L –limit for thespatialcorrelationsinthesystem

BCS BCS - - BEC BEC crossover crossover at at finite finite temperature temperature – – Quantum Monte Quantum Monte Carlo Carlo study study

Piotr Magierski

Warsaw University of Technology

Collaborators

Collaborators:: Aurel Bulgac – University of Washington (Seattle), Joaquin E. Drut – Ohio State University (Columbus), Gabriel Wlazłowski (PhD student) – Warsaw University of Technology

S S cattering cattering at at low low energies energies ( ( s s - - wave wave scattering scattering ) )

2

- radius of the interaction potential k R

R

λ = π >>

scattering amplitude

( ) ; -

ik r e ikr

r e f f

ψ G = ^{G G ⋅} + r

2 0 0

1 , - scattering length, r - effective range 1 1

2

f a

ik a r k

=

− − +

If k → 0 then the interaction is determined by the scattering length alone.

¾ ¾ What is the unitary regime? What is the unitary regime?

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

n n - - particle particle density density

n |a|

n |a| ^{3 3}   1 1 n r n r _{0 0} ^{3 3}   1 1

r r a

- - effective effective range range a - - scattering length scattering length

. . 0 0,

i e r → a →±∞

AT FINITE AT FINITE

TEMPERATURE:

TEMPERATURE: ^{( )} ( )

^{, (0)= 0}

T FG

E T = ξ _ε E ξ ξ

NONPERTURBATIVE NONPERTURBATIVE

REGIME REGIME

System

System is is dilute dilute but but strongly

strongly interacting interacting ! !

( ) ( )( )

10 6

1 1 11 2 2 ... + pairing

9 35

3 - Energy of the noninteracting Fermi gas 5

F F

FG

FG F

E E k a k a ln

E N

π π

ε

⎡ ⎤

= + ⎢⎣ + − + ⎥⎦

=

Perturbation series

UNIVERSALITY:

UNIVERSALITY: E = ξ 0 FG E

1/a T

a<0

no 2-body bound state

a>0

shallow 2-body bound state

Expected phases of a two species dilute Fermi system Expected phases of a two species dilute Fermi system

BCS BCS - - BEC crossover BEC crossover

BCS Superfluid BCS Superfluid

Molecular BEC and Molecular BEC and Atomic+Molecular Atomic+Molecular Superfluids

Superfluids

weak interactions weak interactions

Strong interaction Strong interaction UNITARY REGIME UNITARY REGIME

?

Bose molecule

EASY!

EASY! EASY! EASY!

weak interaction weak interaction

In dilute atomic systems experimenters can control nowadays In dilute atomic systems experimenters can control nowadays almost anything:

almost anything:

• • The number of atoms in the tra The number of atoms in the trap: t p: typically ypically about 10 about 10

10 10

atoms atoms divided

divided 50- 50 - 50 among 50 among the lowest two hyperfine states. the lowest two hyperfine states .

• • The density of atoms The density of atoms

• • Mixtures of various atoms Mixtures of various atoms

• • The temperature of the atomic cloud The temperature of the atomic cloud

• •

Who does experiments?

Who does experiments?

•• Jin’s group at Boulder Jin’s group at Boulder

•• Grimm’s group in InnsbruckGrimm’s group in Innsbruck

•• Thomas’ group at DukeThomas’ group at Duke

•• Ketterle’sKetterle’s group at MIT group at MIT

•• Salomon’s group in ParisSalomon’s group in Paris

•• Hulet’sHulet’s group at Ricegroup at Rice

Physics Today, v54, 20 (2001)

One One fermionic fermionic atom in magnetic field atom in magnetic field

F m F

;

F I J J L S G G G G G = + = + G

Nuclear spin Electronic spin

TwoTwo hypefinehypefine statesstates areare populated

populated inin thethe traptrap Collision

Collision of twoof two atoms:atoms: At low energies (low density of atoms) only L=0 (s-wave) scattering is effective.

•• Due to the high diluteness atoms in the same hyperfineDue to the high diluteness atoms in the same hyperfine state do not interact

state do not interact..

•• Atoms in different hyperfine states experience interactions Atoms in different hyperfine states experience interactions only in s

only in s--wave.wave.

Evidence

Evidence for for fermionic fermionic superfluidity

superfluidity : : vortices! vortices !

