Static Properties and Dynamics Static Properties and Dynamics of Strongly Interacting Many Fermion Systemsof Strongly Interacting Many Fermion Systemsof Strongly Interacting Many Fermion Systemsof Strongly Interacting Many Fermion Systems

Aurel BulgacAurel BulgacAurel BulgacAurel BulgacUniversity of Washington, Seattle, USAUniversity of Washington, Seattle, USA

Collaborators:Collaborators:M.M. Forbes, P. M.M. Forbes, P. MagierskiMagierski, Y., Y.--L. (Alan) L. (Alan) LuoLuo, K.J. Roche, I. , K.J. Roche, I. StetcuStetcu, S. Yoon, Y. Yu, S. Yoon, Y. Yu

Funding: DOE grants No. DEFunding: DOE grants No. DE--FG02FG02--97ER41014 (UW NT Group)97ER41014 (UW NT Group)DEDE--FC02FC02--07ER41457 (07ER41457 (SciDACSciDAC--UNEDF)UNEDF)

Slides to be posted at: Slides to be posted at: http://www.phys.washington.edu/users/bulgac/http://www.phys.washington.edu/users/bulgac/

I will discuss properties of two different systems, I will discuss properties of two different systems, which surprisingly share a lot in common:which surprisingly share a lot in common:

Cold atomic gases Cold atomic gases

Nuclear systemsNuclear systems

I am in big trouble:I am in big trouble:I am in big trouble:I am in big trouble:

I have too many slides!I have too many slides!ave too a y s des!ave too a y s des!

I do not have enough slides!!!I do not have enough slides!!!do o ve e oug s des!!!do o ve e oug s des!!!

Both statements are correct!!!!!Both statements are correct!!!!!

First lecture First lecture –– Static propertiesStatic properties

How to implement Density Functional Theory (DFT) How to implement Density Functional Theory (DFT) in the Case of Superfluid Fermion Systemsin the Case of Superfluid Fermion Systems

Second Lecture Second Lecture -- Implementation of TimeImplementation of Time--Dependent DFT Dependent DFT for Superfluid Fermion Systems in 3Dfor Superfluid Fermion Systems in 3D

Applications to unitary gas and nuclei and how all this was Applications to unitary gas and nuclei and how all this was implemented on implemented on JaguarPfJaguarPf (the largest supercomputer in the world) (the largest supercomputer in the world)

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 Fermions at unitarityunitarity in a harmonic trap within SLDA and comparison in a harmonic trap within SLDA and comparison withwith abab intiointio resultsresultswith with abab intiointio resultsresults

•• Challenges one has to face in order to implement DFT in nucleiChallenges one has to face in order to implement DFT in nuclei

http://http://www.unedf.orgwww.unedf.org

From a talk given by G.F. From a talk given by G.F. BertschBertsch,,INT at 20, July 1INT at 20, July 1--2, 20102, 2010

Part IPart I

Ch dCh d B t hB t h Ph R A 76 021603(R) (2007)Ph R A 76 021603(R) (2007)

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

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

Why would one want to study this system?Why would one want to study this system?

One reason:One reason:One reason: One reason:

(for the nerds, I mean the hard(for the nerds, I mean the hard--core theorists,core theorists,(for the nerds, I mean the hard(for the nerds, I mean the hard core theorists, core theorists, not for the not for the phenomenologistsphenomenologists) )

Bertsch’sBertsch’s ManyMany--Body X challenge, Seattle, 1999Body X challenge, Seattle, 1999yy y gy g

What are the ground state properties of the manyWhat are the ground state properties of the many--body system body system composed of spin ½ fermions interacting via a zerocomposed of spin ½ fermions interacting via a zero range infiniterange infinitecomposed of spin ½ fermions interacting via a zerocomposed of spin ½ fermions interacting via a zero--range, infinite range, infinite scatteringscattering--length contact interaction. length contact interaction.

Wh h i l h d h ff i ?Wh h i l h d h ff i ?What are the scattering length and the effective range?What are the scattering length and the effective range?

20 0

1 1cotank r kδ = − + +0 0 cotan 2

k r ka

δ + +

2 202

4 sin 4 akπσ δ π= + = +

k

If the energy is small only the sIf the energy is small only the s--wave is relevant.wave is relevant.

Let us consider a very old example: the hydrogen atom.Let us consider a very old example: the hydrogen atom.

The ground state energy could only be a function of:The ground state energy could only be a function of:

Electron chargeElectron chargeElectron massElectron massPlanck’s constantPlanck’s constantPlanck s constantPlanck s constant

and then trivial dimensional arguments lead toand then trivial dimensional arguments lead to

4 1e m2

12gs

e mE = ×

Only the factor ½ requires some hard work.Only the factor ½ requires some hard work.

