no small expansion parameter no reliable theoretical framework From: Regal, cond-mat/

Introduction  Nonperturbative, but universal thermodynamics at the unitarity  Theoretical challenges to describe such strongly correlated Fermi gas

Gas in Unitary limit: nonperturbative but with universality X 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  |a s ( 1 S 0 )| =18 fm X Non-perturbative information condenses in the universal parameter x 1. Green’s function Monte Carlo simulation: 2. Extrapolation of infinite ladder sum in the NSR split: 3. e-expansion around 4-space dimension: 4. Experiment: Nishida, Son, PRL97 (2006) : Next-to-leading order, x =0.475 H. Heiselberg, PRA 63, (‘01); T. Schafer et al, NPA762, 82 (‘05), x =0.32 Carlson-Chang-Pandharipande-Schmidt, PRL91, (’03), x =0.44(1) Astrakharchik-Boronat-Casulleras-Giorgini, PRL93, (’04), x =0.42(1) Bourdel et al., PRL91, (’03); x  0.7 but for T/T F > 0.5 and also in a finite trap the universal dimensionless constant

1/N expansion applied to Fermi gas  fluctuation effects are important!  systematic, controlled expansion possible when spin SU(2) generalized to SP(2N) X Nikolic, Sachidev, PRA75 (2007) (NS) 1. T C at unitarity X Veillette, Sheehy, Radzihovsky, PRA75 (2007) (VSR) 1. T C at unitarity 2. T=0, x parameter at and off the unitality

1/N expansion  In this work, X T c at and off the unitarity and analytic asymptotic behavior 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 expansion in 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 ordering of corrections beyond MFA

Pressure up to NLO (VSR)  Thermodynamic potential at NLO  At NLO, bosons contribute Anderson-Bogoliubov (phason), and Sigma mode (ampliton), they are mixed Fermion one loopBoson one loop D=  f 

Coupled equations to be solved  Equations that have to be solved:  For T=0  For T c  

Gapless-Conserving dichotomy  Self-consistent solutions to these coupled equations? … Dangerous! Violation of Goldstone theorem Universal artifact in common with “ conserving ” approximation (Luttinger-Ward, Kadanoff-Baym ’ s F -derivable): Well-known longstanding problem: Gapless-conserving dichotomy X Haussmann et al, PRA75 (2007) X Strinati and Pieri, Europphys. Lett (2005) X T. Kita, J. Phys. Soc. Jpn. 75, (2006)

The way to bypass the problem: order by order expansion  What to be solved is of type:  We also expand …  … to find solution order by order O(1): (MFA) O(1/N):

Order by order expansion  Detailed form of NLO equations … for T=0: for T c :

Relation to other approaches (1)  Nozieres-Schmitt-Rink theory 1/N correction to Thouless criterion missing Not really systematic expansion about MF: Solve the number equation in (m, T) non- perturbatively in 1/N 1/N (NLO) term in # eq. dominates in the strong coupling and recovers the BEC limit The phase diagram in (m, T) -plane unaffected: Only affects the equal density contours in the (m, T) -plane

Relation to other approaches (2)  Haussmann ’ s self-consistent theory besed on Luttinger-Ward formalism 1/N correction to thouless criterion included Solve the coupled equations self-consistently Leads several problems related to “ gapless- conserving dichotomy ” : LO pair propagator gets negative “ mass ” even above T c  Negative weight to partition function! X Haussmann et al, PRA75 (2007)

The results: Unitarity NSVSR 1/N corrections to (T C, m C ), formally equivalent, but they are large! Corrections are a bit smaller at T=0 T=0

The results: Off the unitarity at T=0 from VSR BEC BCS Monte Calro results at unitarity are located between MF(LO) and the NLO result 1/N corrections seem to work at least in the correct direction But the obtained value x=0.28 not satisfactory Monte Calro: Carlson et al, PRL91 (2003) x(MF)= (Leggett) x(MC)=0.44(1) ( Carlson ) x(1/N)=0.28 (VSR) MF : MC : /N : 0.49

Mid-Summary  Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion  T C useless at unitarity, even negative! Only qualitative conclusion, fluctuation lower T C  1/T C -based extrapolation yields T C /E F =0.14, close to MC result 0.152(7): E.Burovski et al., PRL96 (2006)  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 T C : Off the unitarity 1/N to b C (1/T C ) 1/N to T C NSR LO (MFA) T C reduced by a constant factor in the BCS limit! Chemical potential in the BCS limit governed by perturbative corrections: Reproduces second-order analytic formula c.f. Fetter, Walecka’s textbook 1 st 2 nd 0.218

Why 1/N reproduces perturbative m? g 0, O(N) g 2, O(1) g, O(1) g 2, O(1/N) (a) is LO in 1/N (b)(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 T C then? Weak coupling analytical evaluation possible in the deep BCS The BCS limit: k F a s  -0 Pair (fluctuation) propagator extremely sensitive to variation of m Singularity in  mD 2 W and slow convergence of m C to E F responsible!

1/N expansion in dense, relativistic Fermi system, Color superconductivity  Motivation 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 superconductivity  take NJL (4-Fermi) model Several species with equal mass, equal chemical potential qq pairing in total spin zero, Arbitrary color-flavor structure: Different fluctuation channels 2SC: CFL: 3 diquark “ flavor ” 9 diquark “ flavor ”

Economical way to introduce expansion parameter N possible?  What about extending N C =3 to N C =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 control This type of planer (ladder) graph will have growing power of N With # of loops!

No way but to introduce new “flavor”, taste of quarks  q  q i (i=1,2,3, …,N)  Lagrangian has SU(3) C  SO(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 T C  Only interested in shift of T C 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 alone Pair fluctuation becomes massless at T C

1/N expansion to inverse boson propagator, NLO Thouless criterion  Boson propagator at LO:  NLO correction to boson self energy PmPm fafa c pair : Cooperon P m =0 LO  O(N) NLO  O(1)  O(1) vertex:

 Information of color/flavor structure of pairing pattern condenses in simple algebraic factor N B /N F c d NLO correction to boson self energy ab flavor-structure of the graph gives

 Information of flavor structure of pairing pattern condenses in simple algebraic factor N B /N F  NLO fluctuation effect in CFL is twice as large as 2SC  Mean field Tc ’ s split at NLO Pairing pattern dependent algebraic factor pairingNBNB NFNF N B /N F “ BCS ” 111 2SC361/2 CFL991

 NLO 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 D 0 or mean field T c (0) High density approximation

 In this framework  In the weak coupling limit T C (0) =0.567 D 0  Use T C (0) / m as parameter for coupling strength  gives 1/N correction to Tc, final result

Numerical results for universal function T C (0) / m f NLO Fluctuation suppresses T C significantly Suppression of order of 30% at phenomenologically interesting coupling strength weak strong

Implication to QCD phase diagram  Suppression of T C is phase dependent: CFL T C is more suppressed than 2SC one  Schematic phase diagram: There is quantum-fluctuation driven 2SC window even if Ms=0 is assumed. Suppression of Tc is order of 10% : Non-negligible

Summary  General remarks on 1/N expansion Perturbative extrapolation based on MF values of D, m, T, … Avoids problems with self-consistency, technically very easy Only 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 matter Fluctuation corrections non-negligible Different suppressions in T C according to pairing pattern  competition of various phases Improvement necessary: Fermi surface mismatch, Color neutrality, etc. Generalization below the critical temperature Application to pion superfluid, # of color is useful