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Fermion-Boson RG for the Ground- state of Fermionic Superfluids

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. Key Results.

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Fermion-Boson RG for the Ground- state of Fermionic Superfluids

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  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. 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 • Technical finesse • Frequency cutoff : diagonal for fermions and bosons • Smooth flow in the vicinity of critical scale • Judicious choice of fermion and boson cutoff • 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

  3. Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

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

  5. Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

  6. 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) Attractive Hubbard Model as Prototype • Objectives: • Compute effect of quantum fluctuations on: • Fermionic gap, vertex (non-universal) • Infrared behavior (universal) • Experimentally realized in 3d optical lattice with cold fermion atoms (Zurich, Boston)

  7. Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

  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. Truncation functional RG • Leads to truncated flow equations: SYM: SSB:

  10. Truncation functional RG SSB: • Total number of variables in SYM: 4 • Total number of variables in SSB: 9 13 flow equations total

  11. Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

  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: Göttingen, Aachen • Judicious choice of relative cutoff scales: • Consistent treatment of IR sector by equating: SYM: • 2d-integration to be performed numerically • Results robust under cutoff changes. SSB:

  13. Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

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

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

  17. Agenda History of Fermionic Superfluids Summary Attractive Hubbard Model Truncation functional RG Frequency Cutoff Results

  18. Summary Infrared/Universal quantities Fermion gap reduction/connection to MFT Technical finesse Outlook + 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 + 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

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