Is a system of fermions in the crossover BCSIs a system of fermions in the crossover BCS--BEC BEC regime a new type of regime a new type of superfluidsuperfluid??

Finite temperature properties of a Fermi gas Finite temperature properties of a Fermi gas in the unitary regimein the unitary regime

AurelAurel BulgacBulgac, Joaquin E. , Joaquin E. DrutDrut, , PiotrPiotr MagierskiMagierskiUniversity of Washington, Seattle, WAUniversity of Washington, Seattle, WA

Also in WarsawAlso in Warsaw

OutlineOutline

Some general remarksSome general remarks

Path integral Monte Carlo for many fermions on the lattice at Path integral Monte Carlo for many fermions on the lattice at finite temperatures finite temperatures

Thermodynamic properties of a Fermi gas in the unitary Thermodynamic properties of a Fermi gas in the unitary regimeregime

ConclusionsConclusions

Superconductivity and Superconductivity and superfluiditysuperfluidity in Fermi systemsin Fermi systems

20 orders of magnitude over a century of (low temperature) physi20 orders of magnitude over a century of (low temperature) physicscs

Dilute atomic Fermi gases Dilute atomic Fermi gases TTcc ≈≈ 1010--1212 –– 1010-9 eVeV

Liquid 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 superconductivity QCD color superconductivity TTcc ≈ 10107 7 –– 10108 8 eVeV

units (1 eV ≈ 104 K)

A little bit of historyA little bit of history

BertschBertsch ManyMany--Body X challenge, Seattle, 1999Body X challenge, Seattle, 1999What 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 relevanWhy? Besides pure theoretical curiosity, this problem is relevant to neutron stars! t to neutron stars!

In 1999 it was not yet clear, either theoretically or experimentIn 1999 it was not yet clear, either theoretically or experimentally, ally, whether such whether such fermionfermion matter is stable or not! A number of people argued thatmatter is stable or not! A number of people argued thatunder such conditions under such conditions fermionicfermionic matter is unstable.matter is unstable.

- 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)

• 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.

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

Bertsch’sBertsch’s regime is nowadays called regime is nowadays called the unitary regimethe unitary regime

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

n |a|n |a|33 11n rn r003 3 11

rr0 0 nn--1/3 1/3 ≈≈ λλF F /2/2 |a||a|

rr00 -- range of interactionrange of interaction a a -- scattering lengthscattering length

n n -- number densitynumber density

1/a

T

a0shallow 2-body bound state

halo halo dimersdimers

BCS BCS SuperfluidSuperfluid

High T, normal atomic (plus a few molecules) phase

Molecular BEC andMolecular BEC andAtomic+MolecularAtomic+MolecularSuperfluidsSuperfluids

Expected phases of a two species dilute Fermi system Expected phases of a two species dilute Fermi system BCSBCS--BEC crossoverBEC crossover

weak interactionweak interaction

weak interactionsweak interactions

Strong interactionStrong interaction

Dyson (?), Eagles (1969), Leggett (1980) …Dyson (?), Eagles (1969), Leggett (1980) …Early theoretical approach to BCSEarly theoretical approach to BCS--BEC crossoverBEC crossover

( )† †k k

2

2

u +v BCS wave function

1 1 gap equation4 2 2E

2 1 number density equationE

8 exp 2e

π ε

ε µ

πε

↑ − ↓=

⎛ ⎞= −⎜ ⎟

⎝ ⎠⎛ ⎞−

= −⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

∆ ≈

∏

∑

∑

k kk

k k k

k

k k

FF

gs a a vacuum

ma

n

k a2 2

2 22 2 2k k k

2 2 32 2total

2

E ( ) quasi-particle energy

1, u v 1, v = 1 2 2 E

pairing gap

E 3 3 = ... , N 5 2 8 3

ε µ

ε µε

πµ π

= − + ∆

⎛ ⎞−= + = −⎜ ⎟

⎝ ⎠

∆+ + − =

k k

kk

k

F F

km

k ka n nmmNeglected/overlookedNeglected/overlooked Exponentially suppressed in BCSExponentially suppressed in BCS

Consequences:Consequences:

•• Usual BCS solution for small and negative scattering lengths,Usual BCS solution for small and negative scattering lengths,with exponentially small pairing gap with exponentially small pairing gap

