Local Density Functional Theory for Superfluid Fermionic SystemsLocal Density Functional Theory for Superfluid Fermionic Systems

The Unitary Fermi GasThe Unitary Fermi Gas

A.A. Bulgac, University of WashingtonBulgac, University of Washington

arXiv:condarXiv:cond--mat/0703526, mat/0703526, PRA, R , in press (2007)PRA, R , in press (2007)

Unitary Fermi gas in a harmonic trapUnitary Fermi gas in a harmonic trap

Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)

Outline:Outline:

•• What is a unitary Fermi gasWhat is a unitary Fermi gas

•• Very brief/skewed summary of DFTVery brief/skewed summary of DFT

•• BogoliubovBogoliubov--de Gennes equations, renormalizationde Gennes equations, renormalization

•• Superfluid Local Density Approximation (SLDA) Superfluid Local Density Approximation (SLDA) for a unitary Fermi gasfor a unitary Fermi gas

•• Fermions at unitarity in a harmonic trap within Fermions at unitarity in a harmonic trap within SLDA and comparison with SLDA and comparison with abab intiointio resultsresults

What is a unitary Fermi gasWhat is a unitary Fermi gas

Bertsch ManyBertsch Many--Body X challenge, Seattle, 1999Body X challenge, Seattle, 1999

What are the ground state properties of the manyWhat are the ground state properties of the many--body system composed of body system composed of spin spin ½½ fermions interacting via a zerofermions interacting via a zero--range, infinite scatteringrange, infinite scattering--length contactlength contactinteraction. interaction.

In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not.

- systems of bosons are unstable (Efimov effect)- systems of three or more fermion species are unstable (Efimov effect)

• Baker (winner of the MBX challenge) concluded that the system is stable.See also Heiselberg (entry to the same competition)

• Chang et al (2003) Fixed-Node Green Function Monte Carloand Astrakharchik et al. (2004) FN-DMC provided best the theoretical estimates for the ground state energy of such systems.

• Thomas’ Duke group (2002) demonstrated experimentally that such systemsare (meta)stable.

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, PRC 68, 025802 (2003), with Green Function Monte Carlo (GFMC)

54.053

, == NFNN

NE αεα

∞→<<<<→ || 00 ar Fλ

normal state

Carlson, Chang, Pandharipande and Schmidt,PRL 91, 050401 (2003), with GFMC

44.053

, == SFSS

NE αεα 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.

Solid line with open circles Solid line with open circles –– Chang Chang et al. et al. PRA, 70, 043602 (2004)PRA, 70, 043602 (2004)Dashed line with squares Dashed line with squares -- Astrakharchik Astrakharchik et al.et al. PRL 93, 200404 (2004)PRL 93, 200404 (2004)

BEC sideBEC side BCS sideBCS side

mkE F

FG 253 22

=

Green Function Monte Carlo with Fixed NodesChang, Carlson, Pandharipande and Schmidt, PRL 91, 050401 (2003)

⎟⎟⎠

⎞⎜⎜⎝

⎛=∆

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=∆

akmk

e

akmk

e

F

FBCS

F

FGorkov

2exp

28

2exp

22

22

2

223/7

π

π

Fixed node GFMC results, S.Fixed node GFMC results, S.--Y. Chang Y. Chang et al.et al. PRA 70, 043602 (2004)PRA 70, 043602 (2004)

BCS →BEC crossoverLeggett (1980), Nozieres and Schmitt-Rink (1985), Randeria et al. (1993),…

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

F

F

FFF

F

F

kkak

akmk

e11 and 1|| iff ,

2exp

22 223/7

>>∆

=<<<<⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛≈∆

εξεπ

If |a|=∞ and nr03 1 a Fermi system is strongly coupled and its properties

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

( ) )( , and 5344.0 ,

5354.0 uperfluidnormal

FFFs

F OON

EN

E ελξεε =∆=≈≈

If a>0 (a r0) and na3 1 the system is a dilute BEC of tightly bound dimers

23

2 2 and 1, where and 0.6 02

fb b bb

nn a n a a

maε = − << = = >

CarlsonCarlson’’s talk at Pack Forest, WA, August, 2007s talk at Pack Forest, WA, August, 2007

