Strong-coupling properties of a superfuid Fermi gas and application - PowerPoint PPT Presentation

November 21-24 (2016), Neutron Star Matter (NSMAT2016), Tohoku Univ. Strong-coupling properties of a superfuid Fermi gas and application to neutron-star EOS Yoji Ohashi Department of Physics, Keio University, Japan Collaborators: P. van Wyk, H. Tajima, A. Ohnishi Introduction ultracold Fermi gas and neutron star Strong-coupling theory of a superfluid Fermi gas ground state properties in the BCS-BEC crossover region comparison with experiment ( 6 Li gas) challenge to neutron star EoS Summary

Importance of neutron star in “material science” pressure neutron proton quark nucleus electron quark liquid crystal atom gas neutron liquid Earth + neutron star = complete condensed matter physics 地球

Current possible approach by human beings theorists on the earth Equation of state (EoS) internal structure + TOV eq. “ Mass-radius (MR) ” relation Obserbavle! Obserbavle! experimentalists on the earth

Standard approach to “Neutron star EoS ” EoS in the intermediate density region Phase shift data of NN interaction Unitarity Weak Coupling - - - S. Gandolfi et al, Eur. Phys. J. A 50 (2014) effective interaction potential with Low Density High Density “32” fitting parameters (AV 18) FP(`81) : B.Friedman et al, Nucl. Phys, A 361 (1981) 502 APR(`98, AV18) : A. Akmal et al, Phys. Rev. C 58 (1998) QMC, variational calculation, …. GCRSW(`13, AV8) : S. Gandolfi et al, Eur. Phys. J. A 50 (2014)

Approach to neutron star interior from the earth many-body physics few-body physics Experimental support Theoretical challenge (phase shift data → AV18) No experimental support! neutron superfluid   NN 1 ( k a ) ~ 0.0 5 1! F s unitary regime Experimental support Theoretical challenge Ultracold Fermi gas

Ultracold Fermi gas as a testing ground for neutron star physics  9/ 2, 7 / 2 40 K superfluid Fermi gas  9/ 2, 9/ 2 1 S 0 pairing state Fermi atom    9/ 2, 7 / 2 9/ 2, 9/ 2 molecule   N  5 T 0.35 K 10 F Feshbach-induced tunable pairing interaction C. A. Regal, et al. PRL 92 (2004) 040403.

Phase diagram of an ultracold Fermi gas “BCS - BEC crossover” weak-coupling strong-coupling (BCS) (BEC) Fermi wave-length s-wave scattering length

Phase diagram of an ultracold Fermi gas “BCS - BEC crossover” BCS BEC neutron star T ~ 0 Fermi wave-length s-wave scattering length Fermi gas tunable    NN tunable neutron star a 18.5 fm 0 s

EoS observed in a superfluid 6 Li Fermi gas (Horikoshi 2015) internal energy BCS-Leggett (mean field theory) ! 6 Li (experiment) T  0 3  N  E FG F 5 Usually, it is believed that the mean-field-based BCS-Leggett theory is valid for the BCS-BEC crossover at T=0. However, even at T=0 (where thermal fluctuations are absent), strong-coupling corrections are actually important.

Crucial difference between cold Fermi gas and neutron star The magnitude of effective range r e is VERY different between the two systems. ultracold Fermi gas neutron star p r  3 p r ~ 0 ~ F e F e r  ( 2.7fm) e We need to theoretically include a finite effective range r e , to approach neutron star EoS starting from cold Fermi gas physics.

Today’s talk: our strategy We first construct a reliable strong-coupling theory which can quantitatively explain the observed EoS in a superfluid 6 Li Fermi gas. We then extend this theory to include a finite effective range ( r e =2.7fm), to see to what extent we can discuss the neutron star EoS in the low density region by this approach. Ultracold Fermi gas ③ r  0 ② e + r  2.7fm e 6 Li experiment theory Neutron star EoS Strong-coupling ① theory with r e =0

Ground state properties of a superfluid Fermi gas in the unitary regime Effective range r  0 e

Model superfluid Fermi gas BCS Hamiltonian under the Nambu representation U                         † H ( ) ( ) q ( q ) ( ) q ( q )   p p 3 1 p 1 1 2 2 2 p q   c  p       ( : pseudospins describing atomic hyperfine states) : Nambu field ,   † p c     p  : superfluid order parameter  4 a U     U s : tunable s-wave pairing interaction   2 m 1 U /(2 ) p p p        † ( ) q : generalized density operator   j p p q /2 j p q /2 p 2 1        r 0 p e c p  p 2 1 ( p / p ) c c

