The Incredible Many Facets of the Unitary Fermi...

The Incredible Many Facets of the Unitary Fermi GasThe Incredible Many Facets of the Unitary Fermi GasThe Incredible Many Facets of the Unitary Fermi GasThe Incredible Many Facets of the Unitary Fermi Gas

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

Collaborators: Joaquin E. Drut (Seattle, now in Columbus) Collaborators: Joaquin E. Drut (Seattle, now in Columbus) Michael McNeil Forbes (Seattle soon at LANL)Michael McNeil Forbes (Seattle soon at LANL)Michael McNeil Forbes (Seattle, soon at LANL)Michael McNeil Forbes (Seattle, soon at LANL)Piotr Magierski (Warsaw/Seattle)Piotr Magierski (Warsaw/Seattle)AchimAchim Schwenk (Seattle, now at TRIUMF)Schwenk (Seattle, now at TRIUMF)Gabriel Wlazlowski (Warsaw)Gabriel Wlazlowski (Warsaw)Gabriel Wlazlowski (Warsaw)Gabriel Wlazlowski (Warsaw)SukjinSukjin Yoon (Seattle)Yoon (Seattle)Yongle Yu (Seattle, now in Wuhan)Yongle Yu (Seattle, now in Wuhan)

Outline:Outline:

What is a unitary Fermi gas?What is a unitary Fermi gas?

Thermodynamic propertiesThermodynamic properties

Pairing gap and pseudoPairing gap and pseudo‐‐gapgap

EOS for spin imbalanced systemsEOS for spin imbalanced systems

PP‐‐wave pairingwave pairingPP‐‐wave pairingwave pairing

Small systems in traps and (A)SLDASmall systems in traps and (A)SLDA

Unitary Fermi supersolid: the LarkinUnitary Fermi supersolid: the Larkin‐‐OvchinnikovOvchinnikov phasephase

TimeTime‐‐dependent phenomenadependent phenomena

What else can we expect?What else can we expect?

What What is the unitary regime?is the unitary regime?

A gas of interacting fermions is in the unitary regime A gas of interacting fermions is in the unitary regime if the average separation between particles is large if the average separation between particles is large compared to their size (range of interaction), but compared to their size (range of interaction), but small compared to their scattering length. small compared to their scattering length.

The system is very dilute, but strongly interacting!The system is very dilute, but strongly interacting!

n n |a||a|33 11n n rr003 3 11b d ib d i

rr0 0 nn‐‐1/3 1/3 ≈ ≈ λλF F /2/2 |a||a|n n ‐‐ number densitynumber density

r0 ‐ range of interaction a ‐ scattering length

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 ofbody system composed ofWhat 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 contactinteraction. interaction.

Why? Besides pure theoretical curiosity this problem is relevant to neutron stars!Why? Besides pure theoretical curiosity this problem is relevant to neutron stars!

In 1999 it was not yet clear, either theoretically or experimentally, In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not! A number of people argued thatwhether such fermion matter is stable or not! A number of people argued that

ff

Why? Besides pure theoretical curiosity, this problem is relevant to neutron stars! Why? Besides pure theoretical curiosity, this problem is relevant to neutron stars!

under such conditions fermionic matter is unstable.under such conditions fermionic matter is unstable.

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

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

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

Superconductivity and superfluidity in Fermi systems

20 orders of magnitude over a century of (low temperature) physics20 orders of magnitude over a century of (low temperature) physics

Dilute atomic Fermi gases Dilute atomic Fermi gases TTcc ≈≈ 1010--1212 –– 1010‐‐99 eVeV

Li idLi id 33HH TT 1010 77 VVLiquid Liquid 33He He TTcc ≈≈ 1010--77 eVeV

Metals, composite materials Metals, composite materials TTcc ≈≈ 1010--3 3 –– 1010--22 eVeV

Nuclei, neutron stars Nuclei, neutron stars TTcc ≈≈ 101055 –– 101066 eVeV

•• QCD color superconductivityQCD color superconductivity TTcc ≈≈ 10107 7 –– 10108 8 eVeVQCD color superconductivity QCD color superconductivity TTcc ≈≈ 1010 1010 eVeV

units (1 eV ≈ 104 K)