Let us turn now to dilute fermion matterLet us turn now to dilute fermion matterThe ground state energy is given by a function:The ground state energy is given by a function:

( )E f N V m a r= 0( , , , , , )gsE f N V m a r=Taking the scattering length to infinity and the rangeTaking the scattering length to infinity and the rangeTaking the scattering length to infinity and the range Taking the scattering length to infinity and the range of the interaction to zero, we are left with:of the interaction to zero, we are left with:

33( , , , )5gs FE F N V m Nε ξ= = ×

3 2 2

5

F Fk kN ε= =2 , 3 2FV m

επ

= = Pure numberPure number(dimensionless)(dimensionless)

BEC sideBEC side BCS sideBCS side

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 -- AstrakharchikAstrakharchik et al.et al. PRL 93, 200404 (2004)PRL 93, 200404 (2004)

What is a What is a unitary Fermi unitary Fermi gasgas

BertschBertsch ManyMany--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 ½ fermions interacting via a zerospin ½ fermions interacting via a zero--range, infinite scatteringrange, infinite scattering--length contactlength contacti t tii t tiinteraction. interaction.

In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or notwhether 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 Consider Bertsch’sBertsch’s MBX challenge (1999): “Find the ground state of infinite MBX challenge (1999): “Find the ground state of infinite homogeneous neutron matter interacting with an infinite scattering length.” homogeneous neutron matter interacting with an infinite scattering length.”

Carlson, Morales, Carlson, Morales, PandharipandePandharipande and Ravenhall, and Ravenhall, PRC 68 025802 (2003) ith G F ti M t C l (GFMCPRC 68 025802 (2003) ith G F ti M t C l (GFMC)

∞→<<<<→ || 00 ar Fλ

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

3E normal statenormal state54.0

53

, == NFNN

NE αεα

normal statenormal state

3E

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

44.053

, == SFSS

NE αεα superfluid statesuperfluid state

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

mkE F

FG 253 22

=

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

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=Δ

akmk

eF

Gorkov 2exp

22 223/7 π

⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

⎟⎠

⎜⎝⎠⎝

kk

akme

FBCS

F

2exp

28

2222

2

π⎟⎠

⎜⎝ akme F

CS 222

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 crossoverBCS →BEC crossover

Eagles (1960) LeggettEagles (1960) Leggett (1980)(1980) NozieresNozieres and Schmittand Schmitt--Rink (1985)Rink (1985)

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

Eagles (1960), Leggett Eagles (1960), Leggett (1980), (1980), NozieresNozieres and Schmittand Schmitt Rink (1985), Rink (1985), RanderiaRanderia et al.et al. (1993),…(1993),…

F

F

FFF

F

F

kkak

akmk

e11 and 1|| iff ,

2exp

22 223/7

>>Δ

=<<<<⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛≈Δ

εξεπ

If If |a|=|a|=∞∞ and and nrnr0033 11 a Fermi system is strongly coupled and its properties a Fermi system is strongly coupled and its properties

are universal. Carlson are universal. Carlson et alet al. PRL . PRL 9191, 050401 (2003), 050401 (2003)

( ) )( , and 5344.0 ,

5354.0 uperfluidnormal

FFFs

F OON

EN

E ελξεε =Δ=≈≈

If If a>0 (aa>0 (a rr00)) and and nana33 11the the system is a dilute BEC of tightly bound system is a dilute BEC of tightly bound dimersdimers

23

2 2 and 1, where and 0.6 02

fb b bb

nn a n a a

maε = − << = = >

Carlson’s talk at Carlson’s talk at UNEDF meeting, Pack UNEDF meeting, Pack Forest, WA, August, 2007Forest, WA, August, 2007

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

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

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

Density Functional Theory (DFT) Density Functional Theory (DFT) H h bH h b d K h 1964d K h 1964HohenbergHohenberg and Kohn, 1964and Kohn, 1964

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

Local Local Density Approximation (LDA) Kohn and Density Approximation (LDA) Kohn and

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

Sham, 1965 Sham, 1965 2

3 ( ) [ ( )] ( )2gsE d r r n r n r

mτ ε

⎧ ⎫= +⎨ ⎬

⎩ ⎭∫

The energy density is typically The energy density is typically determined in determined in abab initioinitio calculationscalculations

2 2

1 1

2

( ) | ( ) | ( ) | ( ) |N N

i ii i

m

n r r r rψ τ ψ

⎩ ⎭

= = ∇

∫

∑ ∑

of infinite homogeneous matter.of infinite homogeneous matter.

