1. Fermion-Boson RG for the Ground- state of Fermionic Superfluids Philipp Strack, W. Metzner, R. Gersch Max-Planck-Institute for Solid State Research, Stuttgart (to appear in Phys. Rev. B) Discussions: FOR 723, Heidelberg, Rome Heidelberg, July 1st, 2008

2. 2 Key Results Fermion gap reduction/connection to MFT –Included vertex corrections –Deviations to MFT transparent Infrared/universal quantities -Exact momentum scaling of longitudinal and Goldstone propagator present in any fermionic superfluid Structure of fermion- boson theory in phase with broken symmetry –Careful separation of Goldstone and longitudinal fluctuations coupled to fermions –Distinguished gap from order parameter Technical finesse –Frequency cutoff : diagonal for fermions and bosons –Smooth flow in the vicinity of critical scale –Judicious choice of fermion and boson cutoff

3. Heidelberg, July 1st, 2008 3 Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

4. Heidelberg, July 1st, 2008 4 (partial) History of Fermionic Superfluids Development of understanding for fermionic superfluids 1969 Eagles, 1980 Leggett BCS-BEC Crossover 1993 Randeria, functional integral 1957, BCS Theory 1970‘s, Kosterlitz- Thouless, Helium 1986 Nozieres, Schmitt- Rink, Crossover for lattice fermions/finite T 1997/2004, Castellani, di Castro: IR properties of Bose gas Since 2005 fRG efforts underway (Manchester, Heidelberg, Würzburg, Stuttgart)

5. Heidelberg, July 1st, 2008 5 Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

6. Heidelberg, July 1st, 2008 6 Attractive Hubbard Model as Prototype Attractively interacting lattice fermions Superfluid ground state for average fermion density per lattice site 0< n < 2 Consider quarter-filling (away from van-Hove filling) Experimentally realized in 3d optical lattice with cold fermion atoms (Zurich, Boston) Objectives: Compute effect of quantum fluctuations on: -Fermionic gap, vertex (non-universal) -Infrared behavior (universal)

7. Heidelberg, July 1st, 2008 7 Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

8. Heidelberg, July 1st, 2008 8 Truncation functional RG Replacing the four-fermion interaction with Hubbard-Stratonovich field in the BCS pairing channel: Exact flow equation in superfield formalism: Truncation inspired by MFT in symmetric (left) and superfluid (right) regime Baier, Bick, Wetterich 2004

9. Heidelberg, July 1st, 2008 9 Truncation functional RG Leads to truncated flow equations: SYM:SSB:

10. Heidelberg, July 1st, 2008 10 Truncation functional RG Total number of variables in SYM: 4 Total number of variables in SSB: 9 13 flow equations total SSB:

11. Heidelberg, July 1st, 2008 11 Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

12. Heidelberg, July 1st, 2008 12 Frequency Cutoff Implement frequency cutoff for both: fermions and bosons. „Diagonal“ for both particle species; point singularity at origin of frequency axis Equations free of regulator: Judicious choice of relative cutoff scales: Consistent treatment of IR sector by equating: SYM: SSB: 2d-integration to be performed numerically Results robust under cutoff changes. Göttingen, Aachen

13. Heidelberg, July 1st, 2008 13 Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

14. Heidelberg, July 1st, 2008 14 Numerical Results Gap and order parameter split when increasing attraction Vertex renormalization relatively weak Gap is reduced by approx. Factor 4 compared to MFT.

15. Heidelberg, July 1st, 2008 15 Analytical Results Functional RG flow delivers exact IR scaling: Exact results (proof in Castellani, et. al. PRB 69, 024513, 2004) Goldstone propagator remains unrenormalized as a result of Ward identities So far not emergent from functional RG. Numerical results confirm analytical calculation.

16. Heidelberg, July 1st, 2008 16 Numerical Results At critical scale, „boson splits into longitudinal and Goldstone mode“ Flow continuous across critical scale IR scaling confirms analytical calculation

17. Heidelberg, July 1st, 2008 17 Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

18. Heidelberg, July 1st, 2008 18 Summary Infrared/Universal quantities Fermion gapreduction/connection to MFT Technical finesse + Exact momentum scaling of longitudinal and Goldstone propagator present in any fermionic superfluid + Deviations to MFT transparent + Quantum fluctuations incorporated + Vertex corrections relatively weak + Distinguished gap from order parameter + Utilized T=0 frequency cutoff + Consistent choice of relative cutoff scale between fermions and bosons Outlook + Treat small T in Kosterlitz- Thouless + BEC-BCS crossover for lattice fermions + Goldstone mode with real frequencies (Kopietz, et. al.) Thank you for your attention! ??? Questions ??? p.strack@fkf.mpg.de To be published in Phys. Rev. B, arXiv:0804.3994