M.W. Zwierlein et al., Nature, 435, 1047 (2005)

system of fermionic

Li atoms

Feshbach

Feshbach resonance: resonance: B=834G

B=834G

BEC side:

a>0

BCS side:

a<0

UNITARY REGIME UNITARY REGIME

- Spin up fermion - Spin down fermion

External conditions:

- te m p e ra tu re

- c h e m ic a l p o te n tia l T

µ

cut ;

k x

x

= π ∆

∆

L –limit for thespatial correlationsinthesystem

Coordinate

Coordinate spacespace Volume L³

lattice spacing x

=

= ∆

Periodic boundary conditions imposed

π π

ε

Λ =

∆

Λ =

Λ < < ∆ < < Λ

= =

U V

I R

2 2 2 2

U V m o m e n t u m c u t o f f I R m o m e n t u m c u t o f f 2

,

2 2

I R U V

F

x L

m m

k k

k k

2π/L

x π

∆

x π

∆

kk

kk_cutcut==π/π/''xx

22π/Lπ/L

n(k)n(k)

Momentum

Momentum spacespace

( )

3 † 2 3

3 †

ˆ ˆ ˆ ˆ ( ) ˆ ( ) ˆ ( ) ( ) ˆ

2

ˆ ˆ ( ) ˆ ( ) ; ˆ ( ) ˆ ( ) ˆ ( )

s s

s

s s s

H T V d r r r g d r n r n r

m

N d r n r n r n r r r

ψ ψ

ψ ψ

↑ ↓

=↑↓

↑ ↓

⎛ ∆ ⎞

= + = ⎜ − ⎟ −

⎝ ⎠

= + =

∫ ∑ ∫

∫

G = G G G

G G G G G

2 2 2

1

4 2

mk

m

g ^{= −} π ⁼ a ⁺ π ⁼

2

1 - U N IT A R Y L IM IT 2

m g ⁼

π

( ) ( ) ( )

3

( ) 1

1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

exp exp /2 exp( )exp /2

( )

ˆ 1 ˆ ˆ

exp( ) 1 ( ) ( ) 1 ( ) ( ) , exp( ) 1 2

ˆ ( ) ˆ ( );

ˆ (

r r

N j j

j

H N T N V T N

O

V r An r r An r A g

U W

W

τ

σ

τ µ τ µ τ τ µ

τ

τ σ σ τ

σ σ

σ

↑ ↓

=±

=

⎡ − − ⎤ ≈ ⎡ − − ⎤ − ⎡ − − ⎤

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

+

− = ⎡ ⎣ + ⎤⎡ ⎦⎣ + ⎤ ⎦ = −

=

∏ ∑

∏

G G

G G G G

( ^{ˆ ˆ} ) ^{ˆ ˆ} ( ^{ˆ ˆ} )

) exp /2 1 ( ) ( ) 1 ( ) ( ) exp /2

r

T N r An r r An r T N

τ µ σ

σ

τ µ

⎡ ⎤ ⎡ ⎤

= ⎣ − − ⎦ ∏

⎡ ⎣ + ^{G G} ⎤⎡ ⎦⎣ + ^{G G} ⎤ ⎦ ⎣ − − ⎦

Discrete Hubbard

Discrete Hubbard--StratonovichStratonovich transformationtransformation

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

τ

σ σ σ τ

β τ

σ τ σ

σ τ τ

σ τ σ µ

σ τ σ σ

σ

σ σ

=± =± =±

↑

=

≡ =

= − −

⎡ ⎤

⎣ ⎦

=

= + =

∫

∑ ∑ ∑

∫

∫

∫

G G G

G G

G

{ ( ,1) 1} { ( ,2) 1} { ( , ) 1}

0

2

( ) ( , ) Tr ({ }); ˆ

( , ) ... ; 1

ˆ ({ }) exp{ [ ({ }) ˆ ]}

Tr ˆ ˆ ({ }) ( , )Tr ({ }) ˆ

( ) ( ) Tr ({ }) ˆ

ˆ ˆ

Tr ({ }) {det[1 ( )]}

r r r N

Z T D r U

D r N

T

U T d h

D r U HU

E T Z T U

U U σ

ψ σ ψ ψ

σ

↑ ↓

<

− >

⎡ ⎤ ⋅

= = ∑

⎢ ⎣ + ⎥ ⎦

= ^{G G}

G G G G G

G G

, 3

exp[ ({ })] 0

({ }) exp( )