1 12

( ) ( ) ( ) ( )2

i i

i i i ir U r r rmψ ψ εψ

= =

Δ− + = KohnKohn--Sham equationsSham equations

KohnKohn--Sham theoremSham theorem

( ) ( ) ( ) ... ( )N N N N

exti i j i j k i

H T i U ij U ijk V i< < <

= + + + +∑ ∑ ∑ ∑

0 0 0(1,2,... ) (1,2,... )

( ) ( )

j j

N

H N E N

δ

Ψ = Ψ

Ψ Ψ∑0 0

0

( ) ( )

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

ii

ext

n r r r

N V r n r

δ= Ψ − Ψ

Ψ ⇔ ⇔

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

[ ]2

30 ( )

min ( ) ( ) ( ) ( )2 extn r

E d r r n r V r n rmτ ε

⎧ ⎫= + +⎨ ⎬

⎩ ⎭∫

(( ))

2( ) ( ) , in r rϕ=2

( ) ( )N N

ii i

r rτ ϕ= ∇∑ ∑

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 ofGiven a many body Hamiltonian determine the properties ofGiven 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)gy y ( )gy y ( )

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

A l th d it f ti l t i h tA l th d it f ti l t i h tApply 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 (Sham approach (perturbativeperturbative result)result)extended Kohnextended Kohn Sham approach (Sham approach (perturbativeperturbative result)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 effectsrange effectsrange effects.range effects.

Bhattacharyya and Bhattacharyya and FurnstahlFurnstahl, , 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.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 ( ) [ ( )] ( )E d⎧ ⎫

+⎨ ⎬∫ 3*

2 2

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

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

gs

N N

E d r r n r n rm n r

n r r r r

τ ε

ψ τ ψ

= +⎨ ⎬⎩ ⎭

∇

∫

∑ ∑1 12

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

( ) ( ) ( ) ( )

i ii i

n r r r r

r U r r r

ψ τ ψ

ψ ψ εψ

= =

= = ∇

−∇ ∇ + =

∑ ∑

* ( ) ( ) ( ) ( )2 [ ( )] i i i ir U r r r

m n rψ ψ εψ∇ ∇ + =

N l F i t l !Normal Fermi systems only!

H t i l!H t i l!However, not everyone is normal!However, not everyone is normal!

S d ti it d fl idit i F i tS d ti it d fl idit i F i tSuperconductivity 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 eVq c

• 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 units (1 eVeV ≈≈ 101044 K) K)

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

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

3 ( ( ), ( ), ( ))gsE d r n r r rε τ ν= ∫

p yp y

2 2k k

*

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

( ) ( ) ( )k k

n r r r rτ= = ∇∑ ∑

∑ k k( ) u ( )v ( )k

r r rν = ∑

*

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

k kk

r rT U r rE

r rr T U rμ

μ+ − Δ ⎛ ⎞ ⎛ ⎞⎛ ⎞

=⎜ ⎟ ⎜ ⎟⎜ ⎟Δ − + −⎝ ⎠⎝ ⎠ ⎝ ⎠v ( ) v ( )( ) ( ( ) ) k kr rr T U r μΔ +⎝ ⎠⎝ ⎠ ⎝ ⎠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 ΔΔ diverges. diverges.

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.

10

−=≅ FF

kp

r

1⎛ ⎞

radius of interaction inter-particle separation

1 Fε1exp| |D FV N

ω ε⎛ ⎞Δ = − <<⎜ ⎟

⎝ ⎠0

1 rk

F

F

>>Δ

≈εξ

coherence lengthcoherence lengthsize of the Cooper pair size of the Cooper pair

Nature of the problemNature of the problem

*1 2 k 1 k 2

0 1 2

1( , ) v ( )u ( )

kE

r r r rr r

ν>

= ∝−∑ at small separationsat small separations

1 2 1 2 1 2( , ) ( , ) ( , )r r V r r r rνΔ = −

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

)exp(u)(u ),exp(v)(v 2k2k1k1k rkirrkir ⋅=⋅=

2 ,

2 ,1vu ,

)(1

21v

222

k2k

2k22

k

k2k m

Umk

=Δ+==+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ+−

−−=

δεμεμε

||)sin()(

)(

2

k

kkrdkmΔ

⎠⎝

∫∞

μ

|| ,)(

)sin(2

)( 2122220

22 rrrkkk

krkrdkmr

F

−=+−

Δ= ∫ δπ

ν

PseudoPseudo--potential approach potential approach (appropriate for very slow particles, very transparent,(appropriate for very slow particles, very transparent,b t h t diffi lt t i )b t h t diffi lt t i )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)Lee, Huang and Yang (1957)Lee, Huang and Yang (1957)