Finite TemperaturesFinite Temperatures

2 2ˆ ˆ ˆ ⎡ ⎤⎛ ⎞ ⎛ ⎞Δ Δ

∫ ∫

Grand Grand Canonical PathCanonical Path‐‐Integral Monte CarloIntegral Monte Carlo

3 † † 3

3 †

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )2 2

ˆ ˆ ˆ ˆ ( ) ( ) , ( ) ( ) ( ), ,

ψ ψ ψ ψ

ψ ψ

↑ ↑ ↓ ↓ ↑ ↓

↑ ↓

⎡ ⎤⎛ ⎞ ⎛ ⎞Δ Δ= + = − + − −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

= + = =↑ ↓⎡ ⎤⎣ ⎦

∫ ∫

∫ s s s

H T V d x x x x x g d x n x n xm m

N d x n x n x n x x x s

( )ˆ ˆ ˆ ˆ( ) Tr exp Tr expβ β μ τ μ⎡ ⎤= =⎣ ⎦Z H N H( ){ } 1τ

β τ⎡ ⎤ = =⎣ ⎦N

N N

Trotter expansion (Trotter expansion (trotterizationtrotterization of the propagator)of the propagator)

( )( ) Tr exp Tr expβ β μ τ μ⎡ ⎤= − − = − −⎣ ⎦Z H N H( ){ }

( )

,

1 ˆ ˆ ˆ( ) Tr exp

τβ τ

β μ

⎡ ⎤ = =⎣ ⎦

⎡ ⎤= − −⎣ ⎦

N NT

E T H H N( )

( )

( ) Tr exp( )1 ˆ ˆ ˆ( ) Tr exp( )

β μ

β μ

⎡ ⎤= ⎣ ⎦

⎡ ⎤= − −⎣ ⎦

E T H H NZ T

N T N H NZ T

No approximations so far, except for the fact that the interaction is not well defined!No approximations so far, except for the fact that the interaction is not well defined!

Recast the propagator at each time slice and put the system on a 3d‐spatial lattice,in a cubic box of side L=N l with periodic boundary conditionsin a cubic box of side L=Nsl, with periodic boundary conditions

( ) ( ) ( ) 3ˆ ˆ ˆ ˆ ˆ ˆ ˆexp exp / 2 exp( ) exp / 2 ( )τ μ τ μ τ τ μ τ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− − ≈ − − − − − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦H N T N V T N O

Di t H bb dDi t H bb d St t i hSt t i h t f tit f ti

( ) 1

1ˆ ˆ ˆexp( ) 1 ( ) ( ) 1 ( ) ( ) , exp( ) 12σ

τ σ σ τ± ±↑ ↓±

− = + + = −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∑∏xx

V x An x x An x A g

Discrete HubbardDiscrete Hubbard‐‐StratonovichStratonovich transformationtransformation

( ) 1 2σ± =±xx

σσ--fields fluctuate both in space and imaginary timefields fluctuate both in space and imaginary time

2 2 2

1 , 4 2

ππ π

= − + <cc

mkm ka g l

Running coupling constant g defined by lattice Running coupling constant g defined by lattice

n(k)n(k)

22ππ/L/L L L –– box sizebox size

l l ‐‐ lattice spacinglattice spacing

kk ==ππ//ll

kk

kkmaxmax ππ//ll

How to choose the lattice spacing and the box How to choose the lattice spacing and the box size?size?

kkyy

π/l

π/l

2 2

2, ,

2F Tmlπε Δ

kkxx

/

2 2

2

22

ml

mLπδε >

2 2

2

2, F

mL

mLπε Δ

2π/L

sL N lξ =Momentum spaceMomentum space 2

pLπδ >

ˆ( ) ( , ) Tr ({ })Z T D x Uσ τ σ= ∏∫,

ˆ ˆ({ }) exp{ [ ({ }) ]}

x

U T h

τ

ττ

σ τ σ μ= − −

∫

∏OneOne‐‐body evolutionbody evolutionoperator in imaginary timeoperator in imaginary time

ˆ( , )Tr ({ }) ˆ ˆTr ({ })D x U HUσ τ σ σ⎡ ⎤⎣ ⎦∏∫ ,

({ })( ) ˆ( ) Tr ({ })

xE TZ T U

τ

σ⎣ ⎦= ∫

2ˆ ˆTr ({ }) {det[1 ({ })]} exp[ ({ })] 0U U Sσ σ σ= + = − > No sign problem!No sign problem!