•• For small and positive scattering lengths this equations descFor small and positive scattering lengths this equations describe ribe a gas a weakly repelling (weakly bound/shallow) molecules, a gas a weakly repelling (weakly bound/shallow) molecules, essentially all at rest (almost pure BEC state)essentially all at rest (almost pure BEC state)

In BCS limit the particle projected manyIn BCS limit the particle projected many--body wave functionbody wave functionhas the same structure (BEC of spatially overlapping Cooper pairhas the same structure (BEC of spatially overlapping Cooper pairs)s)

•• For both large positive and negative values of the scatteringFor both large positive and negative values of the scatteringlength these equations predict a smooth crossover from BCS to BElength these equations predict a smooth crossover from BCS to BEC,C,from a gas of spatially large Cooper pairs to a gas of small mofrom a gas of spatially large Cooper pairs to a gas of small moleculeslecules

( ) [ ]1 2 3 4 12 34, , , ,... ( ) ( )...r r r r r rϕ ϕΨ ≈ A

What is wrong with this approach:

• The BCS gap (a0 and small) the molecule repulsion is overestimated

• The approach neglects of the role of the “meanfield (HF) interaction,”which is the bulk of the interaction energy in both BCS and unitary regime

• All pairs have zero center of mass momentum, which is reasonable in BCS and BEC limits, but incorrect in theunitary regime, where the interaction between pairs is strong !!!(this situation is similar to superfluid 4He)

3

0

8 , , 0.623

exm mm m mm

n nn a n a an π

= = ≈Fraction of nonFraction of non--condensedcondensedpairs (pairs (perturbativeperturbative result)!?!result)!?!

( ) [ ]1 2 3 4 12 34, , , ,... ( ) ( )...r r r r r rϕ ϕΨ ≈ A

From a talk of Stefano From a talk of Stefano GiorginiGiorgini ((TrentoTrento))

What is the best, the most accurate theory What is the best, the most accurate theory (so far) for the T=0 case?(so far) for the T=0 case?

Fixed-Node Green Function Monte Carlo approach at T=0

BCS 2

7 /3

Gorkov

8 exp e 2

2 exp 2

FF

FF

k a

e k a

πε

πε

⎛ ⎞∆ ≈ ⎜ ⎟

⎝ ⎠

⎛ ⎞⎛ ⎞∆ ≈ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

Carlson et al. PRL 91, 050401 (2003)Chang et al. PRA 70, 043602 (2004)AstrakharchikAstrakharchik et et al.al.PRLPRL 93, 200404(2004)200404(2004)

Theory for fermions at T >0Theory for fermions at T >0in the unitary regimein the unitary regime

Put the system on a spatio-temporal lattice and usea path integral formulation of the problem

Grand Canonical PathGrand Canonical Path--Integral Monte CarloIntegral Monte Carlo

( )

2 23 † † 3

3 †

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )2 2

ˆ ˆ ˆ ˆ ( ) ( ) , ( ) ( ) ( ), ,

ˆ ˆ ˆ ˆ( ) Tr exp Tr exp

ψ ψ ψ ψ

ψ ψ

β β µ τ µ

↑ ↑ ↓ ↓ ↑ ↓

↑ ↓

⎡ ⎤⎛ ⎞ ⎛ ⎞∆ ∆= + = − + − −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

= + = =↑ ↓⎡ ⎤⎣ ⎦

⎡ ⎤= − − = − −⎣ ⎦

∫ ∫

∫ 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

Z H N H( ){ }

( )

( )

1,

1 ˆ ˆ ˆ( ) Tr exp( )1 ˆ ˆ ˆ( ) Tr exp( )

τ

τβ τ

β µ

β µ

⎡ ⎤ = =⎣ ⎦

⎡ ⎤= − −⎣ ⎦

⎡ ⎤= − −⎣ ⎦

N

N NT

E T H H NZ T

N T N H NZ T

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

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

Recast the propagator at each time slice and put the system on aRecast the propagator at each time slice and put the system on a 3d3d--spatial lattice,spatial lattice,in a cubic box of side L=in a cubic box of side L=NNssll, with periodic boundary conditions, with periodic boundary conditions

( ) ( ) ( ) 3

( ) 1

2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆexp exp / 2 exp( ) exp / 2 ( )