RrrVrErrVrr if 0)( ),()()()(2

ψψψ >≈=+Δ

−

krOra

rfikr

rfrkir

m

)(1... 1)exp()exp()(ψ +−≈++≈+⋅=

ikamagikkr

af

rrr

...)1(

4 ,211 2

20

1 π+

+=−+−=−

[ ]rrr

rgrrVkr

ikama

)()()()( then 1 if

)1(2

0 ψδψ∂∂

⇒<<

+

[ ] Brrrr

rBrAr )()()( ...)( :Example δψδψ =

∂∂

⇒++=

( ) ( ) ( ) ( ){ }3 , , gs N SE d r n r r n r rε τ ε ν= +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∫

The SLDA (renormalized) equationsThe SLDA (renormalized) equations

{ }( ) ( ) ( ) ( ) ( ) ( ) 2

eff ,

g

def

S c cn r r r r g r rε ν ν ν

⎣ ⎦ ⎣ ⎦

= − Δ =⎡ ⎤⎣ ⎦

∫

2

i i i*

i i i

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

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

i

h(r) h r r r r E rm

r r h r r E rμ

μ= −∇− + Δ =⎧

⎨Δ − − =⎩ ( ) ( )eff

( )( )

( ) c

U rr

r g r rν

⎧∇ +⎪

⎨⎪ Δ = −⎩ ( ) ( )eff

2 2

( )

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

c

c F c F

g

m r k r k r k r k rk k k

⎩

⎧ ⎫+= − −⎨ ⎬

⎩ ⎭2 2

2 *

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

( ) 2 v ( ) ( ) v (c

eff c c F

E

g r g n r k r k r k r

r r ν r

π

ρ

⎨ ⎬−⎩ ⎭

= =∑ )u ( )cE

r r∑i i0

( ) 2 v ( ) , ( ) v (i

c cE

r r ν rρ≥

= =∑ i0

2 2 2 2

)u ( )

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

iE

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).

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

The naïve SLDA energy density functional The naïve SLDA energy density functional suggested by dimensional argumentssuggested by dimensional arguments

22 2/3 5/3

1/3

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

2 5 ( )

rr n rr

n r

ντ πε α β γ= + +

2

2 5 ( )n r

2

k

2

( ) 2 v ( )k

n r r= ∑2

k

*

( ) 2 v ( )

( ) ( ) ( )k

r rτ = ∇∑

∑ *k k( ) u ( )v ( )

k

r r rν = ∑

The The SLDA SLDA energy density functional energy density functional at unitarityat unitarityfor equal numbers of spinfor equal numbers of spin‐‐up and spinup and spin‐‐down fermions down fermions 

Only this combination is cutoff independent Only this combination is cutoff independent 

τ πε α ν β⎡ ⎤

= − Δ +⎢ ⎥2 2/3 5/3( ) 3(3 ) ( )

( ) ( ) ( )cr n r

r r r

y py p

ε α ν β⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑22

( ) ( ) ( )2 5c

r r r

τ

ν< < < <

= = ∇

=

∑ ∑∑

22

k k0 0

*k k

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

( ) u ( )v ( )k c k c

cE E E E

c

n r r r r

r r r< <

Δ

∑ k k0

22 2/3 2/3 ( )( ) ( )

c

cE E

πβγ

Δ= − + +

2 2/3 2/3

2/3

( )(3 ) ( )( ) ( ) small correct

2 3 ( ) ext

rn rU r V r

n rνΔ = −

ion

( ) ( ) ( )r g r rνΔ =( ) ( ) ( )eff c

r g r r

aa can take any positive value, can take any positive value, but the best results are obtained when but the best results are obtained when aa is fixed by the is fixed by the qpqp‐‐spectrumspectrum

Fermions at unitarity in a harmonic Fermions at unitarity in a harmonic traptrapTotal energies E(N)Total energies E(N)

GFMC     GFMC     ‐‐ Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)FNFN‐‐DMCDMC ‐‐ vonvon StecherStecher Greene andGreene and BlumeBlume PRLPRL 9999 233201 (2007)233201 (2007)FNFN DMC DMC  von von StecherStecher, Greene and , Greene and BlumeBlume, , PRL PRL 9999, 233201 (2007) , 233201 (2007) 

PRA PRA 7676, 053613 (2007), 053613 (2007)

Bulgac, PRA  76, 040502(R) (2007)Bulgac, PRA  76, 040502(R) (2007)

NB P i l j iNB P i l j iNB Particle projectionNB Particle projectionneither required nor neither required nor needed in SLDA!!!needed in SLDA!!!

