Ground State and Vortex State Properties of Strongly Coupled Fermi Superfluids

Aurel Bulgac

Transparencies (both in ppt and pdf format) shall become shortly available at http://www.phys.washington.edu/~bulgac

Colaborators: Yongle Yu (for the most part), P.F. Bedaque, A.C. Fonseca

Why would one study vortices in neutral Fermi superfluids?

They are perhaps just about the only phenomenon in which one can have a true stable superflow!

What shall I cover in this talk?

• SLDA - Superfluid LDAA brief introduction into the extension of the Kohn-Shall LDA to superfluid

fermionic systems (A. Bulgac and Y. Yu, Phys. Rev. Lett. 88, 042504 (2002)Y. Yu and A. Bulgac, Phys. Rev. Lett. 90, 222501 (2003))

• Vortices in the crust of neutron stars

• Vortices in dilute superfluid Fermi gases and some related issues

• Density profiles of dilute normal and superfluid Fermi gases in traps

Density Functional Theory (DFT) Hohenberg and Kohn, 1964

Local Density Approximation (LDA) Kohn and Sham, 1965

Normal Fermi systems only!

)]([3 rrdEgsrρε∫= particle density

2

1

2

1

3

|)(v|)(

|)(v|)(

)](),([

∑

∑

∫

=

=

∇=

=

=

N

ii

N

ii

gs

rr

rr

rrrdE

rrr

rr

rr

τ

ρ

τρε

BABA

AAB

BBA

BBBi

iBB

AAAi

iAA

kjiijk

jiij

ii

NBNA

NB

NA

EEEE

UHE

VHE

UHUHE

VHVHE

VVTHrrrr

rrrrrr

+<+

+ΨΨ<

+ΨΨ<

+ΨΨ=Ψ+Ψ=

+ΨΨ=Ψ+Ψ=

+++=

Ψ≠Ψ⇒Ψ⇒Ψ

∑

∑

∑∑∑<<<

)Tr(||

)Tr(||

)Tr(||||

)Tr(||||

...),...,(),...,(

)(),...,()(),...,(

11

1

1

ρ

ρ

ρ

ρ

ρρ

rrrr

rrr

rrr

Nonsense!

Assume that there are two different many-body wave functions, corresponding to the same number particle density!

=

−+−∆

∆−+

=

∇==

=

∑

∑∑∫

)(v)(u

)(v)(u

))(()()()(

)(v)(u)(

|)(v|2)( ,|)(v|2)(

))(),(),((

*

*kk

2k

2k

3

rr

Err

rUTrrrUT

rrr

rrrr

rrrrdE

k

kk

k

k

k

kk

gs

r

r

r

r

rr

rr

rrr

rrrrr

rrr

λλ

ν

τρ

ντρε

LDA (KohnLDA (Kohn--Sham) for Sham) for superfluidsuperfluid fermifermi systemssystems((BogoliubovBogoliubov--de de GennesGennes equations)equations)

There is a little problem! The pairing field There is a little problem! The pairing field DD diverges.diverges.

MeanMean--field and pairing field are both local fields!field and pairing field are both local fields!(for sake of simplicity spin degrees of freedom are not shown)(for sake of simplicity spin degrees of freedom are not shown)

FD NVExp εω <<

−=∆

||1

10

−=≅ FF

kp

r h

radiusradius ofof interactioninteraction interinter--particleparticle separationseparation

01 rk

F

F

>>∆

≈εξ

coherence lengthcoherence lengthsize of the Cooper pairsize of the Cooper pair

Why would one consider a local pairing field?Why would one consider a local pairing field?

Because it makes sense physically!Because it makes sense physically!The treatment is so much simpler!The treatment is so much simpler!Our intuition is so much better also.Our intuition is so much better also.

