Copenhagen, June 15, 2006 Unitary Polarized Fermi Gases Erich J. Mueller Cornell University Sourish...
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Transcript of Copenhagen, June 15, 2006 Unitary Polarized Fermi Gases Erich J. Mueller Cornell University Sourish...
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?]