Fermions at unitarity in a harmonic Fermions at unitarity in a harmonic traptrapPairing gapsPairing gaps

GFMC     GFMC     ‐‐ Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)FNFN DMCDMC St hSt h G dG d BlBl PRLPRL 9999 233201 (2007)233201 (2007)

( 1) 2 ( ) ( 1)( ) , for odd 2

E N E N E NN N+ − + −Δ =

Bulgac, PRA  76, 040502(R) (2007)Bulgac, PRA  76, 040502(R) (2007)

FNFN‐‐DMC DMC ‐‐ von von StecherStecher, Greene and , Greene and BlumeBlume, , PRL PRL 9999, 233201 (2007) , 233201 (2007) PRA PRA 7676, 053613 (2007), 053613 (2007)

Quasiparticle spectrum in homogeneous matterQuasiparticle spectrum in homogeneous matter

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 circlesred circles GFMC due to Carlson and ReddyGFMC due to Carlson and Reddyred 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 JuilletSLDA, homogeneous MC due to Juilletblack dashedblack dashed--dotted line dotted line –– meanfield at unitaritymeanfield at unitarity

Two more universal parameter characterizing the unitaryTwo more universal parameter characterizing the unitary Fermi gas and its excitation spectrum: effective mass, meanfield potential Bulgac, PRA  76, 040502(R) (2007)Bulgac, PRA  76, 040502(R) (2007)

Agreement Agreement between GFMC/FNbetween GFMC/FN--DMC DMC and and SLDA extremely good, 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 in spite of the strong restrictions imposed by unitarityunitarity

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

•• Absence of polarization effectsAbsence of polarization effects•• 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 includedSpin number densities not includedSpin number densities not included Spin number densities not included extension from SLDA to SLSD(A) neededextension from SLDA to SLSD(A) neededabab initioinitio results for asymmetric system neededresults for asymmetric system needed

•• Gradient corrections not included Gradient corrections not included

Until now we kept the numbers of spinUntil now we kept the numbers of spin--up up p pp p ppand and spinspin--down equal.down equal.

What happens when there What happens when there are not are not enough partners enough partners ff tt i ith?i ith?for for everyone everyone to to pair with? pair with?

(I ti l thi i h t t t h i(I ti l thi i h t t t h i(In particular this is what one expects to happen in (In particular this is what one expects to happen in color superconductivity, due to a heavier strange color superconductivity, due to a heavier strange

k)k)quark)quark)

What theory tells us?What theory tells us?What theory tells us?What theory tells us?

Green   Green   –– Fermi sphere of spinFermi sphere of spin‐‐up fermionsup fermionsYellowYellow –– Fermi sphere of spinFermi sphere of spin‐‐down fermionsdown fermions

If the same solution as for μ μ μ μ↑ ↓ ↑ ↓

Δ− < =

Yellow  Yellow   Fermi sphere of spinFermi sphere of spin down fermionsdown fermions

If the same solution as for 2

μ μ μ μ↑ ↓ ↑ ↓<

LOFF/FFLO  LOFF/FFLO  solution (1964)solution (1964)Pairing gap becomes a spatially varying functionPairing gap becomes a spatially varying functionPairing gap becomes a spatially varying functionPairing gap becomes a spatially varying functionTranslational invariance brokenTranslational invariance broken

MuetherMuether and and SedrakianSedrakian (2002)(2002)Translational invariant solutionTranslational invariant solutionRotational invariance brokenRotational invariance broken

What What we think is happening in spin imbalanced systems?we think is happening in spin imbalanced systems?Induced Induced PP--wave wave superfluiditysuperfluidityTwo new superfluid phases where before they were not expectedTwo new superfluid phases where before they were not expected

One Bose superfluid coexisting with one POne Bose superfluid coexisting with one P--wave Fermi superfluidwave Fermi superfluidTwo coexisting PTwo coexisting P--wave Fermiwave Fermi superfluidssuperfluids

Bulgac, Forbes, Bulgac, Forbes, Schwenk, Phys. Rev. Schwenk, Phys. Rev. LettLett. . 9797, 020402 (2006), 020402 (2006)Two coexisting PTwo coexisting P wave Fermi wave Fermi superfluidssuperfluids

A refined EOS for spin unbalanced systemsA refined EOS for spin unbalanced systems

Bulgac and Forbes, Bulgac and Forbes, Phys. Rev. Phys. Rev. LettLett. . 101101, 215301 (2008), 215301 (2008)

Red line: LarkinRed line: Larkin--OvchinnikovOvchinnikov phasephase

5/3

Black line: normal part of the energy density Black line: normal part of the energy density Blue points: DMC calculations for normal state, Lobo et al, PRL Blue points: DMC calculations for normal state, Lobo et al, PRL 9797, 200403 (2006), 200403 (2006)Gray crosses: experimental EOS due to Shin, Phys. Rev. A 77, 041603(R) (2008)Gray crosses: experimental EOS due to Shin, Phys. Rev. A 77, 041603(R) (2008)

5/32 2/3 23 (6 )( , )