),(),(21),(

1)()u(v),(

212121

2102k1

*k21

rrrrVrr

rrrrrr

kE

rrrrrr

rrrrrr

ν

ν

=∆

−∝= ∑

>

Nature of the problemNature of the problem

It is easier to show how this singularity appears in infinite homogeneous matter (BCS model)

2222

2

022

222

k2k

2k22

k

k2k

kkkk

)()sin(

2)(

2 ,

2 ,1vu ,

)(1

21v

)exp(u)(u ),exp(v)(v

δπν

δελελε

+−

∆=

=∆+==+

∆+−

−−=

⋅=⋅=

∫∞

Fkkk

krkrdkmr

mU

mk

rkirrkir

h

hr

h

rrrrrr

at small separations

Solution of the problem in the case of the homogeneous matter (Lee, Huang and Yang and others)

Gap equation

223

3

)(1

)2(21

∆+−−= ∫ λεπ

k

kdg

Lippmann-Schwinger equation (zero energy collision)

T = V + VGT k

kdga

mgεππ1

)2(21

4 3

3

2 ∫−=+−h

Now combine the two equations and the divergence is (magically) removed!

−∆+−

−= ∫kk

kda

mελεππ1

)(1

)2(21

4 223

3

2h

)()( 2121 rrgrrV rrrr−=− δ

How people deal with this problem in finite (nuclei) systems?

Introduce an explicit energy cut-off, which can vary from 5 MeV to 100 MeV (sometimes significantly higher) from the Fermi energy.Use a particle-particle interaction with a finite range, the most popular one being Gogny’s interaction.

Both approaches are in the final analysis equivalent in principle, as a potential with a finite range r0 provides a (smooth) cut-off at an energy Ec =ħ2/mr0

2

The argument that nuclear forces have a finite range is superfluous, because nuclear pairing is manifest at small energies and distances of the order of the coherence length, which is much smaller than nuclear radii.

Moreover, LDA works pretty well for the regular mean-field.

A similar argument fails as well in case of electrons, where the radius of the interaction is infinite and LDA is fine.

PseudoPseudo--potential approach potential approach (appropriate for very slow particles, very transparent(appropriate for very slow particles, very transparentbut somewhat difficult to improve)but somewhat difficult to improve)

Lenz (1927), Fermi (1931), Lenz (1927), Fermi (1931), BlattBlatt and and WeiskopfWeiskopf (1952)(1952)Lee, Huang and Yang (1957)Lee, Huang and Yang (1957)

[ ]

[ ] Brrrr

rBrAr

rrr

rgrrVkr

ikamagikkr

af

krOra

rfikr

rfrkir

RrrVrErrVrm

r

)()()( ...)( :Example

)()()()( then 1 if

...)1(

4 ,211

)(1... 1)exp()exp()(

if 0)( ),()()()(

0

22

01

2

rrrr

rrrr

h

rrr

rrrrrh r

δψδψ

ψδψ

π

ψ

ψψψ

=∂∂

⇒++=

∂∂

⇒<<

++

=−+−=

+−≈++≈+⋅=

>≈=+∆

−

−

How to deal with an inhomogeneous/finite system?

[ ][ ] ( ) ( )',',)(

0)()(

'2),',(lim),(

),(2

)()(2

)()()()(u)(v)(

2'

**

rrrrGrhrrh

rrmrrGrG

rGrrrrrrr

ii

rr

def

reg

regi i

iiii

def

reg

rrrrr

rr

rrh

rrr

rrrrr

rrr

rr

−=−=−

−+=

∆−

−

∆+=

→

∑

δλλψε

πλλ

λελψψν

There is complete freedom in choosing the Hamiltonian hand we are going to take advantage of this!

)(2

)( ,)(2

)(

)'O(2

)('2

'2)')(exp(

),',(

2222

22

2

ccF

F

F

ErUm

rkrUm

rk

rrmrikrr

mrr

rrrikmrrG

+=+=+

−+−−

−≈

−−

−=

λλ

ππ

πλ

rr

hrr

h

rr

h

r

rrh

rrh

rrrrr

We shall use a “Thomas-Fermi” approximation for the propagator G.