ˆ({ })( , ) ( , ) ( ) ˆ1 ({ }k

Un x y n x y x

U

σϕσ↑ ↓= =

+*

,

exp( )( ), ( )

)c

l kk l k k l

ik xy x

Vϕ ϕ

<

⎡ ⎤ ⋅=⎢ ⎥

⎣ ⎦∑

({ }, )

c k l⎣ ⎦All traces can be expressed through these singleAll traces can be expressed through these single‐‐particle density matricesparticle density matrices

More details of the calculations:More details of the calculations:• Typical lattice Typical lattice sizes used from sizes used from 883 3 x 300x 300 (high Ts) to (high Ts) to 883 3 x x 18001800 (low Ts(low Ts))

•• Effective use of FFT(W) makes all imaginary time propagators diagonal (either in Effective use of FFT(W) makes all imaginary time propagators diagonal (either in real space or momentum space) and there is no need to store large matricesreal space or momentum space) and there is no need to store large matrices

•• Update field configurations using the Metropolis importanceUpdate field configurations using the Metropolis importance•• Update field configurations using the Metropolis importance Update field configurations using the Metropolis importance sampling algorithmsampling algorithm

•• Change randomly at a fraction of all space and time sites the signs the auxiliary Change randomly at a fraction of all space and time sites the signs the auxiliary fields fields σσ(x,(x,ττ) so as to maintain a running average of the acceptance rate between) so as to maintain a running average of the acceptance rate between0.4 and 0.6 0.4 and 0.6

•• ThermalizeThermalize for 50,000for 50,000 –– 100,000 MC steps or/and use as a start100,000 MC steps or/and use as a start‐‐upupThermalizeThermalize for 50,000 for 50,000 100,000 MC steps or/and use as a start100,000 MC steps or/and use as a start upupfield configuration a field configuration a σσ(x,(x,ττ))‐‐field configuration from a different Tfield configuration from a different T

•• At low temperatures use Singular Value Decomposition of the At low temperatures use Singular Value Decomposition of the l i U({l i U({ })}) bili hbili h iievolution operator U({evolution operator U({σσ}) }) to stabilize the to stabilize the numericsnumerics

•• Use 100,000Use 100,000‐‐2,000,000 2,000,000 σσ(x,(x,ττ))‐‐ field configurations for calculationsfield configurations for calculations

••MC correlation “time” MC correlation “time” ≈ 250 ≈ 250 –– 300 time steps300 time steps at T at T ≈ ≈ TTcc

a = a = ±±∞∞

BogoliubovBogoliubov--Anderson phononsAnderson phonons

Normal Fermi Normal Fermi GasGas(with vertical offset, solid line)(with vertical offset, solid line)

BogoliubovBogoliubov Anderson phononsAnderson phononsand quasiparticle contributionand quasiparticle contribution(dot(dot--dashed lien )dashed lien )

BogoliubovBogoliubov--Anderson phonons Anderson phonons contribution only contribution only

44

phonons 3/ 2

3 3( ) , 0.445 16

πε ξξ

⎛ ⎞= ≈⎜ ⎟

⎝ ⎠F s

TE T N

QuasiQuasi--particles contribution only particles contribution only (dashed line)(dashed line)

phonons 3/ 2

3

quasi-particles 4

5 16

3 5 2( ) exp5 2

ξ ε

πεε

⎜ ⎟⎝ ⎠

Δ Δ⎛ ⎞= −⎜ ⎟⎝ ⎠

F ss F

FF

TE T NT

µ µ -- chemical potential chemical potential ((circles)circles)