Fermi gas near unitarity has a very complex phase diagram (T=0)Fermi gas near unitarity has a very complex phase diagram (T=0)

Bulgac, Forbes, Schwenk, PRL 97, 020402 (2007)Bulgac, Forbes, Schwenk, PRL 97, 020402 (2007)

Very brief/skewed summary of DFTVery brief/skewed summary of DFT

Density Functional Theory (DFT) Density Functional Theory (DFT) HohenbergHohenberg and Kohn, 1964and Kohn, 1964

Local Density Approximation (LDA) Local Density Approximation (LDA) Kohn and Sham, 1965Kohn and Sham, 1965

[ ( )]gsE n r= Ε particle density only!particle density only!

23

2 2

1 12

( ) [ ( )] ( )2

( ) | ( ) | ( ) | ( ) |

( ) ( ) ( ) ( )2

gs

N N

i ii i

i i i i

E d r r n r n rm

n r r r r

r U r r rm

τ ε

ψ τ ψ

ψ ψ εψ

= =

⎧ ⎫= +⎨ ⎬

⎩ ⎭

= = ∇

∆− + =

∫

∑ ∑

The energy density is typically determined in ab initio calculationsof infinite homogeneous matter.

KohnKohn--Sham equationsSham equations

KohnKohn--Sham theoremSham theorem

[ ]

0 0 0

0 0

0

23

0 ( )

2

( ) ( ) ( ) ... ( )

(1,2,... ) (1,2,... )

( ) ( )

(1,2,... ) ( ) ( )

min ( ) ( ) ( ) ( )2

( ) ( ) ,

N N N N

exti i j i j k i

N

ii

ext

extn r

i

H T i U ij U ijk V i

H N E N

n r r r

N V r n r

E d r r n r V r n rm

n r r

δ

τ ε

ϕ

< < <

= + + + +

Ψ = Ψ

= Ψ − Ψ

Ψ ⇔ ⇔

⎧ ⎫= + +⎨ ⎬

⎩ ⎭

=

∑ ∑ ∑ ∑

∑

∫2

( ) ( )N N

ii i

r rτ ϕ= ∇∑ ∑

Injective mapInjective map(one(one--toto--one)one)

Universal functional of densityUniversal functional of densityindependent of external potentialindependent of external potential

How to construct and validate an How to construct and validate an abab initioinitio EDF?EDF?

Given a many body Hamiltonian determine the properties of Given a many body Hamiltonian determine the properties of the infinite homogeneous system as a function of densitythe infinite homogeneous system as a function of density

Extract the energy density functional (EDF)Extract the energy density functional (EDF)

Add gradient corrections, if needed or known how (?)Add gradient corrections, if needed or known how (?)

Determine in an Determine in an abab initioinitio calculation the properties of a calculation the properties of a select number of wisely selected finite systems select number of wisely selected finite systems

Apply the energy density functional to inhomogeneous systemsApply the energy density functional to inhomogeneous systemsand compare with the and compare with the abab initioinitio calculation, and if lucky declarecalculation, and if lucky declareVictory! Victory!

One can construct however an EDF which depends both on One can construct however an EDF which depends both on particle density and kinetic energy density and use it in a particle density and kinetic energy density and use it in a extended Kohnextended Kohn--Sham approach (perturbative result)Sham approach (perturbative result)

Notice that dependence on kinetic energy density and on the Notice that dependence on kinetic energy density and on the gradient of the particle density emerges because of finite gradient of the particle density emerges because of finite range effects.range effects.

Bhattacharyya and Furnstahl, Bhattacharyya and Furnstahl, NuclNucl. Phys. A 747, 268 (2005). Phys. A 747, 268 (2005)

The singleThe single--particle spectrum of usual Kohnparticle spectrum of usual Kohn--Sham approach is unphysical, Sham approach is unphysical, with the exception of the Fermi level.with the exception of the Fermi level.The singleThe single--particle spectrum of extended Kohnparticle spectrum of extended Kohn--Sham approach has physical meaning.Sham approach has physical meaning.