22

)(

0222

*reg 4

)()(

)(2

14

)()(u)(v)(νh

rr

rh

rrrr

r

πγλπmrkridkk

irUmk

rrrr Frk

EEii

def c

ci

∆+

+−−

∆+= ∫∑

≤

Regular part of GRegularized anomalous density

SLDA equations for SLDA equations for superfluidsuperfluid Fermi systems:Fermi systems:

Typo: replace m by m(r)

Energy Density (ED) describing the normal phaseAdditional contribution to ED due to superfluid correlations

Y.Yu and A. Bulgac, PRL 90, 222501 (2003)

Andreev reflection

Peculiarity of the finite systems:deep hole states are continuum states.

outside inside

=−+∆−

∆−−−

1001

22211211

)*(*)(

GGGG

hEhE

λλ

Integration contours used to construct the normal and anomalous densities from the Gorkov Green functions

Belyaev et al, Sov. J. Nucl. Phys. 45, 783 (1987).

C

Re ε

Im ε

× × ×

× × × −µ

εF ε

c

× × × × −µ ε

F ε

c Re ε

Im ε

α0 α

0

C

Even systems Odd systems

Let me backtrack a bit and summarize some of the SLDA ingredientLet me backtrack a bit and summarize some of the SLDA ingredients.s.

{ }

=

=

+= ∫

)](),(),(),([)](),(),(),([

)](),([)](),([

)](),(),(),([)](),([ 3

rrrrrrrr

rrrr

rrrrrrrdE

npnpSpnpnS

npNpnN

pnpnSpnNgs

rrrrrrrr

rrrr

rrrrrr

ννρρεννρρε

ρρερρε

ννρρερρε

EnergyEnergy Density (ED) describing the normal systemDensity (ED) describing the normal systemED contribution due to ED contribution due to superfluidsuperfluid correlationscorrelations

IsospinIsospin symmetry symmetry ((Coulomb energy and other relatively small terms not shown here.)Coulomb energy and other relatively small terms not shown here.)

[ ]

np

npnpnpnpS

gg

ggnpnp

ρρ

ννννννρρερρρρ

and on wellas depend could and

||||,,,

10

2

like

12

like

0 321321−+

−++=

Let us consider the simplest possible ED compatible with nuclearLet us consider the simplest possible ED compatible with nuclear symmetriessymmetriesand with the fact that nuclear pairing and with the fact that nuclear pairing corrrelationscorrrelations are relatively weak.are relatively weak.

In the end one finds that a suitable superfluidnuclear EDF has the following structure:

[ ]

),(),( and

),(),( where

]|||)[|,(

]|||)[|,(,

22

22

pnnp

pnnp

np

npnpnp

npnpnpS

ff

gg

f

g

ρρρρ

ρρρρ

ρρρρ

ννρρ

ννρρννε

=

=

+

−−+

+=

Isospin symmetric

Charge symmetric

Let us now remember that there are more neutron rich nuclei and let me estimatethe following quantity from all measured nuclear masses:

1473.0=−A

ZN

Conjecturing now that Goriely et al, Phys. Rev. C 66, 024326 (2002) have as amatter of fact replaced in the “true” pairing EDF the isospin density dependence simply by its average over all masses, one can easily extract from their pairing parameters the following relation:

[ ]

0 and 0390 where

]|||[|

]|||[|,

22

22

<>≈

+

−−+

+=

g g. -f

f

g

np

npnp

npnpS

ρρρρ

νν

ννννε

repulsion attraction

How can one determine the density dependence How can one determine the density dependence of the coupling constant g? I know two methods.of the coupling constant g? I know two methods.

In homogeneous low density matter one can compute the pairing gaIn homogeneous low density matter one can compute the pairing gap as a p as a function of the density. function of the density. NB this is not a BCS or HFB result!NB this is not a BCS or HFB result!

