Copenhagen, June 15, 2006 Unitary Polarized Fermi Gases Erich J. Mueller Cornell University Sourish Basu Theja DeSilva NSF, Sloan, CCMR Outline: Interesting

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?


Questions

1. Nature of normal fluid at zero T at unitarity?





b) How to probe?

Simplest assumptions (ex. Fermi-liquid) -- calculate properties

Controlled approximations: Limit n=0


Questions

1. Nature of normal fluid at unitarity?





b) How to probe?

Homogeneous:•Mean Field Phase Diagram•Fluctuations and Stability•Scaling Arguments

Trap:•Surface tension•Collective modes

How to explain experiments


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]


Homogeneous Phases

Mean Field Theory:

a>0 a>0 [near resonance] a<0

Red: 1st order transition Normal-FFLO: Continuous SF-FFLO: Discontinuous(Schematic)


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


Beyond Mean Field Theory + LDA• Discrepancies with experiments:

PData: Rice/MIT


LDA in harmonic trap:

Integrate:

Axial density should be monotonic [in LDA]

Beyond Mean Field Theory + LDA• Wings -- A violation of LDA


• Simple Explanation:

Beyond Mean Field Theory + LDA

Large Aspect Ratio Trap:LDA:

Surface Tension:

z

Unitarity constrains form:


• 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()


• 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


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]


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


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]

?????


Experimental Clues

Chevy [cond-mat/0605751]Bulgac and Forbes [cond-mat/0606043]

LDA:

Upper bound from self-energy


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


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


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


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?]

Documents

Copenhagen, June 15, 2006 Unitary Polarized Fermi Gases Erich J. Mueller Cornell University Sourish Basu Theja DeSilva NSF, Sloan, CCMR Outline: Interesting