Superfluidity and magnetism in multicomponent ultracold fermions

1
Superfluidity and magnetism in multicomponent
ultracold fermions

• Robert Cherng - Harvard
• Gil Refael - Caltech
• Eugene Demler - Harvard

arXiv0705.0347
2
Fermionic superfluidity and magnetization?

• Superfluidity pairing of different states
• Magnetization imbalance of different states

http//picturethis.pnl.gov/picturet.nsf/All/44CT2Y
?opendocument
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Fermionic superfluidity and magnetization?

• Ultracold atoms tunable model system
• Coexistence? Hard with two components, natural
  with three or more

Zwierlein et al., Nature 435, 1047 (2005)
4
Outline

• Two components BCS theory and beyond
• Two components ultracold atoms
• Classifying mean-field states WT identities
• Mean-field global phase diagrams
• Experimental signatures
• RG and fluctuations (preliminary)

5
BCS Theory 50th Anniversary
http//www.brown.edu/Departments/Physics/50YearsBC
S/
Phys. Rev. 108, 1175 (1957) Nobel Prize 1972
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BCS, FFLO, BP/Sarma

FS
FS
7
BCS, FFLO, BP/Sarma


FS
FS
BCS
8
BCS, FFLO, BP/Sarma


FS
FS
BCS

FS
FS

FS
FS
9
BCS, FFLO, BP/Sarma


FS
FS
BCS


FS
FS
FFLO

FS
FS
10
BCS, FFLO, BP/Sarma


FS
FS
BCS


FS
FS
FFLO


FS
FS
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BCS, FFLO, BP/Sarma

• Bardeen, Cooper, Schreiffer
• Phys. Rev. 108, 1175 (1957)

BCS

• Fulde-Ferrel, Larkin-Ovchinnikov
• Breaks translational symmetry
• Phys. Rev. 135, A550 (1964)
• ZETP 47, 1136 (1964)

FFLO

• Liu-Wilczek (Breached-Pair), Sarma
• Phase separation in k space
• Gapless quasiparticles
• PRL 90, 047002 (2003)
• J. Phys. Chem. Solids 24, 1029 (1963)

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Outline

• Two components BCS theory and beyond
• Two components ultracold atoms
• Classifying mean-field states WT identities
• Mean-field global phase diagrams
• Experimental signatures
• RG and fluctuations (preliminary)

13
Fermionic superfluids with cold atoms

• Isolated and dilute gases of alkali atoms
• Tunable interactions and populations
• Two component superfluids JILA, MIT, Innsbruck,
  Duke,

Zwierlein et al., Nature 435, 1047 (2005)
Regal et al., PRL 92, 040403 (2004)
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Three components using 6Li
6Li (I,L,S)(1,0,1/2)
E
B
PRL 94, 103201 (2005)
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Three or more components

• Each component individually conserved
• N densities na
• N(N-1)/2 scattering lengths aaß

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Outline

• Two components BCS theory and beyond
• Two components ultracold atoms
• Classifying mean-field states WT identities
• Mean-field global phase diagrams
• Experimental signatures
• RG and fluctuations (preliminary)

17
Pairing Three Components?
FS
FS
FS
Honerkamp and Hofstetter PRL 92, 170403 (2004)
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Pairing Three Components?
FS
FS
FS
FS
Honerkamp and Hofstetter PRL 92, 170403 (2004)
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Pairing Three Components?
FS
FS
FS
FS
FS
FS
Honerkamp and Hofstetter PRL 92, 170403 (2004)
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Model action
Imaginary Time Action
Coupling Constants
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Physical symmetries and symmetry breaking
Symmetric µaµ, ?aß?
Normal State
Superfluid State
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Field Redefinition Invariance
Start from
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Field Redefinition Invariance
Start from
Then redefine fields
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Field Redefinition Invariance
Start from
Then redefine fields
But remember to redefine coupling constants
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Field Redefinition Invariance
Start from
Then redefine fields
But remember to redefine coupling constants
Leaving Z invariant
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Field Redefinition Invariance
Start from
Then redefine fields
But remember to redefine coupling constants
Leaving Z invariant
Or infinitesimally (WT identity)
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Physical Interpretation
Rotation
Rotation
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Physical Interpretation
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Physical Interpretation