Extended KohnExtended Kohn--Sham equationsSham equations

Position dependent mass Position dependent mass

23

*

2 2

1 12

*

( ) [ ( )] ( )2 [ ( )]

( ) | ( ) | ( ) | ( ) |

( ) ( ) ( ) ( )2 [ ( )]

gs

N N

i ii i

i i i i

E d r r n r n rm n r

n r r r r

r U r r rm n r

τ ε

ψ τ ψ

ψ ψ εψ

= =

⎧ ⎫= +⎨ ⎬

⎩ ⎭

= = ∇

−∇ ∇ + =

∫

∑ ∑

Normal Fermi systems only!

However, not everyone is normal!However, not everyone is normal!

Superconductivity and superfluidity in Fermi systemsSuperconductivity and superfluidity in Fermi systems

• Dilute atomic Fermi gases Tc ≈ 10-12 – 10-9 eV

• Liquid 3He Tc ≈ 10-7 eV

• Metals, composite materials Tc ≈ 10-3 – 10-2 eV

• Nuclei, neutron stars Tc ≈ 105 – 106 eV

• QCD color superconductivity Tc ≈ 107 – 108 eV

units (1 eV ≈ 104 K)

BogoliubovBogoliubov--de Gennes equations and renormalizationde Gennes equations and renormalization

3

2 2k k

*k k

*

( ( ), ( ), ( ))

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

( ) u ( )v ( )

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

gs

k k

k

k kk

k k

E d r n r r r

n r r r r

r r r

r rT U r rE

r rr T U r

ε τ ν

τ

ν

µµ

=

= = ∇

=

+ − ∆ ⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟∆ − + −⎝ ⎠⎝ ⎠ ⎝ ⎠

∫∑ ∑

∑

SLDA - Extension of Kohn-Sham approach to superfluid Fermi systems

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)

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

10

−=≅ FF

kp

r

radiusradius ofof interactioninteraction interinter--particleparticle separationseparation

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.

01 r

kF

F

>>∆

≈εξ

coherence lengthcoherence lengthsize of the Cooper pairsize of the Cooper pair

FD NVExp εω <<⎟⎟

⎠

⎞⎜⎜⎝

⎛−=∆

||1

Nature of the problemNature of the problem

*1 2 k 1 k 2

0 1 2

1 2 1 2 1 2

1( , ) v ( )u ( )

( , ) ( , ) ( , )kE

r r r rr r

r r V r r r r

ν

ν>

= ∝−

∆ = −

∑ at small separations

It is easier to show how this singularity appears in infinite homogeneous matter.

|| ,)(

)sin(2

)(

2 ,

2 ,1vu ,

)(1

21v

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

212222

2

022

222

k2k

2k22

k

k2k

2k2k1k1k

rrrkkk

krkrdkmr

mU

mk

rkirrkir

F

−=+−

∆=

=∆+==+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆+−

−−=

⋅=⋅=

∫∞

δπν

δεµεµε

PseudoPseudo--potential approach potential approach (appropriate for very slow particles, very transparent,(appropriate for very slow particles, very transparent,but 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

δψδψ

ψδψ

π

ψ

ψψψ

=∂∂

⇒++=

∂∂

⇒<<

++

=−+−=

+−≈++≈+⋅=

>≈=+∆

−

−

( ) ( ) ( ) ( ){ }( ) ( ) ( ) ( ) ( ) ( )

3

2eff

2

i i i*

i i i

, ,

,

[ ( ) ]u ( ) ( )v ( ) u ( )2

( )u ( ) [ ( ) ]v ( ) v ( )

gs N S

def

S c c

i

i

E d r n r r n r r

n r r r r g r r

h(r) h r r r r E rm

r r h r r E r

ε τ ε ν

ε ν ν ν

µµ

= +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= − ∆ =⎡ ⎤⎣ ⎦

= −∇− + ∆ =⎧⎨∆ − − =⎩

∫

( ) ( )eff

2 2

2 *i i

0

( )( )

( )

1 1 ( ) ( ) ( ) ( ) ( ) 1 ln( ) [ ( )] 2 2 ( ) ( ) ( )

( ) 2 v ( ) , ( ) v (c

i

c

c F c F

eff c c F

E

c cE

U rr

r g r r

m r k r k r k r k rg r g n r k r k r k r

r r ν r

ν

π

ρ≥

⎧∇ +⎪

⎨⎪ ∆ = −⎩

⎧ ⎫+= − −⎨ ⎬−⎩ ⎭

= =∑ i0

2 2 2 2

)u ( )

( ) ( ) ( ), ( ) 2 ( ) 2 ( )

c

i

E

E

c Fc

r r

k r k rE U r U rm r m r

µ µ

≥

+ = + = +

∑

Position and momentum dependent running coupling constantPosition and momentum dependent running coupling constantObservables are (obviously) independent of cutObservables are (obviously) independent of cut--off energy (when chosen properly).off energy (when chosen properly).