7 /3

5

2 exp2

ε

πε

⎝ ⎠

⎛ ⎞⎛ ⎞Δ = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

F

FFe k a

Bulgac, Drut, and MagierskiBulgac, Drut, and MagierskiPRL PRL 9696, 090404 (2006), 090404 (2006)

Measurement of the Entropy and Critical Temperature of a Strongly Interacting FermiMeasurement of the Entropy and Critical Temperature of a Strongly Interacting FermiMeasurement of the Entropy and Critical Temperature of a Strongly Interacting Fermi Measurement of the Entropy and Critical Temperature of a Strongly Interacting Fermi GasGas, , LuoLuo, Clancy, Joseph, , Clancy, Joseph, KinastKinast, and Thomas, PRL , and Thomas, PRL 9898, 080402 (2007), 080402 (2007)

Bulgac, Drut, and MagierskiBulgac, Drut, and MagierskiPRL PRL 9999, 120401 (2007), 120401 (2007)

Long range order and condensate fractionLong range order and condensate fraction

23 3 † †2( ) ( ) ( ) ( ) ( )d d⎛ ⎞

⎜ ⎟ ∫ ∫3 3 † †2 1 2 1 2 2 2

2 3 †2 1 1 1

2( ) ( ) ( ) ( ) ( )

2lim ( ) ( ) , ( ) ( ) ( )2r

g r d r d r r r r r r rN

N g r n r n r d r r r rN

ψ ψ ψ ψ

α ψ ψ

↑ ↓ ↓ ↑

→∞ ↑ ↑

⎛ ⎞= + +⎜ ⎟⎝ ⎠

= − = +

∫ ∫

∫2 1 1 1( ) ( ) , ( ) ( ) ( )2r g

Nψ ψ→∞ ↑ ↑∫

Bulgac, Drut, and Magierski, arXiv:0803:3238Bulgac, Drut, and Magierski, arXiv:0803:3238

Critical temperature for superfluid to normal transitionCritical temperature for superfluid to normal transition

AmherstAmherst‐‐ETH:ETH: BurovskiBurovski et al arXiv:0805:3047et al arXiv:0805:3047


AmherstAmherst‐‐ETH: ETH: BurovskiBurovski et al. arXiv:0805:3047et al. arXiv:0805:3047Hard and soft bosons: Hard and soft bosons: PilatiPilati et al. PRL et al. PRL 100100, 140405 (2008, 140405 (2008)

Ground state energyGround state energy

Bulgac, Drut, Magierski, and Bulgac, Drut, Magierski, and WlazowskiWlazowski, arXiv:0801:1504, arXiv:0801:1504Bulgac, Drut, and Magierski, arXiv:0803:3238Bulgac, Drut, and Magierski, arXiv:0803:3238

Pairing Gap and PseudoPairing Gap and Pseudo‐‐gapgapPairing Gap and PseudoPairing Gap and Pseudo‐‐gapgap

Response of the twoResponse of the two‐‐component Fermi gas in the unitary regime component Fermi gas in the unitary regime

†

00

Tr{exp[- (H- N+g (p))] (p)}( ) G( , )Tr{exp[- (H- N+g (p))]} g

dp T d pdg

ββ μ ψ ψχ τ τβ μ ψ

=

= − = −∫

OneOne‐‐body temperature (Matsubara) Green’s functionbody temperature (Matsubara) Green’s function

1 2= − 1 02Fk a

= 0Fk a

=


Q iQ i i li l d i id i iQuasiQuasi‐‐particle particle spectrum and pairing gap spectrum and pairing gap

1 exp[ ( )] 1( ) G( , )( ) [ ( )] 1

E pp d pE E

β βχ τ τβ

−= − =

+∫


0 ( ) exp[ ( )] 1E p E pβ +∫

QuasiQuasi‐‐particle spectrumparticle spectrum at unitarityat unitarity


1 2, 1, 0Fk a

= − −

222( )