• Physics same under change of basis
• Dependence on explicit symmetry breaking not
  arbitrary

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Mean-field theory
Order Parameters
Gap Equations
Greens Functions
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Diagonal Pairing States
Solve WT Identity
By diagonalizing order parameters
And finding the eigenvectors
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Microscopic Pairing Wavefunctions
N2 P1
N3 P1
N4 P1
N4 P2
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Outline

• Two components BCS theory and beyond
• Two components ultracold atoms
• Classifying mean-field states WT identities
• Mean-field global phase diagrams
• Experimental signatures
• RG and fluctuations (preliminary)

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Ginzburg-Landau Free Energy
Expansion from U(N) symmetric superfluid
transition
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Ginzburg-Landau Free Energy
Coupling of magnetization and pairing
Quadratic symmetry breaking
Expansion from U(N) symmetric superfluid
transition
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Ginzburg-Landau Free Energy
Coupling of magnetization and pairing
Quadratic symmetry breaking
Particle-hole symmetric
Expansion from U(N) symmetric superfluid
transition
Particle-hole symmetry breaking
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N3 Phase Diagrams
TgtTcSYM Fixed µ
TltTcSYM Fixed µ
TgtTcSYM Fixed n
TltTcSYM Fixed n
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N4 Phase Diagrams
TgtTcSYM, fixed µ
TltTcSYM, fixed µ
Legend
Global minimum
1st meta- stable
?, f parameterize anisotropies in µ
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N4 Phase Diagrams
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Outline

• Two components BCS theory and beyond
• Two components ultracold atoms
• Classifying mean-field states WT identities
• Mean-field global phase diagrams
• Experimental signatures
• RG and fluctuations (preliminary)

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State Selective Imaging
BEC
BCS
Less Imbalance
More Imbalance
Science 311, 492 (2006)
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Phase-contrast Imaging
PRL 97, 030401 (2006)
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RF Spectroscopy
3
RF
1
2
Science 305, 1128 (2004)
BEC
BCS
Unitary
Higher T
Lower T
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RF Spectroscopy
3
RF
1
2
Science 305, 1128 (2004)
BEC
BCS
Unitary
Higher T
Normal
Lower T
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RF Spectroscopy
3
RF
1
2
Science 305, 1128 (2004)
BEC
BCS
Unitary
Higher T
Normal
Paired
Lower T
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Outline

• Two components BCS theory and beyond
• Two components ultracold atoms
• Classifying mean-field states WT identities
• Mean-field global phase diagrams
• Experimental signatures
• RG and fluctuations (preliminary)

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U(N) Symmetric Superfluid Transition
Ginzburg-Landau Action
Fields
Symmetric
Symmetry Breaking
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RG in e-Expansion
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Fixed Points and Stability

• Three couplings, eight fixed points
• For N3
• Eight physical FP
• One IR stable FP with ?M2, ?I, ?H e
• For Ngt3
• 4 physical FP (4 with imaginary couplings)
• No IR stable FP

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Conclusions

• Superfluidity drives magnetization for
  multicomponent fermions
• Classification of microscopic pairing
  wavefunctions via Ward-Takahashi identities
• Rich phase diagrams first/second order
  transitions, metastability/phase separation,
  multicritical

arXiv0705.0347
51
State Selective Imaging
Less Imbalance
Majority/Minority/Difference
Majority Species
Minority Species
Difference
Science 311, 503 (2006)
More Imbalance
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Three or more components

• Each component individually conserved
• N densities na, N(N-1)/2 scattering lengths aaß
• Caveats Nuclear spin flips, inelastic losses

Elastic collisions
N2
Dipole-dipole interaction
N3