The SLDA (renormalized) equations

Superfluid Local Density Approximation (SLDA) Superfluid Local Density Approximation (SLDA) for a unitary Fermi gasfor a unitary Fermi gas

The naThe naïïve SLDA energy density functional ve SLDA energy density functional suggested by dimensional argumentssuggested by dimensional arguments

22 2/3 5/3

1/3

2

k

2

k

*k k

( )( ) 3(3 ) ( )( )

2 5 ( )

( ) 2 v ( )

( ) 2 v ( )

( ) u ( )v ( )

k

k

k

rr n rr

n r

n r r

r r

r r r

ντ πε α β γ

τ

ν

= + +

=

= ∇

=

∑

∑

∑

The renormalized SLDA energy density functional The renormalized SLDA energy density functional

2 2/3 5/32

2 *k k k

1/30 0

20

2

( ) 3(3 ) ( )( ) ( ) ( )

2 5

( ) 2 v ( ) , ( ) u ( )v ( )

1 ( ) ( ) ( ) ( ) ( )1 ln

( ) 2 2 ( ) ( ) ( )

( )2

c c

ceff c

c cE E E E

c c

eff c c

cc

r n rr g r r

r r r r r

n r k r k r k r k rg r k r k r k r

k rE

τ πε α β ν

τ ν

γ π α

µ α

< <

= + +

= ∇ =

⎡ ⎤+= − −⎢ ⎥−⎣ ⎦

+ =

∑ ∑

20

22 2/3 2/3

2/3

( )( ), ( )

2

( )(3 ) ( )( ) ( ) small correction

2 3 ( )

( ) ( ) ( )

ext

eff c

k rU r U r

rn rU r V r

n r

r g r r

µ α

πβγ

ν

+ = +

∆= − + +

∆ = −

How to determine the dimensionless parameters How to determine the dimensionless parameters α, β α, β andand γ ?γ ?

( )α β µ

π π α β µ

επ

εε ξ ε β α

π

γ π α

⎛ ⎞⎜ ⎟+ −

= = −⎜ ⎟⎜ ⎟+ − + ∆⎝ ⎠

⎛ ⎞= −⎜ ⎟

⎝ ⎠⎡ ⎤⎛ ⎞ ∆

= + − −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎛ ⎞= −⎜ ⎟

⎝ ⎠

∫

∫

∫

∫

3 2 23

2 3 22 2 2

3

3

3 2 2

3

1/3 3

3 2

/ 2 / 21

3 (2 ) / 2 / 2

1(2 )

3 32 1

5 5 2 2(2 )

1 12(2 )

F F

F

k

k

kF S F

k k

k

k k kd kn

k k

d kE

d k kn n

E E

n d kEk

One thus obtains:One thus obtains:

50.42(2)

3 1 .14

0 .504(24) 0.553

10.0906 0 .42(2)

sF

F

F

EN

ξε α

η βεµς γε

⎧ ⎧= =⎪ ⎪ =⎪ ⎪∆⎪ ⎪= = ⇒ = −⎨ ⎨⎪ ⎪⎪ ⎪ = −= =⎪ ⎪⎩⎩

solid/dotted blue line solid/dotted blue line -- SLDA, homogeneous GFMC due to Carlson et al SLDA, homogeneous GFMC due to Carlson et al red circles red circles -- GFMC due to Carlson and Reddy GFMC due to Carlson and Reddy dashed blue line dashed blue line -- SLDA, homogeneous MC due to SLDA, homogeneous MC due to JuilletJuilletblack dashedblack dashed--dotted line dotted line –– meanfield at unitaritymeanfield at unitarity

Quasiparticle spectrum in homogeneous matterQuasiparticle spectrum in homogeneous matterBonus!Bonus!

Two more universal parameter characterizing the unitary Fermi gas and its excitation spectrum: effective mass, meanfield potential

Extra Bonus!Extra Bonus!