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

1 , 4 2

σ

τ µ τ µ τ τ µ τ

τ σ σ τ

ππ π

±

± ±↑ ↓=±

⎡ ⎤ ⎡ ⎤ ⎡ ⎤− − ≈ − − − − − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

− = + + = −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= − + <

∑∏xx

cc

H N T N V T N O

V x An x x An x A g

mkm ka g l

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

Discrete HubbardDiscrete Hubbard--StratonovichStratonovich transformationtransformation

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

kkyyπ/l

π/l

2 2

2

2 2

2

2 2

2

, , 2

2

2,

2

F

F

s

Tml

mL

mL

L N l

pL

πε

πδε

πε

ξ

πδ

∆

>

∆

=

>

kkxx

2π/L

Momentum spaceMomentum space

22ππ/L/L

n(kn(k))

kk

kkmaxmax==ππ//ll

l l -- lattice spacinglattice spacing

L L –– box sizebox size

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

From a talk of J.E. From a talk of J.E. DrutDrut

,

,

2

ˆ( ) ( , ) Tr ({ })

ˆ ˆ({ }) exp{ [ ({ }) ]}

ˆ( , )Tr ({ }) ˆ ˆTr ({ })( ) ˆ( ) Tr ({ })

ˆ ˆTr ({ }) {det[1 ({ })]} exp[ ({ })] 0

ˆ({ })( , ) ( , ) ( ) ˆ1 ({ }

x

x

k

Z T D x U

U T h

D x U HUE T

Z T U

U U S

Un x y n x y x

U

τ

ττ

τ

σ τ σ

σ τ σ µ

σ τ σ σ

σ

σ σ σ

σϕσ↑ ↓

=

= − −

⎡ ⎤⎣ ⎦=

= + = − >

= =+

∏∫

∏

∏∫

*

,

exp( )( ), ( )

)c

l kk l k k l

ik xy x

Vϕ ϕ

<

⎡ ⎤ ⋅=⎢ ⎥

⎣ ⎦∑

No sign problem!No sign problem!

OneOne--body evolutionbody evolutionoperator in imaginary timeoperator in imaginary time

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:• Lattice sizes used from 6Lattice sizes used from 63 3 x (300x (300--1361), 81361), 833 x (250x (250--1750) 1750) and 10and 1033

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

•• Update field configurations using the Metropolis importance sampUpdate field configurations using the Metropolis importance sampling algorithmling algorithm

•• Change randomly at a fraction of all space and time sites the sChange randomly at a fraction of all space and time sites the signs the auxiliary fields igns the auxiliary fields σσ(x,(x,ττ) so as to maintain a running average of the acceptance rate of ) so as to maintain a running average of the acceptance rate of about 0.5about 0.5

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

Superfluid to Normal Fermi Liquid Transition

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

BogoliubovBogoliubov--Anderson phononsAnderson phononsand and quasiparticlequasiparticle contributioncontribution(dot(dot--dashed lien )dashed lien )

BogoliubovBogoliubov--Anderson phonons Anderson phonons contribution only (little crosses)contribution only (little crosses)People never consider this ???People never consider this ???

44

phonons 3/ 2

3

quasi-particles 4

7 /3

3 3( ) , 0.445 16

3 5 2( ) exp5 2

2 exp2

πε ξξ ε

πεε

πε

⎛ ⎞= ≈⎜ ⎟

⎝ ⎠

∆ ∆⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞∆ = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

F ss F

FF

FF

TE T N

TE T NT

e k a

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

a = a = ±±∞∞

µµ -- chemical potential chemical potential (circles)(circles)

3 2 2

2

3 - = ( ) ( )

5 ( )

, ( )3 2

5( ) 23 , ( )

( ) 3

FF

F FF

F

TE N PV TS n N e n nV

n

N k kn nV m

e n TS N N P e n n

T n

µ ε εε

επ

µσ

ε

⎛ ⎞= + =⎜ ⎟

⎝ ⎠

= = =

− ⎛ ⎞= = =⎜ ⎟

⎝ ⎠

SS

EE

µµ

The known dependence of the entropy on temperature atThe known dependence of the entropy on temperature atunitarityunitarity S(T) can be used to devise a S(T) can be used to devise a thermometer!thermometer!

How? How?