2pE p Uα μ

⎛ ⎞= + − + Δ⎜ ⎟

⎝ ⎠ Bulgac, Drut, and Magierski, arXiv:0801:1504Bulgac, Drut, and Magierski, arXiv:0801:1504

Dynamical MeanDynamical Mean‐‐Field Theory applied to a Unitary Fermi Field Theory applied to a Unitary Fermi gasgas( t i i fi it b f di i )( t i i fi it b f di i )(exact in infinite number of dimensions)(exact in infinite number of dimensions)

Superfluid to insulator phase transition in a unitary Fermi gasSuperfluid to insulator phase transition in a unitary Fermi gasBarneaBarnea arXiv:0803:2293arXiv:0803:2293BarneaBarnea, arXiv:0803:2293, arXiv:0803:2293

2 2k2s

kE hm

ν φ− + = +

( ) ( 1) sE N E N E= − +

Using photoemission spectroscopy to probe a strongly interacting Fermi gasUsing photoemission spectroscopy to probe a strongly interacting Fermi gasStewart, Stewart, GaeblerGaebler and Jin, arXiv:0805:0026 and Jin, arXiv:0805:0026


Normal phaseNormal phase(unpaired spin(unpaired spin‐‐up fermions)up fermions)(unpaired spin(unpaired spin up fermions)up fermions)

Fully symmetric phaseFully symmetric phase LOFF phasesLOFF phasesFully symmetric phaseFully symmetric phase

Gapless BEC superfluidGapless BEC superfluid(BEC of (BEC of dimersdimers + unpaired spin+ unpaired spin‐‐up fermions)up fermions)

LOFF phasesLOFF phases

Son and Son and StephanovStephanov, PRA , PRA 7474, 013614 (2006), 013614 (2006)

What What we think is we think is happening in spin imbalanced systems?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 superfluid

Two coexisting PTwo coexisting P‐‐wave Fermiwave Fermi superfluidssuperfluids

Bulgac, Forbes, Bulgac, Forbes, Schwenk, PRL Schwenk, PRL 9797, 020402 (2006), 020402 (2006)

Two coexisting PTwo coexisting P wave Fermi wave Fermi superfluidssuperfluids

Going beyond the naïve BCS approximationGoing beyond the naïve BCS approximation

EliashbergEliashberg approx. (red)approx. (red)

BCS approx. (black)BCS approx. (black)

Full momentum and frequency dependence of the selfFull momentum and frequency dependence of the self‐‐consistent equations (blue)consistent equations (blue)

Bulgac and Yoon, unpublished (2007)

Full momentum and frequency dependence of the selfFull momentum and frequency dependence of the self‐‐consistent equations (blue)consistent equations (blue)

What happens at unitarity?What happens at unitarity?

Predicted quantum first phase order transition, subsequently observed Predicted quantum first phase order transition, subsequently observed

Red points with error bars Red points with error bars –– subsequent DMC calculations for normal statesubsequent DMC calculations for normal statedue to Lobo et al, PRL due to Lobo et al, PRL 9797, 200403 (2006) , 200403 (2006)

in MIT experiment , Shin in MIT experiment , Shin et alet al. arXiv:0709:3027. arXiv:0709:3027

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

5 2b

a b a a bnE n n n g n nπ ⎡ ⎤⎛ ⎞

= ≥⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

( )( )

5 2 am n⎜ ⎟⎝ ⎠⎣ ⎦

Bulgac and Forbes, PRA Bulgac and Forbes, PRA 7575, 031605(R) (2007) , 031605(R) (2007)

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

ff

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, Gray crosses: experimental EOS due to Shin, arXivarXiv :0801:1523:0801:1523

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

5 2b

a b aa

nE n n n gm n

π ⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦ Bulgac and Forbes, arXiv:0804:3364Bulgac and Forbes, arXiv:0804:3364

h f d f b l dh f d f b l dHow this new refined EOS for spin imbalanced systems How this new refined EOS for spin imbalanced systems was obtained?was obtained?

