Copenhagen, June 15, 2006
Unitary Polarized Fermi Gases
Erich J. Mueller
Cornell University
Sourish Basu
Theja DeSilva
NSF, Sloan, CCMR
Outline:
•Interesting Questions
•What goes wrong with mean field theory and LDA
•Speculations about the phase diagram
•Collective modes [no time, but ask me about them]
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 2 of 18
Questions
1. Nature of normal fluid at zero T at unitarity?
2. Existence of exotic phases?a) Modulated Order parameters [FFLO]
b) Deformed Fermi surfaces
c) Polarized s-wave superfluids [Sarma, Breeched gap]
3. Properties?a) Equation of state
b) How to probe?
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 3 of 18
Questions
1. Nature of normal fluid at zero T at unitarity?
2. Existence of exotic phases?a) Modulated Order parameters [FFLO]
b) Deformed Fermi surfaces
c) Polarized s-wave superfluids [Sarma, Breeched gap]
3. Properties?a) Equation of state
b) How to probe?
Simplest assumptions (ex. Fermi-liquid) -- calculate properties
Controlled approximations: Limit n=0
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 4 of 18
Questions
1. Nature of normal fluid at unitarity?
2. Existence of exotic phases?a) Modulated Order parameters [FFLO]
b) Deformed Fermi surfaces
c) Polarized s-wave superfluids [Sarma, Breeched gap]
3. Properties?a) Equation of state
b) How to probe?
Homogeneous:•Mean Field Phase Diagram•Fluctuations and Stability•Scaling Arguments
Trap:•Surface tension•Collective modes
How to explain experiments
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 5 of 18
Summary
• Hulet’s “wings” can be explained by surface tension in the normal-superfluid interface
• Hulet’s low polarization data is a mystery
• Preliminary theoretical studies indicate:Partially polarized normal phase at unitarityMay be partially polarized superfluid phase at unitarity
Probably breaks translation/rotation symmetry [eg. FFLO]
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 6 of 18
Homogeneous Phases
Mean Field Theory:
a>0 a>0 [near resonance] a<0
Red: 1st order transition Normal-FFLO: Continuous SF-FFLO: Discontinuous(Schematic)
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 7 of 18
Application to TrapLDA
P. Pieri, and G.C. Strinati, PRL 96, 150404 (2006) ; W. Yi, and L. -M. Duan, cond-mat/0604558; M. Haque and H.T.C. Stoof, cond-mat/0601321; Zheng-Cheng Gu, Geoff Warner and Fei Zhou, cond-mat/0603091 ; C.-H. Pao, S.-K. Yip, J. Phys.: Condens. Matter 18 (2006) 5567;Theja N. De Silva, Erich J. Mueller, Phys. Rev. A 73, 051602(R) (2006)
BECBCS
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 8 of 18
Beyond Mean Field Theory + LDA• Discrepancies with experiments:
PData: Rice/MIT
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 9 of 18
LDA in harmonic trap:
Integrate:
Axial density should be monotonic [in LDA]
Beyond Mean Field Theory + LDA• Wings -- A violation of LDA
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 10 of 18
• Simple Explanation:
Beyond Mean Field Theory + LDA
Large Aspect Ratio Trap:LDA:
Surface Tension:
z
Unitarity constrains form:
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 11 of 18
• Calculating :Beyond Mean Field Theory + LDA
x
E
Gradient expand quadratic term Take rest to be local, but go to all order in
MFT minimizes
Equivalent to approximate solution of BdG eqns.
Ansatz:
Find by minimizing E()
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 12 of 18
• Surface tension resultBeyond Mean Field Theory + LDA
=0.9 10-3 at unitarity
P= 0.14
P= 0.53
P= 0.72
Theja De-Silva and EJM, Cond-mat/0603068Data: Hulet
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 13 of 18
Beyond Mean Field Theory + LDA• Discrepancies with experiments:
PData: MIT
Experimentalist interpretation:Evidence of partially polarized normal phase
Crude theorist argument:Possibly existence of partially polarized superfluid phase
[Vortex experiments seem to not be consistent with this]
a>0 [near resonance]
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 14 of 18
Normal State at T=0
Start fully polarized:
Energy to add single spin down =
Ignore Pauli blocking [Leggett ~1999]
Include Pauli Blocking
=(0,0)= + + + …
(Generalization of Hartree to include beyond-Born scattering)
[Part of a systematic self-consistent theory -- but appears to be important bit]-- NSR+approximate self-consistency
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 15 of 18
Normal State at T=0
20 1.04
Thouless criterion:Normal state is unstable to pairing with q=0.6k [k=0.44k (n/n=0.09)]
1.18
Fully Polarized Normal [Exact]
1.885
Unpolarized Superfluid[Monte-Carlo]
?????
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 16 of 18
Experimental Clues
Chevy [cond-mat/0605751]Bulgac and Forbes [cond-mat/0606043]
LDA:
Upper bound from self-energy
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 17 of 18
Normal State at T=0
20
1.8851.04
Thouless criterion:Normal state is unstable to pairing with q=0.6k [k=0.44k
(n/n=0.09)]
1.18
Transition to unpolarized superfluid must be right of this line
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 18 of 18
Alternative Pictures
Gubbels, Romans and Stoof, cond-mat/0606330
Finds MIT data is consistent with Finite T + polarized superfluidExplains temp dep of Rice critical polarization
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 19 of 18
Differences
MIT• Resolves 3 shells• No distortion of aspect
ratio• No “critical polarization”
for phase separation– Compares lowest T data
• Superfluid-normal transition at P=0.7
Rice• Resolves 2 shells
– May be 3 shells• Interacting normal gas looks a
lot like superfluid
– May be in different regime [above tricritical point]
• Sees distortion of aspect ratio– Surface energies important:
Large aspect ratio + small particle number
• Observes “critical polarization”– Increases with increasing T
CopenhagenJune 15, 2006 Erich Mueller -- Polarized Fermions 20 of 18
Summary
• Surface Tension:– Another illustration of strong interactions in such
dilute gases
• Phase Diagram– Argument that there may be a polarized superfluid --
normal state is only stable for quite low polarizations.– Normal state instability is at finite q [FFLO?]
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