5 2b

a b aa

nE n n n gm n

π ⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

Asymmetric  SLDA (ASLDA)Asymmetric  SLDA (ASLDA)2 2( ) ( ) ( ) ( )∑ ∑2 2

0 0

2 2

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

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

n n

a n b nE E

a n b n

n r r n r r

r r r rτ τ

< >

= =

= ∇ = ∇

∑ ∑

∑ ∑0 0

*1( ) sign( )u ( )v ( ),2

n n

n

E E

n n nE

r E r rν

< >

= ∑

[ ]2

( ) ( ) ( ) ( ) ( ) ( ) ( )2 a a b br r r r r r r

mα τ α τ νΕ = + −Δ +

+( ) [ ]

2/32 25/33 3

( ) ( ) [ ( )],10 a bn r n r x r

mπ

β+

[ ] [ ]( ) ( ) , ( ) 1/ ( ) , ( ) ( ) / ( ),a b b ar x r r x r x r n r n rα α α α= = =

[ ]3 3 P( ) ( ) ( ) ( )a a b bd r r d r r n r n rμ μΩ = − = Ε − −∫ ∫Bulgac and Forbes, arXiv:0804:3364Bulgac and Forbes, arXiv:0804:3364

A Unitary Fermi Supersolid: A Unitary Fermi Supersolid: the Larkinthe Larkin--Ovchinnikov phaseOvchinnikov phasethe Larkinthe Larkin--Ovchinnikov phase Ovchinnikov phase

5/2Bulgac and Forbes Bulgac and Forbes 

5/23/2

2 2

2 2[ , ]30

ba b a

a

mP h μμ μ μπ μ

⎡ ⎤⎛ ⎞⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦

ggPRL PRL 101101, 215301 (2008), 215301 (2008)

Challenges towards implementation of SLDA in nucleiChallenges towards implementation of SLDA in nuclei

BaldoBaldo, , SchuckSchuck, and , and VinasVinas, arXiv:0706.0658, arXiv:0706.0658

GandolfiGandolfi et al. arXiv:0805.2513et al. arXiv:0805.2513

Gezerlis and Carlson, PRC 77, 032801 (2008)Gezerlis and Carlson, PRC 77, 032801 (2008) Bulgac et al. arXiv:0801.1504, arXiv:0803.3238Bulgac et al. arXiv:0801.1504, arXiv:0803.3238

Let Let us summarize us summarize some of the ingredients of some of the ingredients of the SLDA in nucleithe SLDA in nuclei

EnergyEnergy Density (ED) describing the normal systemDensity (ED) describing the normal system

ED contribution due to superfluid correlationsED contribution due to superfluid correlations

{ }⎪⎧

+= ∫)]()([)]()([

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

rrrr

rrrrrrrdE pnpnSpnNgs

ρρερρε

ννρρερρε

⎪⎩

⎪⎨⎧

=

=

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

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

rrrrrrrr

rrrr

npnpSpnpnS

npNpnN

ννρρεννρρε

ρρερρε

I iI iIsospinIsospin symmetry symmetry ((Coulomb energy and other relatively small terms not shown here.)Coulomb energy and other relatively small terms not shown here.)

Let us consider the simplest possible ED compatible with nuclear symmetriesLet us consider the simplest possible ED compatible with nuclear symmetries

[ ] npnpnpnpS gg ννννννρρε ||||,,, 21

20 −++=

and with the fact that nuclear pairing and with the fact that nuclear pairing corrrelationscorrrelations are relatively weak.are relatively weak.

npggnpnp

ρρρρρρ

and on wellas depend could and 10

like like −+

Let us stare at Let us stare at the anomalous part of the anomalous part of the ED for a moment, … or two.the ED for a moment, … or two.

[ ] |||| 22 ++ ννννννε

??SU(2) invariantSU(2) invariant

[ ]][ ' ]|||[|

||||,**22

21

20

gg

gg

pnnpnp

npnpnpS

+++=

−++=

νννννν

ννννννε

' 1010 gggggg −=+=

NB I am dealing here with sNB I am dealing here with s--wave pairing only (S=0 and T=1)!wave pairing only (S=0 and T=1)!

The last term could not arise from a twoThe last term could not arise from a two--body bare interactionbody bare interaction..

In the end one finds that a suitable superfluidIn the end one finds that a suitable superfluid nuclear EDF has the following structure:

IsospinIsospin symmetricsymmetric

[ ] ]|||)[|,(, 22npnpnpS g ννρρννε +=

]|||)[|,( 22 npnpnp

ppp

fρρρρ

ννρρ+

−−+

),(),( where pnnp

np

gg ρρρρ

ρρ

=

+

),(),( and pnnp ff ρρρρ =

Th li t t f b th d ddTh li t t f b th d ddThe same coupling constant for both even and odd The same coupling constant for both even and odd neutron/proton numbers!!!neutron/proton numbers!!!