One compute also the energy of the normal and One compute also the energy of the normal and superfluidsuperfluid phases as a function phases as a function of density, as was recently done by of density, as was recently done by Carlson et al, Phys. Rev. Lett. Carlson et al, Phys. Rev. Lett. 9191, 050401 (2003), 050401 (2003)for a Fermi system interacting with an infinite scattering lengtfor a Fermi system interacting with an infinite scattering length (h (Bertsch’sBertsch’s MBXMBX1999 challenge) 1999 challenge)

=∆

akmk

e F

F

2exp

22 223/7 πh

In both cases one can extract from these results the In both cases one can extract from these results the superfluidsuperfluid contribution to thecontribution to theLDA energy density functional in a straight forward manner. LDA energy density functional in a straight forward manner.

What shall I cover in this talk?

• SLDA - Superfluid LDAA brief introduction into the extension of the Kohn-Shall LDA to superfluid

fermion systems

• Vortices in the crust of neutron stars(Y. Yu and A. Bulgac, Phys. Rev. Lett. 90, 161101 (2003))

• Vortices in dilute superfluid Fermi gases and some related issues

• Density profiles of dilute normal and superfluid Fermi gases in traps

Borrowed from http://www.lsw.uni-heidelberg.de/~mcamenzi/NS_Mass.html, author Dany Page

“meat balls”

“lasagna”

BCS

from Lombardo and Schulzeastro-ph/0012209

““Screening effects” are significant!Screening effects” are significant!

ss--wave pairing gap in infinitewave pairing gap in infiniteneutron matter with realisticneutron matter with realisticNNNN--interactionsinteractions

These are major effects beyond the naïve HFB when it comes to deThese are major effects beyond the naïve HFB when it comes to describingscribingpairing correlations.pairing correlations.

=∆

akmk

e F

F

2exp

22 223/7 πh

from Heiselberg et al Phys. Rev. Lett. 85, 2418, (2000)

An additional factor ofAn additional factor of 1/1/(4(4ee))1/3 1/3 ≈≈ 0.450.45 is due is due to induced interactionsto induced interactionsGorkovGorkov and and MelikMelik--BarkhudarovBarkhudarov in 1961.in 1961.

BCS/HFB in error even when the interaction BCS/HFB in error even when the interaction is very weak, is very weak, unlike HFunlike HF!!

Let us check a simple example, homogeneous dilute Fermi gas withLet us check a simple example, homogeneous dilute Fermi gas with a weaka weakattractive interaction, when pairing correlations occur in the gattractive interaction, when pairing correlations occur in the ground state.round state.

BCS resultBCS result

=∆

akmk

e F

F

2exp

28 22

2

πh

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

k [fm−1]

∆ [M

eV]

phase shifteffective rangeRG

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

k [fm−1]

∆/ε

F

phase shifteffective range

RG- renormalization group calculationSchwenk, Friman, Brown, Nucl.Phys. A713, 191 (2003)R

Landau criterion for superflow stability(flow without dissipation)

Consider a superfluid flowing in a pipe with velocity vs:

pNmpENmE p

p

r

rrr ε

ε <⇒+⋅++<+ s

2s

s0

2s

0 v2vv

2v

One single quasi-particle excitation with momentum p

Fkh∆

<svIn the case of a Fermi superfluid this condition becomes

no internal excitations

Vortex in neutron matter

( ) ( )m

rdrxymr

r

zrrirr

ikznirikznir

rr

C 2V

21 )0,,(

2V

pair)Cooper per (oneon quantizati sOnsager' vortex,Ideal

axissymmetry vortex - Oz s]coordinate cal[cyllindri ),,( ),exp()()(

integer-half -n ,])2/1(exp[)(v])2/1(exp[)(u

)(v)(u

v2v

kn

kn

hrrrhrr

h

rr

r

r

=⋅⇐−=

=∆=∆

−−−+

=

∫π

φφ

φφ

α

α

α

α

Y. Yu and A. Bulgac, PRL 90, 161101 (2003)

Fayans’s FaNDF0

−

=∆

)(tan2exp

22 223/7

F

F

kmk

e δπh

from Heiselberg et al Phys. Rev. Lett. 85, 2418, (2000)

An additional factor of 1/(4e)1/3

is due to induced interactionsAgain, HFB not valid.