( , ) ( , ) ( ) ( ), ( )

1 ({ })

c

k l k

k l k k l

S

U ik x

n x y n x y x y x

U L

No sign problem No sign problem for for unpolarized unpolarized system

system ! ! One- One -body evolution body evolution

operator in imaginary time operator in imaginary time

All traces can be expressed through these single

All 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: 6Lattice sizes used: 6^{3 3} ––1010³³. . Imaginary

Imaginary timetime steps:8steps:8^{3 3} xx300300(high Ts) to 8(high Ts) to 8^{3 3} x 1800x 1800 (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 larg

real space or momentum space) and there is no need to store large matrices.e matrices.

•• Update field configurations using the Metropolis importance sampling algorithmUpdate field configurations using the Metropolis importance sampling 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(r(r,,ττ) so as to maintain a running average of the acceptance rate bet) so as to maintain a running average of the acceptance rate betweenween 0.4 and 0.6

0.4 and 0.6 ..

•• 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-up-up field field configuration a

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 evolution operator At low temperatures use Singular Value Decomposition of the evolution operator U({σU({σ}) }) to stabilize the numericsto stabilize the numerics..

•• Use 2Use 200,00000,000--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 ≈ TT_cc

Deviation

Deviation from from Normal Fermi Gas Normal Fermi Gas

Bogoliubov

Bogoliubov--Anderson phononsAnderson phonons and quasiparticleand quasiparticle contributioncontribution (dashed(dashed lineline ))

Bogoliubov

Bogoliubov--Anderson phonons Anderson phonons contribution only (

contribution only (dotteddotted line)line) Quasi

Quasi--particle contribution onlyparticle contribution only (dotted(dotted line)line)

Normal Fermi Gas

(with vertical offset, solid line) (with vertical offset, solid line)

a =

a = ± ± ∞ ∞

( 0) 0.41(2)T ξ = ≈

3

quasi-particles 4

7 / 3

3 5 2

( ) exp

5 2

2 exp

2

F

F

F

F

E T N T

T

e k a

ε π

ε ε π

∆ ⎛ ∆ ⎞

= ⎜⎝− ⎟⎠

⎛ ⎞

∆ =⎛ ⎞⎜ ⎟⎝ ⎠ ⎜⎝ ⎟⎠

4 4

phonons 3/2

3 3

( ) , 0.44

5

16

E T ε N π T ξ

ξ ε

= ⎛ ⎞ ⎜ ⎟ ≈

⎝ ⎠

A. Bulgac, J.E. Drut, P. Magierski,PRL96,090404(2006)

ε ξ ε π ε

⎛ ⎞

⎜ ⎟

⎝ ⎠

= =

= =

2

= 3 ( )

5 ( )

, ( )

3 2

F

F

F F

F

E n N T

n

N k k

n n

V m

µ µ

E E S S

Ideal Fermi gas entropy

ξ

+

=

= ∂

∫ ∂

∫

3

'( )

( ) 5 ( ) (0)

F

T

T e

y

S T

S T S E dT

T

N dy

T

y

Thermodynamics

Thermodynamics of of the the unitary unitary Fermi Fermi gas gas

ENERGY: ( ) 3 ( ) ;

5

E x ξ ε x N x T

= = ε

0

0

3 3 '( )

'( ) ( )

5 5

( ) 3 '( ) ENTROPY/PARTICLE: ( )

5

x V

x

S E y

C T N x S x N dy

T T y

S x y

x dy

N y

ξ ξ

σ ξ

∂ ∂

= = = ⇒ =

∂ ∂

= =

∫

∫

FREE ENERGY: 3 ( )

5

( ) ( ) ( )

F E TS x

N x x x x

ϕ ε

ϕ ξ σ

= − =

= −

Low Low temperature temperature behaviour behaviour of a Fermi of a Fermi gas gas in in the the unitary unitary regime regime

ε ϕ µ ξ

ε ε

⎛ ⎞

= ⎜ ⎟ = − ≈ ≈ <

⎝ ⎠

3 ( )

