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Particle-number conservation for pairing transition in finite systems

Particle-number conservation for pairing transition in finite systems. K. Kaneko Kyushu Sangyo University, Fukuoka, Japan. Collaborator:. A. Schiller Michigan State University, USA. Motivation. Question: Is the S-shape a signature of the breaking of nucleon Cooper pairs?.

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Particle-number conservation for pairing transition in finite systems

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  1. Particle-number conservation for pairing transition in finite systems K. Kaneko Kyushu Sangyo University, Fukuoka, Japan Collaborator: A. Schiller Michigan State University, USA

  2. Motivation Question: Is the S-shape a signature of the breaking of nucleon Cooper pairs? Particle-number Projection K. Esashika and K. Nakada PRC72, 044303(2005) BCS Theorypredicts Tc=0.57Δ～0.5MeV Oslo group A. Schiller et al., PRC63,021306(R)(2001). (3He,αγ）、(3He,3He’γ) ０．５ Ex

  3. Pairing transition at finite temperature • Pairing correlation is fundamental for many-fermion systems such as electrons in thesuperconducting metal, nucleons in nucleus, and quarks in the color superconductivity. • Infinite systems show superfluid-to-normal sharp phase transition, which is described by the BCS theory. • In finite systems, recent theoretical approaches demonstrate that thermal and quantum fluctuations are important. The BCS theory fails to describe the pairing transition.

  4. Static path approximation (SPA) with number projection • The SPA is a microscopic method for going beyond the mean-field approximation at finite temperature, which avoids the sharp phase transition. • The SPA is an efficient way compared with shell model calculations, and can be applied to heavy nuclei. • In finite systems such as nuclei, the SPA violates particle-number conservation. We need the particle-number projection in the SPA. • In this talk, I present the particle-number projection in the SPA, and report the numerical results and discussions.

  5. Out line • [I] Brief Review of the static path approximation and the particle-number projection • [II] Numerical results and discussions in the heat capacity and the pairing correlation • [III] Conclusion

  6. SPAin monopole pairing model

  7. ●Hubbard-Stratonovich transform No two-body interaction ●Static path approximation

  8. Δ(MeV) BCS Tc T (MeV) Effective mean-field equation The SPA avoids the sharp phase transition in the BCS equation.

  9. Thermal energy Δ2/G (MeV) T (MeV) Heat capacity Cv T (MeV) The S shape is closely related to the drastic decrease of pairing correlation.

  10. Particle-number projection

  11. Treatment of thermo field dyamics K. Tanabe and H. Nakada PRC71, 024314(2005)

  12. Numerical results in heat capacity and pairing correlation Parameters Model space

  13. ・The S shape appears in the SPA. ・The particle-number projection enhances the S shape. Heat Capacity

  14. ・The SPA result does not show the S shape. ・The number projection produces the S shape.

  15. ・The results are a similar to those of 94Mo. ・The number projection shows a more substantial increase compared with those of the heavier nuclei, 172Yb and 94Mo.

  16. Number Projection in no pairing phase transition S shape appears even though there is no pairing phase transition.

  17. ・The SPA curve for 172Yb drastically drops down around the temperature 0.5 MeV, but for the other lighter nuclei they decrease gradually. ・The particle-number projection makes it steeper than the slope of the SPA curve. Pairing gap

  18. Thermal odd-even mass difference Shifted thermal energy Partition function

  19. Shell model calculations sd-shell USD interaction 27,28,29Mg K. Kaneko and M. Hasegawa NPA740, 95(2004) Tc

  20. Derivative of thermal odd-even Mass difference with respect to temperature Odd-even difference of heat capacities The derivative of the thermal odd-even mass difference is identical with the odd-even difference of heat capacities. This peak shows odd-even difference of heat capacity corresponding to the S shape. Tc

  21. Thermal odd-even mass difference for the neutron R. Chankova et al., PRC73, 034311(2006). K. Kaneko et al., PRC74, 024325(2006).

  22. Conclusion • The particle-number projection affects the S shape of the heat capacity in all of the nuclei, 172Yb, 94Mo, and 56Fe. • In the heavy nucleus 172Yb, the particle-number projection enhances the S shape in the SPA, which is regarded as a fingerprint of pairing transition. • However, for the lighter nuclei 94Mo and 56Fe, the S shape appears only in the calculation with particle-number projection, but not in the SPA alone. • The effective pairing gap in 94Mo is in good agreement with experimental thermal odd-even mass difference, which is regarded as a direct measurement of pairing correlations at finite temperature.

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