Distances scale with lF

Distances scale with x>>lF

Y. Yu and A. Bulgac, PRL 90, 161101 (2003)

[ ]

[ ]

well.as affectedtly significan toexpected are properties transportheat, Specific

)()(L

E

by estimates, previous tocomparedwhen ly dramatical changed be togoing islength unit per energy The

)()(

:0energy pinning-anti large a toleads which thus

)()(region density lowIn

motion, calfast vortiextremely 0.12, 2

vv

2vortex

F

F

maxFF

S

•

−≈∆

•

−=

>

>•

∆∝

≈∆

≤•

R

VE

E

ininoutout

ininoutoutVpin

Vpin

ininoutout

πρρερρε

ρρερρε

ρρερρεεξ

λ

ε

Dramatic structural changes of the vortex state naturally lead Dramatic structural changes of the vortex state naturally lead to significant changes in the energy balance of a neutron starto significant changes in the energy balance of a neutron star

Some similar conclusions have been reached recently also by Donati and Pizzochero, Phys. Rev. Lett. 90, 211101 (2003).

What shall I cover in this talk?

• SLDA - Superfluid LDAA brief introduction into the extension of the Kohn-Shall LDA to superfluid

fermion systems

• Vortices in the crust of neutron stars

• Vortices in dilute superfluid Fermi gases and some related issues

(A. Bulgac and Y. Yu, Phys. Rev. Lett. 91, 190404 (2003))

• Density profiles of dilute normal and superfluid Fermi gases in traps

Regal and Jin Phys. Rev. Lett. 90, 230404 (2003)

Feshbach resonance

BSSVSSa

V

VPrVPrVVVpH

nzn

eze

Znehfhf

dZi

i

hfi

r

)(,

)()()(2

2

1100

2

1

2

γγ

µ

−=⋅=

+++++= ∑=

rr

h

rrr

Tiesinga, Verhaar, StoofPhys. Rev. A47, 4114 (1993)

Atomicseperation

Energy

Eres

Eth

Reagal, Ticknor, Bohm and Jin, Nature 424, 47 (2003)

a) Loss of atoms |9/2,-9/2> and |9/2,-5/2> as a function of final B. The initial value of B=227.81 G.

b) Scattering length between hyperfine states |9/2,-9/2> and |9/2,-5/2> as a functionof the magnetic field B.

Number of atoms after ramping B from 228.25 G to 216.15 (black dots) and for ramping B down (at 40 ms/G) and up at various rates (squares).

BCS →BEC crossover

If a<0 at T=0 a Fermi system is a BCS superfluid

Leggett (1980), Nozieres and Schmitt-Rink (1985), Randeria et al. (1993)

F

F

FF

F

F

kkak

akmk

e11 and 1|| iff ,

2exp

22 223/7

>>∆

=<<

≈∆

εξπh

( )FFs

F ON

EN

E λξεε =≈≈ and 5344.0 ,

5354.0 uperfluidnormal

If |a|=∞ and nr03á1 a Fermi system is strongly coupled and its properties

are universal. Carlson et al. PRL 91, 050401 (2003)

If a>0 (a≫r0) and na3á1 the system is a dilute BEC of tightly bound dimers

061.0 and 2

e wher,1 and 32

2

2 >==<<−= aan

nanma bb

fbb

hε

What do we know about dilute normal Fermi systems?

g – spin degeneracy, as - s-wave scattering length, ap – p-wave scattering length, rs – s-wave effective range

kinetic energy HF energy correlation energy

( )∑ −i

HFi

RPAi ωω

21

effective range corrections appears at this order

(For a recent review see Hammer and Furnstahl, Nucl. Phys. A678, 277 (2000))

Lessons:

The first type of correction to be accounted for in both Bose and Fermi systems is the Lee & Yang, Huang, Luttinger (1957) correlation energy, which is still determined by the scattering length.

Effective range corrections appear only much later.

More importantly, the corrections to mean-field are always controlled by the parameter na3

When the parameter na3 becomes large, other methods are required.

“Fundamental” and “effective” Hamiltonians

If one is interested in phenomena with momenta p = ħk << ħ/r0 , ,where r0 is the typical range of the interaction, the “fundamental” Hamiltonian is too complex.