( ) and 0.41(2) for

5

T T

F T N E TS T T

µ ε ϕ ϕ ε ξ

ε ε ε

⎡ ⎛ ⎞ ⎛ ⎞ ⎤

= = ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ≈

⎝ ⎠ ⎝ ⎠

⎣ ⎦

( ) 2

( ) '

F

5

F F F

dF T T T T

T dN

ϕ ϕ ϕ

ε ε

⎛ ⎞ ⎛ ⎞

= +

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

5/2

0 1

F F

T T

Lattice results disfavor Lattice results disfavor either

either n n≥ ≥3 3 or or n n ≤ ≤ 2 2 and suggest

and suggest n=2.5(0.25) n=2.5(0.25)

ε ξ ς

ε

⎡ ⎛ ⎞ ⎤

= ⎢ + ⎜ ⎟ ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

( ) 3

5

n

F s s

F

E T N T

This is the same behavior as for a gas of This is the same behavior as for a gas of

noninteracting

noninteracting (!) bosons below (!) bosons below the condensation temperature.

the condensation temperature.

Experiment Experiment

John Thomas’ group at Duke University, L.Luo, et al. Phys. Rev. Lett. 98, 080402, (2007) Dilute system of fermionic 6Li atoms in a harmonic trap

• • The number of atoms in the tra The number of atoms in the trap: N=1.3(0.2) x p: N=1.3(0.2) x 10 10

atoms atoms divided

divided 50- 50 -50 among 50 among the lowest two hyperfine states. the lowest two hyperfine states .

• • Fermi Fermi energy energy : :

• • Depth Depth of of the the potential potential : :

• • How How they they measure: measure : energy energy , , entropy entropy and and temperature? temperature ?

( )

(3 ) ;

/ 1

Fho x y z

Fho B

N

k K

ε ω ω ω

ε µ

= Ω Ω =

≈

=

0

10

U ≈ ε

2

- virial theorem

3 2

( )

( ) - local density

PV E E

N U P n r U

n r

= ⎫ ⎪⇒ ⎬ =

∇ = − G G ∇ ⎭ G ⎪

G

noninteracting Fermi gas

•For the weakly interacting gas ( ) the energy and entropy is calculated. In this limit one can use Thomas-Fermi approach to relate the energy to the given density distribution.

The entropy can be estimated as for the noninteracting system with 1% accuracy. In practice:

•The magnetic field is changed adiabatically (S=const.) to the value corresponding to the unitary limit:

•Relative energy in the unitary limit is calculated from virial theorem:

•Temperature is calculated from the identity:

1200 1/ _F 0.75 B= G⇒ k a ≈ −

840 1 / _F 0

B = G ⇒ k a ≈

1

2

2 1

2 2

( ) ( )

T T

E T z

E T = z

1 S

T E

= ∂

∂

2

1200

,

z

E S

=

⇒

•The plot S(E) contains a cusp related to the phase transition:

/ _B S k

(E E− (0)) /(Nε_F^ho)

( ) (0) 0.41(5) , / 2.7(2) ,

0.29(3)

c Fho

c B

c Fho

E T E N

S N k

T

ε

ε

⎧ − ≈

⎪ ≈

⎨ ⎪ ≈

⎩

Theory

Theory: : locallocal densitydensity approximationapproximation (LDA)(LDA)

3 ( )

5

F λ N ϕ ε x N λ N

Ω= − = −

Uniform system

3

2 2 2/3

3 ( ) ( ( )) ( ) ( ) 5

( ) ; ( ) 3 ( )

( ) 2

F

F F

d r r x r U r n r

x r T r n r

r m

ε ϕ λ

ε π

ε

⎡ ⎤

Ω= ⎢ ⎣ + − ⎥ ⎦

⎡ ⎤

= = ⎣ ⎦

∫ ^{G G G G}

=

G G G

G

Nonuniform system (gradient corrections

neglected)

( )

( ( )) ( ) 0

( ) ( )

F N

x r U r

n r n r

δ δ λ µ λ

δ δ

Ω = − = G + − =

G G

The overall chemical potential and the temperature T are constant throughout the system. The density profile will depend on the shape of the trap as dictated by:

λ

Using as an input the Monte Carlo results for the uniform system and experimental data (trapping potential, number of particles), we determine the density profiles.

