Eiji Nakano, Dept. of Physics, National Taiwan University

1. Epsilon Expansion Approach for BEC-BCS Crossover J-W Chen+ EN (cond-mat/0610011) Eiji Nakano, Dept. of Physics, National Taiwan University • Outline: • Experimental and theoretical background • Epsilon expansion method at finite scattering length • Application to energy per particle • Summary and outlook

2. 1) Experimental and theoretical background Cold Trapped Atoms Source: C. Regal

3. Superfluidity of 2004 Closed channel Open channel Feshbach resonance:

4. Source: C. Regal Review: Scattering Length Binding energy:

5. BEC-BCS Crossover Changing a at will: Technique of Feshbach Resonance Source: C. Regal

6. Studies on Unitary Fermi gas: • Zero-rang interaction, • Infinite scattering length, • The only parameter akF goes to infinity • (no expansion parameter ) • Physical quantities become universal • (scaled by Fermion density). e.g., Usual diagrammatic method is not reliable. (There is no expansion parameters. )

7. QMC calculations: Astrakharchik. et al. (2004) Chang. et al. (2004)

8. Approach from different spatial dimensions, d>4 (1) Study at arbitrary dimension by Nussinov and Nussinov (cond-mat/0410597) N-body wave function and variational method Twod-body bound state. Its normalization diverges at Free Bose gas at

9. (2) Epsilon expansion at unitary point by Nishida and Son (cond-mat/0604500) (3) Pionless EFT for dilute nuclear matter, specific ladder diagram at d=gN=infinity, by T. Schaefer, C-W Kao, S. R. Cotanch, (cond-mat/0604500)

10. Epsilon Expansion • Computing in dim. • Expanding in • Setting (Nishida and Son)

11. In Unitary limit and at Region of akF>0 Free Bose Gas (approximately) Mean field gives exact solution. Fluctuation develops as one goes to lower dimension Non-trivial vacuum: the unitary Fermi gas If expansion coefficients of epsilon are convergent, extrapolation to d=3 might give reliable results, a la, Wilsonian epsilon approach.

12. 2) Epsilon expansion method at finite scattering length After Hubbard-Stratonovich transformation, Condensation and Bosonic fluctuation: 1) which is determined uniquely so as to make boson wave function be unit. Here we impose the scaling to boson chemical pot.: 2) so that reflecting free Bose gas.

13. Reorganization of Lagrangian: e.g.,

14. Effective Field Theory: Pole:

15. Around the unitary limit: Expansion in B (binding energy)

16. For instance, Chemical potential, Energy/particle, to next-to-leading order in epsilon and up to O(B) Steps to 1, 2, 3,

17. In the Unitary limit: In BEC limit: from large B expansion up to B^2, we find

18. In BCS limit: Since we can not expect that physics at d=4 is trivial as free Bose gas anymore, counting rules should be changed: And B serves as an effective Boson mass at region of akF<0. Mean-field is exponentially small Two-loop gives a slope. Comparable to result by K. Huang and C.N. Yang (1956)

19. Energy per particle relative to that of free gas:

20. Blow-up of around unitary limit:

21. Outlook • 1, Application to Nuclear matter (Neutron star) • 2, Investigation of finite range correction. 4) Summary and outlook • Summary We have extended the epsilon expansion method to finite scattering region. Result, Slope and curvature of E/A and Chemical pot., is in overall good agreement with QMC and other low energy theorems.

22. Why is 4d special? has a singularity at for ground state a free Bose gas