Working with contact couplings requires regularization and renormalization,which can be done in several different, but equivalent ways.

We will show that Heff is over-determined.

Explicit introduction of “Feshbach molecules”

After introducing contact couplings

Example: open channel — two 85Rb atoms in f = 2, mf = -2 state eachclosed channel — two 85Rb atoms in f = 3 each and total Mf = -4

Lesson:

εαλλλ

2

−=⇒ aaeffaaaa

( )

(DR) 14

reg.) off-(cut 2

14

1|1

)()()()()()(

)()()()()()(

222222

00022

2100

012112122

1211

22221212

2

12121111

2

εαλππ

εαλπ

φφ

φη

φη

ψψψψ

ψψψψ

−=+

−=

=++−

+≈+−−

+⇒

=++∆−

=++∆−

aa

c

aa

k

kaa

ammk

am

EVT

ViEE

VVViVTE

VVV

rErrVrrVrm

rErrVrrVrm

hhh

rrrrrrh

rrrrrrh

Köhler, Gasenzer, Jullienne and Burnett, cond-mat/0305028.

Feshbach resonance

Atomicseperation

Energy

Eres

Eth

NB The size of the “Feshbach molecule”(closed channel state) is largely B-independentand smaller than the interparticle separation.

Some simple estimates in case a > 0 and a ≫ r0

[ ]

>>≈

<>

≈=>

≈+=<

≈≈

−

+=

<>

∫

∫∞

000

0

20

22

12

0

30

230

22

22

10

20

0210

01

00

23or 1

43

)()(

2)()(

oscillate if 3

or 3

2)()()(

:atoms twofind y toProbabilit

)()( exp1)(

rrregion in wfs rrat channelopen in wf

0

0

ra

ra

rrPrrP

arArdrrrrP

rArArrdrrrrP

Arrrrrar

arOArrr

r

r

ψ

ψψ

ψψψ

Most of the time the two atoms spend at large separations, y1(r) — open channel (dimer), y2(r) — closed channel (Feshbach molecule)

So far we discussed only interaction between atoms and one needs to include molecules.

If atoms and molecules coexist, it makes sense to introduce moleculesas independent degrees of freedom.

Previously various authors, starting with Timmermans et al. (1998) introduced explicitely the “Feshbach molecules” for several reasons:

There was hope to overcome the restriction na3 á 1 close to a Feshbach resonance, when |a|≫r0, and replace it hopefully with the milder condition nr0

3 á 1 and thus still be able to usethe many-body tools developed for dilute systems.

Develop a formalism for a mixture of atoms and molecules.

In order to decide whether a given program is feasible one has to construct the ground state properties of the system under consideration within the framework of the formalism of choice and then consider higher order corrections. This aspect was largely ignored in previous works.

It is relatively easy to convince oneself that corrections to the energy of a system of either Bose atoms and molecules, or Fermi atoms and Bosemolecules (bound state of two Fermi atoms) are always controlled by the parameter na3 and never by the parameter nr0

3.

(Essentially one has to repeat the old Lee, Young and Huang 1957calculations and compute the correlation energy.)

One can develop a theoretical framework to describe atoms and dimers(not Feshbach molecules) for the case a>0, a≫r0 and na3Ü1 and one canshow that in this regime a mixture of atoms and dimers can be describedby one coupling constant, the scattering length a.

The regime a > 0 and na3≫1 (strong coupling) can be studied as well, but using different methods (ab initio).

The regime a<0 and |a|≫r0, n|a|3Ü1 is also universal and a new class of truly quantum liquids (not gases) appears, see AB Phys. Rev. Lett. 89, 050402 (2002).

mmmmmmammaamaaaaaa

mmaaam

aaaaaaaaaaaaa

mmH

mH

ψψψψλψψψψλψψψψλ

ψεψψψ

ψψψψψψλψψψψλψψ

++++++

++

++++++

+++

+

∇−+

∇−=

++∇

−=

21

21

42

31

21

2

2

2222

32

22

hh

h

In order to develop our program we have at first to have a well defined procedure for constructing an effective Hamiltonian for interacting atoms and dimers starting from the “fundamental” Hamiltonian describing bare interacting atoms.

