fermion-boson rg for the ground- state of fermionic superfluids philipp strack, w. metzner, r....
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![Page 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](https://reader036.vdocuments.us/reader036/viewer/2022062305/56649cba5503460f94981d2f/html5/thumbnails/1.jpg)
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
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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
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3Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
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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)
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5Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
![Page 6: Fermion-Boson RG for the Ground- state of Fermionic Superfluids Philipp Strack, W. Metzner, R. Gersch Max-Planck-Institute for Solid State Research, Stuttgart](https://reader036.vdocuments.us/reader036/viewer/2022062305/56649cba5503460f94981d2f/html5/thumbnails/6.jpg)
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)
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7Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
![Page 8: Fermion-Boson RG for the Ground- state of Fermionic Superfluids Philipp Strack, W. Metzner, R. Gersch Max-Planck-Institute for Solid State Research, Stuttgart](https://reader036.vdocuments.us/reader036/viewer/2022062305/56649cba5503460f94981d2f/html5/thumbnails/8.jpg)
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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9Heidelberg, July 1st, 2008
Truncation functional RG
• Leads to truncated flow equations:
SYM: SSB:
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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:
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11Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
![Page 12: Fermion-Boson RG for the Ground- state of Fermionic Superfluids Philipp Strack, W. Metzner, R. Gersch Max-Planck-Institute for Solid State Research, Stuttgart](https://reader036.vdocuments.us/reader036/viewer/2022062305/56649cba5503460f94981d2f/html5/thumbnails/12.jpg)
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
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13Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
![Page 14: Fermion-Boson RG for the Ground- state of Fermionic Superfluids Philipp Strack, W. Metzner, R. Gersch Max-Planck-Institute for Solid State Research, Stuttgart](https://reader036.vdocuments.us/reader036/viewer/2022062305/56649cba5503460f94981d2f/html5/thumbnails/14.jpg)
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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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15Heidelberg, July 1st, 2008
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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16Heidelberg, July 1st, 2008
Numerical Results
• At critical scale, „boson splits into longitudinal and Goldstone mode“• Flow continuous across critical scale• IR scaling confirms analytical calculation
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17Heidelberg, July 1st, 2008
Agenda
History of Fermionic
Superfluids
SummaryAttractive Hubbard Model
Truncation functional
RG
Frequency Cutoff
Results
![Page 18: Fermion-Boson RG for the Ground- state of Fermionic Superfluids Philipp Strack, W. Metzner, R. Gersch Max-Planck-Institute for Solid State Research, Stuttgart](https://reader036.vdocuments.us/reader036/viewer/2022062305/56649cba5503460f94981d2f/html5/thumbnails/18.jpg)
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