Comparison

Comparison withwith experimentexperiment

John Thomas’ group at Duke University,

L.Luo, et al. Phys. Rev. Lett. 98, 080402, (2007)

( ) _ho3

n r a

Superfluid

2 max

aho

mω

= =

(0) - Fermi energy at the center of the trap εF

Normal THEORY

THEORY

Entropy as a function of energy (relative to the ground state)

for the unitary Fermi gas in the harmonic trap. The radial (along shortest axis) density profiles of the atomic cloud in the Duke group experiment at various

temperatures.

Theory:

THEORY

0 ho

E = N ε

1200 1/

0.75 B = G ⇒ k a ≈ −

Ratio of the mean square cloud size at B=1200G to its value at unitarity (B=840G) as a function of the energy. Experimental data are denoted by point with error bars.

840 1/

0

B = G ⇒ k a ≈

ρ ψ ψ ψ ψ

ρ ρ

ρ α

↑ ↓ ↓ ↑

→ ∞

=

= + +

=

∫

G G G G G G G G

G G G G G G G

G

† †

2 1 2 3 4 1 2 4 3

3 3

2 1 2 2 1 2 1 2

2

ˆ ˆ ˆ ˆ

( , , , ) ( ) ( ) ( ) ( )

( ) 2 ( , , , )

lim ( ) - co n d en sa te fra ctio n

P

P r

r r r r r r r r

r d r d r r r r r r r

N

r

Results off unitary limit:

-Critical temperature -Ground state energy -Pairing gap

Pairing gap, pseudogap and quasi-particle spectrum

Dynamical Mean Field Theory

(exact in infinite number of dimensions)

Quantum Monte Carlo

Preliminary measurements of pseudogap in ultracold atomic gases

40K at T=T_c

Conclusions Conclusions

9 9

system in the unitary regime at finite temperatures are feasible andand apparently the system undergoes a phase transition in the bulk a apparently the system undergoes a phase transition in the bulk at t TT_cc = 0.= 0.1515 (1(1) ) εε_FF..

99 BetweenBetween TT_cc andandTT_{0 0} =0.23(2) ε=0.23(2) ε_FF thethe system issystem is neitherneither superfluidsuperfluid nor nor follows

follows thethenormalnormal FermiFermi gasgas behaviorbehavior. . PossiblyPossiblyduedue to to pairingpairing effectseffects.. 99 ChemicalChemical potentialpotentialisisconstantconstant upupto to thetheTT₀₀ ––notenote

similarity

similarity withwithBose systemsBose systems!!

99 Below the transition temperature,Below the transition temperature, both phonons and fermioniboth phonons and fermionicc quasiparticles

quasiparticles contribute almost equalycontribute almost equaly to the specific heat. In morto the specific heat. In more e thanthanone way the system is at crossover between a Bose and Fermione way the system is at crossover between a Bose and Fermi systems

systems..

99 ResultsResults (energy(energy, , entropyentropy vsvs temperature) temperature) agreeagree withwith recentrecentmeasurmentsmeasurments: : L. L. LuoLuo et al., PRL 98, 080402 (2007)et al., PRL 98, 080402 (2007)

99 ThereThere isis ananevidenceevidence for thefor the existenceexistenceof of pseudogappseudogapatat unitarity.unitarity.

Summary Summary

We presented the first model-independent comparison of recent

measurements of the entropy and the critical temperature, performed by the Duke group: L.Luo, et al. Phys. Rev. Lett. 98, 080402, (2007), with our recent finite temperature Monte Carlo calculations.

( ) (0) 0.41(5) , / 2.7(2) ,

0.29(3)

c Fho

c B

c Fho

E T E N

S N k

T

ε

ε

⎧ − ≈

⎪ ≈

⎨ ⎪ ≈

⎩

( ) (0) 0.34(2) , / 2.4(3) ,

0.27(3)

c Fho

c B

c Fho

E T E N

S N k

T

ε

ε

⎧ − ≈

⎪ ≈

⎨ ⎪ ≈

⎩

EXP.EXP. THEORYTHEORY

A.Bulgac, J.E. Drut, P. Magierski, cond-mat/0701786

The results are consistent with the predicted value of the critical temperature for the uniform unitary Fermi gas: 0.23(2) F

ε