Ha is a low energy reduction of the “fundamental” Hamiltonian, l2 and l3 are determined by the scattering length a and a three-body characteristic (denoted below by a3’). Interaction terms with derivatives are small as long as kr0á1.

Ham is determined by the matching” to be briefly described below.

Matching between the 2--, 3-- and 4--particle amplitudes computed with Ha and Ham.Only diagrams containing l2--vertices are shown.

The effective vertices thus defined (right side) can then be used to compute the ground state interaction energy in the leading order terms in an na3 expansion, which is given by the diagrams after the arrows.

atom-atom vertex(Lippmann-Schwinger eq.)

atom-dimer vertex(Faddeev eqs.)

dimer-dimer vertex(Yakubovsky eqs.)

Ha Ham E/V

In medium the magnitude of the relevant momenta are determined by estimatingThe quantum fluctuations of the mean-field. One thus easily can show thatp = ħk ≈ ħ(na)1/2/m. As long as kr0 µ ka µ (na3)1/2 Ü1 one can use contact couplings.

The accuracy of the mean-field approximation can be ascertained by estimating the magnitude of the quantum fluctuations to the energy density µn (na3)3/2ħ2/ma2.

We shall consider the regime when a ≫ r0 when the relevant momenta satisfyp = ħk ≈ ħ/a ¥ (na3)1/2 Ü ħ/a Ü ħ/r0.

Note that both Hamiltonians Ha and Ham are appropriate for kaÜ1 and kr0Ü1.

However, while perturbation theory is not valid for Ha when p ≈ ħ/a, all the non-perturbative physics at this scale (dimers of size ≈ a and the Efimov effect)Have been encapsulated in the couplings of the Hamiltonian Ham.

The “matching” described here was performed in vaccum, at length scales of order O(a) and this matching is not modified by the many-body physics, whichoccurs at scales O(a/(na3)1/2) ≫ O(a).

Fermi atoms

61.0 ,22.12

179.1 ,537.33

,4

22

22

2

2

2

2

aam

am

a

aam

am

a

mama

mmmm

mm

amam

am

aa

===

===

−==

hh

hh

hh

ππλ

ππλ

επλ

aam was first computed first by Skornyakov and Ter-Martirosian (1957) who studied neutron-deuteron scattering.

amm was computed by Petrov (2003) and Fonseca (2003) .

- ,3

scorrection 62.0537.325

3

2

2

22

3

22

222

222

makn

nnm

annm

anm

anmk

VE

Ff

bbbfffF

h

hhhh

==

+++++=

επ

επππ

Consider now a dilute mixture of fermionic atoms and (bosonic) dimersat temperatures smaller than the dimer binding energy (a>0 and a≫r0)

Induced fermion-fermion interaction

Bardeen et al. (1967), Heiselberg et al. (2000), Bijlsma et al. (2000)Viverit (2000), Viverit and Giorgini (2000)

( )

b

bbbbbb

bbb

bb

sbb

bbbb

fbfbf

bq

b

bbbb

bbbqq

qbfbfbf

mUnsa

ana

sm

rrU

UrU

mq

maU

Unn

UqU

=>>==

−−=

=

==

+−=

2

3

2

2

222

222

,162

exp4

1)(

0at tion representa coordinatein

2 ,4

)2(2

,

πξ

ξπξ

ω

επ

εεωε

ω

h

hh

h

coherence/healing length and speed of sound

One can show that pairing is typically weak!

a = nb-1/3/2.5 (solid line)

a = nb-1/3/3 (dashed line)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

nf/n

b

∆/ε F

The atom-dimer mixture can potentially be a system where relatively strong coupling pairing can occur.