The normal state has been also determined in GFMCThe normal state has been also determined in GFMC

50.55(2)

3NF

EN

ξε

= =

SLDA functional predicts SLDA functional predicts

0.59Nξ α β= + =

Fermions at unitarity in a harmonic trap within SLDA Fermions at unitarity in a harmonic trap within SLDA and comparison with and comparison with abab intiointio resultsresults

GFMC GFMC -- Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)FNFN--DMC DMC -- von von StecherStecher, Greene and , Greene and BlumeBlume, arXiv:0705.0671, arXiv:0705.0671

arXiv:0708.2734arXiv:0708.2734

Fermions at unitarity in a harmonic trapFermions at unitarity in a harmonic trap

GFMC GFMC -- Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)FNFN--DMC DMC -- von von StecherStecher, Greene and , Greene and BlumeBlume, arXiv:0705.0671, arXiv:0705.0671

NB Particle projectionNB Particle projectionneither required nor neither required nor needed in SLDA!!!needed in SLDA!!!

SLDA - Extension of Kohn-Sham approach to superfluid Fermi systems

{

}

3

* *

2 2k k

*k k

*

( ( ), ( ), ( ))

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

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

( ) u ( )v ( )

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

gs

ext ext ext

k k

k

k

k

E d r n r r r

V r n r r r r r

n r r r r

r r r

rT U r rr T U r

ε τ ν

ν ν

τ

ν

µµ

= +

+ ∆ + ∆

= = ∇

=

+ − ∆⎛ ⎞⎜ ⎟∆ − + −⎝ ⎠

∫

∑ ∑

∑

u ( )) v ( )

kk

k

rE

r r⎛ ⎞ ⎛ ⎞

=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

universal functionaluniversal functional(independent of external potential) (independent of external potential)

[ ]2

1( ) ( ) ( 1) ( 1)

2E N E N E N E Nδ = − + + −

Densities for N=8 (solid), N=14 (dashed) and N=20 (dotDensities for N=8 (solid), N=14 (dashed) and N=20 (dot--dashed)dashed)GFMC (red), SLDA (blue)GFMC (red), SLDA (blue)

New extended FNNew extended FN--DMC resultsDMC resultsD. D. BlumeBlume, J. von , J. von StecherStecher, and C.H. Greene, arXiv:0708.2734, and C.H. Greene, arXiv:0708.2734

New extended FNNew extended FN--DMC resultsDMC resultsD. D. BlumeBlume, J. von , J. von StecherStecher, and C.H. Greene, arXiv:0708.2734, and C.H. Greene, arXiv:0708.2734

•• Agreement between GFMC/FNAgreement between GFMC/FN--DMC and SLDA extremely good, DMC and SLDA extremely good, a few percent (at most) accuracya few percent (at most) accuracy

Why not better?Why not better?A better agreement would have really signaled big troubles!A better agreement would have really signaled big troubles!

•• Energy density functional is not unique, Energy density functional is not unique, in spite of the strong restrictions imposed by unitarity in spite of the strong restrictions imposed by unitarity

•• SelfSelf--interaction correction neglectedinteraction correction neglectedsmallest systems affected the mostsmallest systems affected the most

•• Absence of polarization effects Absence of polarization effects spherical symmetry imposed, odd systems mostly affectedspherical symmetry imposed, odd systems mostly affected

•• Spin number densities not included Spin number densities not included extension from SLDA to SLSD(A) neededextension from SLDA to SLSD(A) neededab initioab initio results for asymmetric system neededresults for asymmetric system needed

•• Gradient corrections not included Gradient corrections not included

OutlookOutlook

Extension away from unitarity Extension away from unitarity -- trivialtrivial

Extension to (some) excited states Extension to (some) excited states -- easyeasy

Extension to time dependent problems Extension to time dependent problems –– appears easyappears easy

Extension to finite temperatures Extension to finite temperatures -- easy, but one more parameter easy, but one more parameter is needed, the pairing gap dependence as a function of Tis needed, the pairing gap dependence as a function of T

Extension to asymmetric systems straightforward (at unitarityExtension to asymmetric systems straightforward (at unitarityquite a bit is already know about the equation of state) quite a bit is already know about the equation of state)