•• The temperature can be measured either in the BCS The temperature can be measured either in the BCS or BEC limits, where interactions are weak, and easily be or BEC limits, where interactions are weak, and easily be related to the entropy S(T) related to the entropy S(T)

•• The system can then adiabatically (S= constant) be brought The system can then adiabatically (S= constant) be brought into the unitary regime and from the calculated S(T) one can into the unitary regime and from the calculated S(T) one can readread TT

SS

EE

µµ

NB The chemical potential is essentially constant below NB The chemical potential is essentially constant below the critical temperature!the critical temperature!

5/2

3( )

5

2( )

5

( ) 2( ) 0.44(2)

5

( ) , 11(1)

FF

F

F s F FF F F

s s s

TE T N

TS T N

dE T T T TT

dN

x x

ε ξε

ξε

µ ε ξ ξ ξ ε εε ε ε

ξ ξ ς ς

⎛ ⎞= ⎜ ⎟

⎝ ⎠

⎛ ⎞′= ⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞′= = − ≈ ≈⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

⇒ = + ≈

This is the same behavior as for a gas of This is the same behavior as for a gas of noninteractingnoninteracting (!) bosons (!) bosons below the condensation temperature. below the condensation temperature.

2

2 2

24 2 2

2 3 3

3/2

5/21/2 2 3

2

assuming a Bogoliubov like spectrum ( ) 14

3 (3) (3) v( ) , if ,

5 4 3

3 33 2 2( ) , 5 2

if for

B

FF s B s

B

F s

B

pp pc

m c

E T N T V T m c cc

mE T N T V

T m c

ε

ςε ξ ξπ

ςε ξ

π

= +

Γ≈ + =

⎛ ⎞ ⎛ ⎞Γ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠≈ +

an ideal Bose gas

and fitting to lattice results 3Bm m⇒ ≈

•• Why this value for the Why this value for the bosonicbosonic mass?mass?

•• Why these bosons behave like Why these bosons behave like noninteractingnoninteracting particles?particles?

Let us consider other power law behaviorsLet us consider other power law behaviors

3( )

5

( ) 2( ) 1

5

( ) 50

2

n

F s sF

n

F s sF

TE T N

dE T n TT

dN

d Tn

dT

ε ξ ςε

µ ε ξ ςε

µ

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞⎛ ⎞= = + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

≤ ⇒ ≥

Lattice results disfavor either Lattice results disfavor either nn≥≥33 or or nn≤≤22and suggest and suggest n=2.5(0.25)n=2.5(0.25)

Functional form for an ideal Bose gas!Functional form for an ideal Bose gas!

From a talk of J.E. From a talk of J.E. DrutDrut

ConclusionsConclusions

The present The present calculationalcalculational scheme is free of the scheme is free of the fermionfermion sign problem sign problem (unlike the Quantum MC T=0 results). The T=0 limit of the presen(unlike the Quantum MC T=0 results). The T=0 limit of the present t results confirms the QMC results.results confirms the QMC results.

Fully nonFully non--perturbativeperturbative calculations for a spin ½ many calculations for a spin ½ many fermionfermionsystem in the unitary regime at finite temperatures are feasiblesystem in the unitary regime at finite temperatures are feasible..This system undergoes a phase transition in the bulk at This system undergoes a phase transition in the bulk at TTcc = 0.22 (3) = 0.22 (3) εεFF

The phase transition is observed in various thermodynamic potenThe phase transition is observed in various thermodynamic potentials tials (total energy, entropy, chemical potential) as well as in the pr(total energy, entropy, chemical potential) as well as in the presenceesenceof offof off--diagonal long range order in the twodiagonal long range order in the two--body density matrix. One can body density matrix. One can define also a thermometer.define also a thermometer.

Below the transition temperature the system behaves as a free cBelow the transition temperature the system behaves as a free condensedondensedBose gas (!), which is Bose gas (!), which is superfluidsuperfluid at the same time! No thermodynamic hint ofat the same time! No thermodynamic hint ofFermionicFermionic degrees of freedom!degrees of freedom!Above the critical temperature one observes the thermodynamic beAbove the critical temperature one observes the thermodynamic behavior ofhavior ofa free Fermi gas! No thermodynamic trace of a free Fermi gas! No thermodynamic trace of bosonicbosonic degrees of freedom! degrees of freedom!

New type of New type of fermionicfermionic superfluidsuperfluid. .