Part A - Comments on the papers of Burovski Part A - Comments on the papers of Burovski et al.et al.

Part BPart B - - On Superfluid Properties of Asymmetric On Superfluid Properties of Asymmetric Dilute Fermi SystemsDilute Fermi Systems

Part APart A

Comments on papers of Comments on papers of

E. Burovski, N. Prokof’ev, B. Svistunov and M. TroyerE. Burovski, N. Prokof’ev, B. Svistunov and M. Troyer

- - Phys. Rev. Lett. Phys. Rev. Lett. 9696, 160402 (2006), 160402 (2006)- cond-mat/0605350 version 2, New Journal of Physics, in e-press, August (2006)- cond-mat/0605350 version 2, New Journal of Physics, in e-press, August (2006)

by A. Bulgac, J.E. Drut and P. Magierski by A. Bulgac, J.E. Drut and P. Magierski

Determinant Diagrammatic Monte CarloDeterminant Diagrammatic Monte Carlo

The partition function is expanded in a power series The partition function is expanded in a power series in the interactionin the interaction

It is notoriously known that the pairing gap is a non-analytical It is notoriously known that the pairing gap is a non-analytical function of the interaction strength and that no power expansion of function of the interaction strength and that no power expansion of pairing gap exists. It is completely unclear why an expansion of thispairing gap exists. It is completely unclear why an expansion of thistype should describe correctly the pairing properties of a Fermi type should describe correctly the pairing properties of a Fermi gas at unitarity.gas at unitarity.

Extrapolation prescription used by Burovski Extrapolation prescription used by Burovski et al.et al.

Argument based on comparing the continuum and lattice T-matrix at unitarity. Argument based on comparing the continuum and lattice T-matrix at unitarity.

However:However:

- T-matrix is governed by the scattering length, which is infinite at unitarityT-matrix is governed by the scattering length, which is infinite at unitarity- many Fermion system is governed by Fermi wave length, which is finite at unitaritymany Fermion system is governed by Fermi wave length, which is finite at unitarity- large error bars and clear non-linear dependence large error bars and clear non-linear dependence

Energy at critical temperatureEnergy at critical temperatureNotice the strong size dependence! Notice the strong size dependence!

Burovski Burovski et al.et al. Bulgac Bulgac et al.et al.

Burovski Burovski et al.et al.

Single particle kinetic energy and occupation probabilitiesSingle particle kinetic energy and occupation probabilities

- We have found that in order to have a reasonable accuracy the highest - We have found that in order to have a reasonable accuracy the highest momentum states should have an occupation probability of less than 0.01! momentum states should have an occupation probability of less than 0.01! -Notice the large difference, and the spread of values, between the kinetic energy Notice the large difference, and the spread of values, between the kinetic energy of the free particle and the kinetic energy in the Hubbard model, even at half the of the free particle and the kinetic energy in the Hubbard model, even at half the maximum momentum (one quarter of the maximum energy)maximum momentum (one quarter of the maximum energy)

Occupation probabilities are from our results, were we treat the kinetic energy exactly. Occupation probabilities are from our results, were we treat the kinetic energy exactly. Burovski Burovski et al.et al. have not considered them this quantity. Since both groups have similar have not considered them this quantity. Since both groups have similar filling factors, we expect large deviations from continuum limit. filling factors, we expect large deviations from continuum limit.

Finite size scalingFinite size scaling

Burovski Burovski et al.et al.

Bulgac Bulgac et al. et al. (new, preliminary results)(new, preliminary results)

0.157(7) c FT

F0.23(3) cT Value consistent with behaviorValue consistent with behaviorof other thermodynamic quantitiesof other thermodynamic quantities

These authors however never displayedThese authors however never displayed the order parameter as function of T andthe order parameter as function of T andwe have to assume that the phase transitionwe have to assume that the phase transitionreally exists in their simulation.really exists in their simulation.

Condensate fractionCondensate fraction

??Burovski Burovski et al.et al.

Preliminary new data!Preliminary new data!Finite size scaling consistent with Finite size scaling consistent with our previously determined valueour previously determined value

F0.23 cT

However, see next slideHowever, see next slide

Two-body correlation function, condensate fractionTwo-body correlation function, condensate fraction

System dependent scaleSystem dependent scaleHere the Fermi wavelengthHere the Fermi wavelength L/2L/2

Green symbols, T=0 results of Astrakharchick et al, PRL 95, 230405 (2005)Green symbols, T=0 results of Astrakharchick et al, PRL 95, 230405 (2005)

Power law critical scaling expected between the Fermi wavelength Power law critical scaling expected between the Fermi wavelength and L/2 !!! and L/2 !!! Clearly L is in all cases too small!Clearly L is in all cases too small!