A single universal  parameter for pairing!A single universal  parameter for pairing!

IntermissionIntermission

Part IIPart II

Time Dependent Phenomena Time Dependent Phenomena The timeThe time--dependent density functional theory is viewed in general as a dependent density functional theory is viewed in general as a reformulation of the exact quantum mechanical time evolution of a manyreformulation of the exact quantum mechanical time evolution of a many--body body system when only singlesystem when only single--particle properties are considered.particle properties are considered.

TDDFT for normal systems:TDDFT for normal systems:A KA K RajagopalRajagopal and J Calla a Ph s Re Band J Calla a Ph s Re B 77 1912 (1973)1912 (1973)A.K. A.K. RajagopalRajagopal and J. Callaway, Phys. Rev. B and J. Callaway, Phys. Rev. B 77, 1912 (1973), 1912 (1973)V. V. PeuckertPeuckert, J. Phys. C , J. Phys. C 1111, 4945 (1978) , 4945 (1978) E. E. RungeRunge and E.K.U. Gross, Phys. Rev. and E.K.U. Gross, Phys. Rev. LettLett. . 5252, 997 (1984), 997 (1984)http://www.tddft.orghttp://www.tddft.org

[ ]3( ) ( ( , ), ( , ), ( , ), ( , )) ( , ) ( , ) ...extE t d r n r t r t r t j r t V r t n r tε τ ν= + +∫TDSLDATDSLDA

u ( )r t∂⎧ ii i

* *i i

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

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

ext ext

ext ext

r th r t V r t r t r t r t r t it

r t r t r t h r t V r t r t

μ

μ

∂+ − + Δ + Δ =

∂

Δ + Δ − + − iv ( , )) r tit

⎧⎪⎪⎨ ∂⎪ =⎪ ∂⎩

t⎪ ∂⎩

For timeFor time--dependent phenomena one has to add currents!dependent phenomena one has to add currents!

From a talk given by I. From a talk given by I. StetcuStetcu, UNEDF 2010, UNEDF 2010

From a talk given by I. From a talk given by I. StetcuStetcu, UNEDF 2010, UNEDF 2010

From a talk given by I. From a talk given by I. StetcuStetcu, UNEDF 2010, UNEDF 2010

From a talk given by K.J. Roche, UNEDF 2010From a talk given by K.J. Roche, UNEDF 2010

A very rare excitation mode: the Higgs pairing mode.A very rare excitation mode: the Higgs pairing mode.

Energy of a Fermi system as a function of the pairing gapEnergy of a Fermi system as a function of the pairing gap

[ ] 0n vn+∇⋅ =2

[ ]2

02

mvmv nμ⎧ ⎫

+∇ + =⎨ ⎬⎩ ⎭

( )2

2( , ) ( , ) ( , ) ( , )4

i r t r t U r t r tmΔ

Ψ =− Ψ + Ψ Ψ

“Landau“Landau--Ginzburg” equationGinzburg” equationQuantum hydrodynamicsQuantum hydrodynamics

Higgs modeHiggs mode

Small amplitude oscillations of the modulus of Small amplitude oscillations of the modulus of the order parameter (pairing gap)the order parameter (pairing gap)

2Ω = Δ02HΩ = Δ

This mode has a bit more complex characterThis mode has a bit more complex charactercf.cf. VolkovVolkov and and KoganKogan (1972) (1972)

Response of a unitary Fermi system to changing Response of a unitary Fermi system to changing the scattering length with timethe scattering length with time

Tool: TD DFT extension to superfluid systems (TDTool: TD DFT extension to superfluid systems (TD--SLDA)SLDA)

•• All these modes have a very low frequency below the pairing gapAll these modes have a very low frequency below the pairing gapand a very large amplitude and excitation energy as welland a very large amplitude and excitation energy as well

N f th d b d ib d ith ithi Q t H d d iN f th d b d ib d ith ithi Q t H d d i

Bulgac and Yoon, Phys. Rev. Lett. 102, 085302 (2009)

•• None of these modes can be described either within Quantum HydrodynamicsNone of these modes can be described either within Quantum Hydrodynamicsor Landauor Landau--Ginzburg like approachesGinzburg like approaches

TDSLDATDSLDATDSLDA TDSLDA (equations TDHFB/(equations TDHFB/TDBdGTDBdG like)like)

( )( )( )

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

( ) ( ) ( )

