fermion-boson rg for the ground- state of fermionic superfluids
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Fermion-Boson RG for the Ground- state of Fermionic Superfluids
Philipp Strack, W. Metzner, R. GerschMax-Planck-Institute for Solid State Research, Stuttgart
(to appear in Phys. Rev. B)
Discussions: FOR 723, Heidelberg, Rome
Heidelberg, July 1st, 2008
2Heidelberg, July 1st, 2008
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
3Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
4Heidelberg, July 1st, 2008
(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)
5Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
6Heidelberg, July 1st, 2008
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)
7Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
8Heidelberg, July 1st, 2008
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
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Truncation functional RG
• Leads to truncated flow equations:
SYM: SSB:
10Heidelberg, July 1st, 2008
Truncation functional RG
• Total number of variables in SYM: 4• Total number of variables in SSB: 9
13 flow equations total
SSB:
11Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
12Heidelberg, July 1st, 2008
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
13Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
14Heidelberg, July 1st, 2008
Numerical Results
• Gap and order parameter split when increasing attraction• Vertex renormalization relatively weak• Gap is reduced by approx. Factor 4 compared to MFT.
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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.
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Numerical Results
• At critical scale, „boson splits into longitudinal and Goldstone mode“• Flow continuous across critical scale• IR scaling confirms analytical calculation
17Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
18Heidelberg, July 1st, 2008
Summary
Infrared/Universal quantities
Fermion gap reduction/connection
to MFTTechnical 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 ???
[email protected]• To be published in Phys. Rev. B, arXiv:0804.3994