Inclusion of strong coupling corrections beyond mean-field BCS theory normal phase ( T > T c ): “pairing” fluctuations -U superfluid phase ( T < T c ): fluctuations of “Δ” U                         † H ( ) ( ) q ( q ) ( ) q ( q )   p p 3 1 p 1 1 2 2 2 p q phase fluc. amplitude fluc. (2) phase fluctuations       e   ˆ i    1 1 1 2        2 1 2 2 ˆ ˆ ˆ (1)-(2) coupling    = 0 0 0 -U (1) amplitude fluctuations

Construction of Gaussian fluctuation (NSR) theory below Tc Thermodynamic potential: W  W MF W fluc Ω fluc Π 𝑗𝑘   1 q q  ˆ ˆ         Tr G p ( ) G p ( )    ij p i p j   2 2 Number Equation p 1 ˆ ( )  G p i          𝑂 = 𝑂 MF − 𝜖Ω fluc ( ) p 3 1 n 𝜖𝜈 𝑊,𝑈 Internal Energy Solve Δ and 𝜈 𝐹 = 𝐹 𝑁𝐺 + Ω fluc − 𝑈 𝜖Ω fluc − 𝜈 𝜖Ω fluc 𝜖𝑈 𝜖𝜈 𝑊,𝜈 𝑊,𝑈 Gap Equation    2 tanh( / 2 ) E T 4 a  1      p  s 1  p   m 2 E 2       p 2 2 p p E p

Self- consistent solutions for Δ and μ in the crossover region Fermi chemical potential superfluid order parameter r  r  0(coldatom) 0(coldatom) e e     F F T BEC T c BEC  1 T / T ( k F a )  1 ( k F a ) F s T BCS S F BCS T c BEC BCS

EoS: Superfluid Fermi gas BCS-Leggett Internal energy our result: NSR Exp.: 6 Li (Horikoshi) T  0 weak coupling unitarity Inclusion of superfluid fluctuations is crucial for the quantitavive evaluation of the internal energy in the unirary regime ((k F a s ) -1 <<0), even at T =0.

Diagrammatic representation of NSR theory and its extension Green’s function to reproduce the NSR results (T >T c )    G G G G 1  G NSR 0 0 0 i    0 n p self-energy describing pairing fluctuations G 0   G G -U = T-matrix approximation (TMA) 1         G G G G G G G ... G    TMA 0 0 0 0 0 0 1 0 Extended T-matrix approximation (ETMA) 1    G i      G ETMA n p

EoS: Superfluid Fermi gas BCS-Leggett Internal energy our result: NSR Exp.: 6 Li (Horikoshi) our result: ETMA T  0 weak coupling unitarity Inclusion of superfluid fluctuations is crucial for internal energy E in the unirary regime ((k F a s ) -1 <<0), even at T =0.

Compressibility k in a superfluid Fermi gas at T=0 our result: ETMA Isothermal compressibility Exp.: 6 Li (Horikoshi) BCS-Leggett T  0 weak coupling unitarity Superfluid fluctuations enhances the compressibility at T =0.

Strong-coupling effects: quantum depletion Some Cooper pairs are kicked out from the condensate, because of this repulsive interaction between them. E q q q 0 Non-condensed Cooper- The binding energy of non- pair molecules enhance the condensed pairs lowers the The BCS theory bosonic character of the internal energy. system, leading to k ↑ .

Application to neutron star EoS in the low density region Finite effective range r  2.7fm e

Two key effects of effective range r e (1) interaction “window”      † † H U c c c c           int p p ' p q /2, p q /2, p ' q /2, p ' q /2, p p , ', q 1 2 2       1 p 0.74fm p  c 2 r 2.7 1 ( p / p ) e c p p p F S-wave pairing The s-wave superfluid phase is suppressed when k F >1 fm -1 . interaction p ~ F 0 0 Low density high density

Two key effects of effective range r e (2) Hartree energy E Hatree      † † H U c c c c E Un n         int p ' q , p ', p q , p , Hartree p p q , ', The Hartree term actually vanishes identically, when r e =0 ( p c = ∞ ), because  4 a U 1       s U 0   p 1 m  p 1 m p  p c c 1 U    2 4 a 2 p s p divergence when p c = ∞ This is, however, not the case when r e >0, because then - U becomes finite. Including these two keys within the framework of NSR, we evaluate the neutron star EoS in the low density region (where the s-wave interaction is dominant) .