Through the use of the (A)SLDA , which is an extension of the Through the use of the (A)SLDA , which is an extension of the KohnKohn‐‐Sham LDA to superfluid systemsSham LDA to superfluid systems

How to construct and validate an How to construct and validate an abab initioinitio Energy DensityEnergy DensityFunctional (EDF)?Functional (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 theExtract the EDFEDFExtract the Extract the EDFEDF

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 systemsselect number of wisely selected finite systemsselect 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 theand compare with the abab initioinitio calculation and if lucky declarecalculation and if lucky declareand compare with the and compare with the abab initioinitio calculation, and if lucky declarecalculation, and if lucky declareVictory! Victory!

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 Only this combination is is cutoff independent cutoff independent

τ πε α ν β⎡ ⎤

= − Δ +⎢ ⎥2 2/3 5/3( ) 3(3 ) ( )

( ) ( ) ( )cr n r

r r r

yy pp

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)

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)

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 rm

α 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

Unitary Fermi Supersolid: the LarkinUnitary Fermi Supersolid: the Larkin‐‐OvchinnikovOvchinnikov phase phase

5/25/23/2

2 2

2 2[ , ]30

ba b a

a

mP h μμ μ μπ μ

⎡ ⎤⎛ ⎞⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦Bulgac and Forbes, arXiv:0804:3364Bulgac and Forbes, arXiv:0804:3364

TD ASLDATD ASLDABulgac, Roche, Yoon Bulgac, Roche, Yoon

ext extn nˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , )u ( , ) u ( , )h r t V r t r t r tr t r t

i⎛ ⎞+ Δ + Δ⎛ ⎞ ⎛ ⎞∂

= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟† †n next ext

ˆˆ ˆ ˆv ( , ) v ( , )( , ) ( , ) ( , ) ( , )i

r t r tt r t r t h r t V r t= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ Δ + Δ − −⎝ ⎠ ⎝ ⎠⎝ ⎠

3

3

( ) ( , , ) ( , , ) exp( )Q d rdtQ r t n r t i tσ

ω σ σ ω=∑∫4 53 x tN N× 4 5

3

, 50...100, 10 ...10

number of ( , , ) ( 40)x t

n x

N N

r t O Nψ σ

≈ ≈

≈ ×

SpaceSpace‐‐time lattice, use of FFTW for spatial derivative time lattice, use of FFTW for spatial derivative No matrix operations (unlike (Q)RPA)No matrix operations (unlike (Q)RPA)All nuclei (odd even spherical deformed)All nuclei (odd, even, spherical, deformed)Any quantum numbers of QRPA modesFully selfconsistent, no self‐consistent symmetries imposed

Higgs mode of the pairing field in a homogeneous unitary Fermi gasHiggs mode of the pairing field in a homogeneous unitary Fermi gas

A remarkable example of extreme A remarkable example of extreme ppLarge Amplitude Collective MotionLarge Amplitude Collective Motion

Bulgac and Yoon, unpublished (2007)Bulgac and Yoon, unpublished (2007)

A zoo of HiggsA zoo of Higgs‐‐like pairing modeslike pairing modesThe frequency of all these modes is below the 2The frequency of all these modes is below the 2‐‐qp gapqp gap

Maximum and minimum oscillation amplitudes versus frequencyMaximum and minimum oscillation amplitudes versus frequency

In case you’ve got lost:In case you’ve got lost: What did I talk about so far?What did I talk about so far?In case you ve got lost: In case you ve got lost: What did I talk about so far?What did I talk about so far?

What is a unitary Fermi gas?What is a unitary Fermi gas?


d dd dPairing gap and pseudoPairing gap and pseudo‐‐gapgap


PP‐‐wave pairingwave pairing

Small systems in traps and (A)SLDASmall systems in traps and (A)SLDASmall systems in traps and (A)SLDASmall systems in traps and (A)SLDA

Unitary Fermi supersolid: the LarkinUnitary Fermi supersolid: the Larkin‐‐OvchinnikovOvchinnikov phasephase

TimeTime‐‐dependent phenomenadependent phenomena

What else can we expect?What else can we expect?

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