Kinast et al. Science, Kinast et al. Science, 307307, 1296 (2005) , 1296 (2005)

The energy of a Fermi gas at unitarity in a trap at the critical The energy of a Fermi gas at unitarity in a trap at the critical temperature is determined experimentally, even though T is not.temperature is determined experimentally, even though T is not.

0.8 K inast et al, Science 307, 1296(2005) ( )

1 0.2 Burovski et al. cond-mat/ 0605350 (0)

0.6 Bulgac et al. PRL 96, 090404 (2006)

cE TE

estimatedestimated

Below the transition temperature the system behaves as a Below the transition temperature the system behaves as a free free condensedcondensedBose gas (!), which is superfluid at Bose gas (!), which is superfluid at the same time!the same time! No thermodynamic hintNo thermodynamic hint of ofFermionic degrees of freedom!Fermionic degrees of freedom!

Above the critical temperature one observes the thermodynamic behavior ofAbove the critical temperature one observes the thermodynamic behavior ofa free Fermi gas! a free Fermi gas! No thermodynamic trace of bosonic degrees of freedom!No thermodynamic trace of bosonic degrees of freedom!

New type of fermionic superfluid.New type of fermionic superfluid.

Our conclusions based on our resultsOur conclusions based on our results

Part B Part B

On Superfluid Properties of Asymmetric Dilute Fermi SystemsOn Superfluid Properties of Asymmetric Dilute Fermi Systems

Aurel Bulgac, Michael McNeil Forbes and Achim SchwenkAurel Bulgac, Michael McNeil Forbes and Achim Schwenk Department of Physics, University of WashingtonDepartment of Physics, University of Washington

Outline:Outline:

Induced Induced pp-wave superfluidity in asymmetric Fermi gases -wave superfluidity in asymmetric Fermi gases Bulgac, Forbes, and Schwenk, cond-mat/0602274, Phys. Rev. Lett. Bulgac, Forbes, and Schwenk, cond-mat/0602274, Phys. Rev. Lett. 9797, 020402 (2006), 020402 (2006)

T=0 thermodynamics in asymmetric Fermi gases at unitarity T=0 thermodynamics in asymmetric Fermi gases at unitarity Bulgac and Forbes, cond-mat/0606043Bulgac and Forbes, cond-mat/0606043

If the same solution as for 2

LOFF (1964) solutionLOFF (1964) solutionPairing gap becomes a spatially varying functionPairing gap becomes a spatially varying functionTranslational invariance brokenTranslational invariance broken

Muether and Sedrakian (2002)Muether and Sedrakian (2002)Translational invariant solutionTranslational invariant solutionRotational invariance brokenRotational invariance broken

Green – spin upGreen – spin upYellow – spin downYellow – spin down

Son and Stephanov, cond-mat/0507586Son and Stephanov, cond-mat/0507586Pao, Wu, and Yip, Pao, Wu, and Yip, PR B 73, 132506 (2006)

Sheeny and Radzihovsky, PRL Sheeny and Radzihovsky, PRL 9696, 060401(2006), 060401(2006)Parish, Marchetti, Lamacraft, SimonsParish, Marchetti, Lamacraft, Simonscond-mat/0605744cond-mat/0605744

Sedrakian, Mur-Petit, Polls, MuetherSedrakian, Mur-Petit, Polls, MuetherPhys. Rev. A 72, 013613 (2005)Phys. Rev. A 72, 013613 (2005)

Bulgac, Forbes, SchwenkBulgac, Forbes, Schwenk

What we predict?What we predict?Induced Induced pp-wave superfluidity in asymmetric Fermi gases-wave superfluidity in asymmetric Fermi gasesTwo new superfluid phases where before they were not expectedTwo new superfluid phases where before they were not expected

One Bose superfluid coexisting with one P-wave Fermi superfluidOne Bose superfluid coexisting with one P-wave Fermi superfluid

Two coexisting P-wave Fermi superfluidsTwo coexisting P-wave Fermi superfluids

BEC regimeBEC regime

all minority (spin-down)all minority (spin-down) fermions form dimers and the dimersfermions form dimers and the dimersorganize themselves in a Bose superfluidorganize themselves in a Bose superfluid

the leftover/un-paired majority (spin-up) fermions will form a the leftover/un-paired majority (spin-up) fermions will form a Fermi seaFermi sea

the leftover spin-up fermions and the dimers coexist and, the leftover spin-up fermions and the dimers coexist and, similarly to the electrons in a solid, the leftover spin-up fermionssimilarly to the electrons in a solid, the leftover spin-up fermionswill experience an attraction due to exchange of Bogoliubov will experience an attraction due to exchange of Bogoliubov phonons of the Bose superfluidphonons of the Bose superfluid

Bulgac, Bedaque, Fonseca, cond-mat/030602Bulgac, Bedaque, Fonseca, cond-mat/030602

p-wave gapp-wave gap

222

22 2

max

exp 0.44 , if 1

6exp , if ,

ln

5.6exp , if

fbp F F

f b

fb Fp F F

f b bfb F

p FF

nnk a

n n

nn kk a x

n n mck a x

k a

0.44 1fF

b

nk a

n !!!!!!

