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1/N expansion for strongly correlated quantum Fermi gas and its application to quark matter

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, 125010 (2008). Outline. Introduction Nonrelativistic Fermi gas Formulation Results

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1/N expansion for strongly correlated quantum Fermi gas and its application to quark matter

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  1. 1/N expansion for strongly correlated quantum Fermi gasand its application to quark matter Hiroaki Abuki (Tokyo University of Science) Tomas Brauner (Frankfurt University) Based on PRD78, 125010 (2008) 27 May 2009, @Komaba

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

  3. 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!! • K40 , Li6 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…

  4. From: Regal, cond-mat/0601054 1924 Unitarity limit Naïve application of BCS leads power law blow up Unitary regime no small expansion parameter no reliable theoretical framework BEC 1957 Smooth crossover BCS/BEC: Eagles (1969), Leggett (1980) Nozieres & Schmitt-Rink (1985) BCS broken symmetry phase +1 0 -1  strong attraction weak attraction

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

  6. the universal dimensionless constant 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|as(1S0)|=18 fm X Non-perturbative information condenses in the universal parameterx • Green’sfunction Monte Carlo simulation: • Extrapolationof infinite ladder sum in the NSR split: • e-expansionaround 4-space dimension: • Experiment: Carlson-Chang-Pandharipande-Schmidt, PRL91, 050401 (’03),x =0.44(1) Astrakharchik-Boronat-Casulleras-Giorgini, PRL93, 200404 (’04),x =0.42(1) H. Heiselberg, PRA 63, 043606 (‘01); T. Schafer et al, NPA762, 82 (‘05),x =0.32 Nishida, Son, PRL97 (2006) 050403: Next-to-leading order, x =0.475 Bourdel et al., PRL91, 020402 (’03); x0.7 but for T/TF > 0.5 and also in a finite trap

  7. 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) 033608 (NS) 1. TC at unitarity XVeillette, Sheehy, Radzihovsky, PRA75 (2007) 043614 (VSR) 1. TC at unitarity 2. T=0, x parameter at and off the unitality

  8. 1/N expansion • In this work, XTc 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?

  9. 1/N expansion, philosophy (1) • Euclidian lagrangian • Extend SU(2)  Sp(2N) by introducing N copies of spin doublet: “flavor”

  10. 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

  11. 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

  12. 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

  13. 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 loop Boson one loop D=f

  14. Coupled equations to be solved • Equations that have to be solved: • For T=0 • For Tc  

  15. 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) 023610 X Strinati and Pieri, Europphys. Lett. 71 359 (2005) X T. Kita, J. Phys. Soc. Jpn. 75, 044603 (2006)

  16. 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):

  17. Order by order expansion • Detailed form of NLO equations … for T=0: for Tc:

  18. 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

  19. 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 Tc  Negative weight to partition function! X Haussmann et al, PRA75 (2007) 023610

  20. The results: Unitarity NS VSR • 1/N corrections to (TC, mC), formally equivalent, but they are large! • Corrections are a bit smaller at T=0 T=0

  21. The results: Off the unitarity at T=0 from VSR 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 x(MF)=0.5906 (Leggett) x(MC)=0.44(1) (Carlson) x(1/N)=0.28 (VSR) MF : 0.6864 MC : 0.54 1/N : 0.49 Monte Calro: Carlson et al, PRL91 (2003) BCS BEC

  22. Mid-Summary • Extrapolation 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

  23. Result for TC : Off the unitarity 2nd LO (MFA) 1st NSR 0.218 1/N tobC(1/TC) 1/N to TC • TC 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

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

  25. What is the origin of asymptotic offset in TC then? Weak coupling analytical evaluation possible in the deep BCS The BCS limit: kFas -0 Pair (fluctuation) propagator extremely sensitive to variation of m Singularity in mD2Wand slow convergence of mC to EF responsible!

  26. 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?

  27. 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”

  28. 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 control This type of planer (ladder) graph will have growing power of N With # of loops!

  29. No way but to introduce new “flavor”, taste of quarks • q  qi (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

  30. 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 alone Pair fluctuation becomes massless at TC

  31. fa Pm 1/N expansion to inverse boson propagator, NLO Thouless criterion • Boson propagator at LO: • NLO correction to boson self energy LO  O(N) NLO  O(1) cpair : Cooperon vertex: Pm=0  O(1)

  32. NLO correction to boson self energy • Information of color/flavor structure of pairing pattern condenses in simple algebraic factor NB/NF flavor-structure of the graph gives c d a b

  33. Pairing pattern dependent algebraic factor • Information of flavor structure of pairing pattern condenses in simple algebraic factor NB/NF • NLO fluctuation effect in CFL is twice as large as 2SC • Mean field Tc’s split at NLO

  34. High density approximation • 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 D0 or mean field Tc(0)

  35. 1/N correction to Tc, final result • In this framework • In the weak coupling limit TC(0)=0.567D0 • Use TC(0)/m as parameter for coupling strength • gives

  36. Numerical results for universal function Fluctuation suppresses TC significantly Suppression of order of 30% at phenomenologically interesting coupling strength fNLO strong weak TC(0)/m

  37. Implication to QCD phase diagram • Suppression of TC is phase dependent: CFL TC 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

  38. 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 TC 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

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