Aurel BulgacAurel BulgacUniversity of Washington, Seattle, WA University of Washington, Seattle, WA

Collaborators: Yuan Lung (Alan) Luo (Seattle)Collaborators: Yuan Lung (Alan) Luo (Seattle) Piotr Magierski (Warsaw/Seattle)Piotr Magierski (Warsaw/Seattle) Kenneth J. Roche (ORNL, now at PNNL-Seattle)Kenneth J. Roche (ORNL, now at PNNL-Seattle) Sukjin Yoon (Seattle)Sukjin Yoon (Seattle)

Funding: DOE grants No. DE-FG02-97ER41014 (UW NT Group)Funding: DOE grants No. DE-FG02-97ER41014 (UW NT Group) DE-FC02-07ER41457 (SciDAC-UNEDF)DE-FC02-07ER41457 (SciDAC-UNEDF) Franklin supercomputer (Cray XT4) at Franklin supercomputer (Cray XT4) at NERSC/DOENERSC/DOE Athena and Tigger clusters at UWAthena and Tigger clusters at UW

Large Amplitude Superfluid Dynamics Large Amplitude Superfluid Dynamics of a Unitary Fermi Gas of a Unitary Fermi Gas

Outline:Outline:

DFT extension to superfluid systems and its furtherDFT extension to superfluid systems and its furtherextension to time-dependent phenomenaextension to time-dependent phenomena A rare excitation mode in unitary Fermi gases: A rare excitation mode in unitary Fermi gases: The Higgs modeThe Higgs mode The birth (and miscarriage) and life of vortices in The birth (and miscarriage) and life of vortices in a unitary Fermi gas witnessed in real timea unitary Fermi gas witnessed in real time

0 0 0

0 0

0

23

0 *( )

( ) ( ) ( ) ... ( )

(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 r

n r

22) , ( ) ( )

N N

ii i

r r r

Universal functional of particle density aloneUniversal functional of particle density aloneIndependent of external potentialIndependent of external potential

Kohn-Sham theoremKohn-Sham theorem

Injective mapInjective map(one-to-one)(one-to-one)

Normal Fermi systems only!

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

Dilute atomic Fermi gases Tc 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

Superconductivity and superfluidity in Fermi systemsSuperconductivity and superfluidity in Fermi systems

units (1 eV 104 K)

2 2/ 3 5/ 3

22

k k0 0

*k k

0

22 2/ 3 2/ 3

2/ 3

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

2 5

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

( ) u ( )v ( )

( )(3 ) ( )( ) ( )

2 3 ( )( ) (

k c k c

c

cc

cE E E E

cE E

ext

eff

r n rr r r

n r r r r

r r r

rn rU r V r

n rr g r

) ( )cr

The SLDA (DFT) energy density functional at unitarityThe SLDA (DFT) energy density functional at unitarityfor equal numbers of spin-up and spin-down fermions for equal numbers of spin-up and spin-down fermions

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

can take any positive value, can take any positive value, but the best results are obtained when but the best results are obtained when is fixed by the qp-spectrum is fixed by the qp-spectrum

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

GFMC - Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)GFMC - Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)FN-DMC - von Stecher, Greene and Blume, PRL FN-DMC - von Stecher, Greene and Blume, 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)

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

GFMC - Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)GFMC - Chang and Bertsch, Phys. Rev. A 76, 021603(R) (2007)FN-DMC - von Stecher, Greene and Blume, PRL FN-DMC - von Stecher, Greene and Blume, PRL 9999, 233201 (2007) , 233201 (2007) PRA PRA 7676, 053613 (2007), 053613 (2007)