( )

+−

=∆

−1

22

22223/7

441ln25.1012exp

22

bF

bF

F

F

kk

akmk

e ξξ

πh

Vortices in dilute atomic Fermi systems in traps

1995 BEC was observed.2000 vortices in BEC were created, thus BEC confirmed un-ambiguously.In 1999 DeMarco and Jin created a degenerate atomic Fermi gas.2002 O’Hara, Hammer, Gehm, Granada and Thomas observed expansion of a Fermi cloud compatible with the existence of a superfluid fermionic phase.

Observation of stable/quantized vortices in Fermi systems would provide the ultimate and most spectacular proof for the existence of a Fermionic superfluidphase.

Consider Bertsch’s MBX challenge (1999): “Find the ground state of infinite homogeneous neutron matter interacting with an infinite scattering length.”

Carlson, Morales, Pandharipande and Ravenhall, Phys.Rev. C68 (2003) 025802 , with Green Function Monte Carlo (GFMC)

54.053

, == NFNN

NE αεα

Carlson, Chang, Pandharipande and Schmidt,Phys. Rev. Lett. 91, 050401 (2003), with GFMC

44.053

, == SFSS

NE αεα

normal state

superfluid state

This state is half the way from BCS→BEC crossover, the pairing correlations are in the strong coupling limit and HFB invalid again.

How can one put in evidence a vortexin a Fermi superfluid?

Hard to see, since density changes are not expected, unlike the case of a Bose superfluid.

What we learned from the structure of a vortex in low densityneutron matter can help however.

If the gap is not small one can expect a noticeable density depletion along the vortex core, and the bigger the gap the bigger the depletion.

One can change the magnitude of the gap by altering the scattering length between two atoms with magnetic fieldsby means of a Feshbach resonance.

Now one can construct an SLDA functional to describe this new state of Fermionic matter

This form is not unique, as one can have either:b=0 (set I) or b≠0 and m*=m (set II).Gradient terms not determined yet (expected minor role).

0 2 4 6 80

0.2

0.4

0.6

0.8

1

kF ρ

n (ρ

)/n ∞

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

kF ρ

Vs(ρ

) /v

F

0 2 4 6 80

0.2

0.4

0.6

0.8

1

∆(ρ)

/∆∞

kF ρ

Solid lines are results for parameter set I, dashed lines for parameter set II(dots – velocity profile for ideal vortex)

The depletion along the vortex coreis reminiscent of the correspondingdensity depletion in the case of a vortex in a Bose superfluid, when the density vanishes exactly along the axisfor 100% BEC.

Extremely fast quantum vortical motion!

What shall I cover in this talk?

• SLDA - Superfluid LDAA brief introduction into the extension of the Kohn-Shall LDA to superfluid

fermion systems

• Vortices in the crust of neutron stars

• Vortices in dilute superfluid Fermi gases and some related issues

• Density profiles of dilute normal and superfluid Fermi gases in traps

m= 0.14µ10-10eV, ħw=0.568 µ10-12eV, a = -12.63nm (when finite)

0 2 4 6 8 10 12−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

r [µm]

U(r

) or

∆(r

) [1

0−10

eV

]

Mean−field and Pairing fields

U∞

Ua

U0∞

U0 a

∆∞(r)

∆a(r)

0 2 4 6 8 10 120

2

4

6

8

10

12

14

r [µm]

n(r)

[1012

cm−

3 ]

Particle number density

no interaction mean−field + pairing mean−field + no pairing

a infinite

a finite

40K (Fermi) atoms in a spherical harmonic trap

Effect of interaction, with and without weak and strong pairing correlations with fixed chemical potential.

ħw=0.568 µ10-12eV, a = -12.63nm (when finite)

40K (Fermi) atoms in a spherical harmonic trap

Effect of interaction, with and without weak and strong pairing correlations with fixed particle number, N = 5200.

0 2 4 6 8 10 12−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

r [µm]

U(r

) or

∆(r

) [1

0−10

eV

]

Mean−field and Pairing field

U∞

Ua

U0∞

U0 a

∆∞(r)

∆a(r)

0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

9

r [µm]

n(r)

[1012

cm−

3 ]

Particle number density

no interaction mean−field + pairing mean−field + no pairing