( )( )( )* * *

ˆ ˆh , h , 0 ,u , u ,ˆ ˆh , h , , 0u , u ,

ˆ ˆ0 h h

n n

n n

r t r t r tr t r tr t r t r tr t r t

it tt

μ

μ↑↑ ↑↓↑ ↑

↓↑ ↓↓↓ ↓

⎛ ⎞− Δ⎛ ⎞ ⎛⎜ ⎟⎜ ⎟ ⎜ ⎟− −Δ∂ ⎜ ⎟ = ⎜ ⎟⎜ ⎟∂ Δ⎜ ⎟

⎞⎜ ⎟⎜ ⎟⎜ ⎟( )

( )( ) ( ) ( )

( ) ( ) ( )( )( )

* * *

* * *

v , v ,0 , -h , -h ,v , v ,ˆ ˆ, 0 -h , -h ,

n n

n n

r t r tt r t r t r tr t r tr t r t r t

μ

μ↑ ↑↑↑ ↑↓

↓ ↓↓↑ ↓↓

⎜ ⎟∂ −Δ +⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝Δ +⎝ ⎠

⎜ ⎟⎜ ⎟⎜ ⎟

⎠

•• The system is placed on a 3D spatial latticeThe system is placed on a 3D spatial lattice•• Derivatives are computed with FFTWDerivatives are computed with FFTW•• Fully selfFully self--consistent treatment with Galilean invarianceconsistent treatment with Galilean invariance•• No symmetry restrictionsNo symmetry restrictions•• Number of quasiparticle wave functions is of the order Number of quasiparticle wave functions is of the order

of the number of spatial lattice pointsof the number of spatial lattice pointsI iti l t t i th d t t f th SLDA (f ll lik HFB/I iti l t t i th d t t f th SLDA (f ll lik HFB/BdGBdG))•• Initial state is the ground state of the SLDA (formally like HFB/Initial state is the ground state of the SLDA (formally like HFB/BdGBdG))

•• The code was implementation on The code was implementation on JaguarPfJaguarPf

I will present a few short movies, illustrating the complex I will present a few short movies, illustrating the complex timetime--dependent dynamics in 3D of a unitary Fermi superfluiddependent dynamics in 3D of a unitary Fermi superfluidp y y pp y y pand of and of 280280Cf excited with various external probes.Cf excited with various external probes.

In each case we solved onIn each case we solved on JaguarPfJaguarPf the TDSLDA equations for athe TDSLDA equations for aIn each case we solved on In each case we solved on JaguarPfJaguarPf the TDSLDA equations for a the TDSLDA equations for a 323233 spatial lattice (approximately 30k to 40k quasiparticle spatial lattice (approximately 30k to 40k quasiparticle wavefunctions) for about 10k to 100k time steps using from about wavefunctions) for about 10k to 100k time steps using from about 30K i l 40K PE30K i l 40K PE30K to approximately 40K PEs30K to approximately 40K PEs

Fully unrestricted calculations!Fully unrestricted calculations!yy

The size of the problem we solve here is The size of the problem we solve here is several orders of magnitudeseveral orders of magnitudelarger than any other similar problem studied by other groups (andlarger than any other similar problem studied by other groups (andlarger than any other similar problem studied by other groups (and larger than any other similar problem studied by other groups (and we plan to further increase the size significantly).we plan to further increase the size significantly).

The movies will be eventually posted atThe movies will be eventually posted atThe movies will be eventually posted atThe movies will be eventually posted at

http://www.phys.washington.edu/groups/qmbnt/index.htmlhttp://www.phys.washington.edu/groups/qmbnt/index.html

SummarySummarySummarySummary•• Created a set of accurate and efficient tools for the petaflop regime Created a set of accurate and efficient tools for the petaflop regime Successfully Successfully implementatedimplementated on leadership class computers (Franklin, on leadership class computers (Franklin, JaguarPFJaguarPF))•• Currently capable of treating nuclear volumes as large as 50Currently capable of treating nuclear volumes as large as 503 3 fmfm33 ,, for up to for up to y p g gy p g g ,, pp10,00010,000--20,000 fermions , and for times up to a fraction of an attosecond 20,000 fermions , and for times up to a fraction of an attosecond fully selffully self--consistently and with no symmetry restrictions and under the action consistently and with no symmetry restrictions and under the action of complex of complex spatiospatio--temporal external probes temporal external probes •• Capable of treating similarly large systems of cold atoms Capable of treating similarly large systems of cold atoms •• The suites of codes can handle systems and phenomena ranging from: The suites of codes can handle systems and phenomena ranging from:

ground states propertiesground states propertiesit d t t i th li iit d t t i th li iexcited states in the linear response regime,excited states in the linear response regime,

large amplitude collective motion, large amplitude collective motion, response to various electromagnetic and nuclear probes,response to various electromagnetic and nuclear probes,

•• There is a clear path towards There is a clear path towards exascaleexascale applications and implementation of the applications and implementation of the Stochastic TD(A)SLDAStochastic TD(A)SLDA