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

2

E N E N E NN N

Fermions at unitarity in a harmonic trapFermions at unitarity in a harmonic trapPairing gapsPairing gaps

solid/dotted blue line - SLDA, homogeneous GFMC due to Carlson et al solid/dotted blue line - SLDA, homogeneous GFMC due to Carlson et al red circles - GFMC due to Carlson and Reddy red circles - GFMC due to Carlson and Reddy dashed blue line - SLDA, homogeneous MC due to Juilletdashed blue line - SLDA, homogeneous MC due to Juillet

black dashed-dotted line – meanfield at unitarityblack dashed-dotted line – meanfield at unitarity

Quasiparticle spectrum in homogeneous matterQuasiparticle spectrum in homogeneous matter

Two 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)

Time Dependent Phenomena and Formalism Time Dependent Phenomena and Formalism

The time-dependent density functional theory is viewed in general as a The time-dependent density functional theory is viewed in general as a reformulation of the exact quantum mechanical time evolution of a many-body reformulation of the exact quantum mechanical time evolution of a many-body system when only one-body properties are considered.system when only one-body properties are considered.

A.K. Rajagopal and J. Callaway, Phys. Rev. B A.K. Rajagopal and J. Callaway, Phys. Rev. B 77, 1912 (1973), 1912 (1973)V. Peuckert, J. Phys. C V. Peuckert, J. Phys. C 1111, 4945 (1978) , 4945 (1978) E. Runge and E.K.U. Gross, Phys. Rev. Lett. E. Runge and E.K.U. Gross, Phys. Rev. Lett. 5252, 997 (1984), 997 (1984)

http://www.tddft.orghttp://www.tddft.org

3

ii i

* *i i

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

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

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

ext

ext ext

ext ext

E t d r n r t r t r t j r t V r t n r t

r th r t V r t r t r t r t r t i

t

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

iv ( , ))

r ti

t

For time-dependent phenomena one has to add currents.For time-dependent phenomena one has to add currents.

Full 3D implementation of TD-SLDA is a petaflop problem and is almost Full 3D implementation of TD-SLDA is a petaflop problem and is almost complete.complete.

Bulgac and Roche, Bulgac and Roche,   J. Phys. Conf. Series J. Phys. Conf. Series 125125, 012064 (2008), 012064 (2008)

A rare excitation mode: the Higgs pairing mode.A 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

2

0

02

n vn

mvmv n

22

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

i r t r t U r t r tm

““Landau-Ginzburg” equationLandau-Ginzburg” 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)

02H

This mode has a bit more complex characterThis mode has a bit more complex charactercf.cf. Volkov and Kogan (1972) Volkov and Kogan (1972)

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

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

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

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

3D unitary Fermi gas confined to a 1D ho potential well (pancake)3D unitary Fermi gas confined to a 1D ho potential well (pancake)

Black solid line – Time dependence of the cloud radiusBlack solid line – Time dependence of the cloud radiusBlack dashed line – Time dependence of the quadrupole moment of momentum distributionBlack dashed line – Time dependence of the quadrupole moment of momentum distribution

New qualitative excitation mode of a superfluid Fermi system(non-spherical Fermi momentum distribution)

About 60,000 PDEs, 48About 60,000 PDEs, 4833 spatial lattice (homogeneity assumed in 3 spatial lattice (homogeneity assumed in 3rdrd direction), direction), Around 2,000,000 times steps, approximately 4-5 CPU-years for the time evolution Around 2,000,000 times steps, approximately 4-5 CPU-years for the time evolution

From Giorgini, Pitaevskii and Stringari, From Giorgini, Pitaevskii and Stringari, Rev. Mod. Phys., 80, 1215 (2008)Rev. Mod. Phys., 80, 1215 (2008)

Study based on BCS/Leggett approximationStudy based on BCS/Leggett approximation

Critical velocity in a unitary gasCritical velocity in a unitary gas

F

F

0.370(5)v

min 0.385(3)

v 0.370(5)v

s

qp

c

c

k

Values obtained using QMC dataValues obtained using QMC data

Miller et al. (MIT, 2007)Miller et al. (MIT, 2007)

v 0.25(3)vc F