BCS regime:BCS regime:

The same mechanism works for the minority/spin-down component

2

2

2 2 22

4 2

2

2max

1exp exp

4

5 2 5 1 2( ) ln 1 ln

15 30 1 15

exp , for 0.77 and fixed 0.11 2

e

p F FF p F

F F pF

p

p F F F F

F

p F

N U kk k a L

k

z z z zL z z

z z z z

k k kk a

2

2

2

2

3xp

2 2 ln for

18exp

2

F

FFF

F FF

Fp F

FF

k

kkk a

k kk

k

kk a

What we think is going on:What we think is going on:

At unitarity the equation of state of a two-component fermion system is subject to At unitarity the equation of state of a two-component fermion system is subject to rather tight theoretical constraints, which lead to well defined predictions for therather tight theoretical constraints, which lead to well defined predictions for thespatial density profiles in traps and the grand canonical phase diagram is: spatial density profiles in traps and the grand canonical phase diagram is:

In the grand canonical ensemble there are only two dimensionfull quantitiesIn the grand canonical ensemble there are only two dimensionfull quantities

T=0 thermodynamics in asymmetric Fermi gases at unitarityT=0 thermodynamics in asymmetric Fermi gases at unitarity

5/ 3

5/ 2

1, 1

3, ( )

52 2

, ( ) , ,5 3

'( ) 1, ( )

( ) '( ) ( ) '( )

'( ) 1, ( )

( ) '( ) ( ) '( )

b b

a a

a b a

a b a a a b b a b a b

nx y

n

E n n n g x

P h y n n E n n E n n

g xy h y

g x xg x g x xg x

h yx g x

h y yh y h y yh y

We use both micro-canonical and grand canonical ensemblesWe use both micro-canonical and grand canonical ensembles

The functions g(x) and h(y) determine fully the thermodynamic properties The functions g(x) and h(y) determine fully the thermodynamic properties and only a few details are relevantand only a few details are relevant

Both g(x) and h(y) are convex functions of their argument.Both g(x) and h(y) are convex functions of their argument.

0

13/ 5

3/ 50 0 1

1 if

( ) 1 if y y ,1

(2 )

"( ) 0

1, (2 ) 1

c c

y y

h y y

h y

y Y y Y y

3/ 5

0 11

(0) 1, ( ) (2 )

"( ) 0

(1) (1)'(0) and '(1) ,

1 2

g g x

g x

g gg Y g

Y

Bounds given by GFMCBounds given by GFMC

Bounds from the energy required to Bounds from the energy required to add a single spin-down particle to a fullyadd a single spin-down particle to a fullypolarized Fermi sea of spin-up particlespolarized Fermi sea of spin-up particles

Non-trivial regions exist!Non-trivial regions exist!

Now put the system in a trapNow put the system in a trap

, ,

3/ 2

3/ 2

( )( ) ( ), ( )

( )2

( ) ( ) ( ( )) ( ( )) ( ) '( ( ))

( ) ( ) ( ( )) '( ( ))

bab ab

a

a b

a a

b a

rr V r y r

r

n r r h y r h y r y r h y r

n r r h y r h y r

• blue - P = 0 regionblue - P = 0 region• green - 0 < P < 1 regiongreen - 0 < P < 1 region• red - P= 1 regionred - P= 1 region

Zweirlein Zweirlein et al.et al. cond-mat/0605258 cond-mat/0605258

SuperfluidSuperfluid

NormalNormal

Column densities (experiment)Column densities (experiment)

Experimental data from Zwierlein Experimental data from Zwierlein et al. et al. cond-mat/0605258cond-mat/0605258

2 21 0

2 20 1

10.70(5)

1vac

vac

y R Ry R R

Main conclusions:Main conclusions:

• At T=0 a two component fermion system is always superfluid, irrespective of the At T=0 a two component fermion system is always superfluid, irrespective of the imbalance and a number of unusual phases should exists.imbalance and a number of unusual phases should exists.

• At T=0 and unitarity an asymmetric Fermi gas has non-trivial partially polarized phasesAt T=0 and unitarity an asymmetric Fermi gas has non-trivial partially polarized phases