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

**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

_{0}

_{0}

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

### . . 0 0,

*i e r* → *a* →±∞

**AT FINITE ** **AT FINITE **

**TEMPERATURE:**

**TEMPERATURE:** ^{( )} ( )

*F*

^{, } ^{(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

^{5-}

^{5}

^{-}

### 10 10

^{6 }

^{6 }

### 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

### • •

**The strength of this interaction is fully tunable!The strength of this interaction is fully tunable!**

**Who does experiments?**

**Who does experiments?**

**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

6*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 th****e****sp****atial** **correlations****in****th****e****sy****stem**

**Coordinate**

**Coordinate** **spacespace** *Volume* *L*^{3}

*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**

_{y}

_{y}**k** **k**

_{x}

_{x}### 2π/L

*x*
π

∆

*x*
π

∆

**kk**

**k****k**_{cut}_{cut}**=****=****π/****π****/**''**x****x**

**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*

*cut*

*m*

*g* ^{= −} π ^{=} *a* ^{+} π ^{=}

Running coupling constant g defined by lattice Running coupling constant g defined by lattice
2

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

*m*
*g* ^{=}

### π

= ∆*x*

### ( ) ( ) ( )

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} ^{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 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: 6****Lattice sizes used****: 6**^{3 }^{3 }**–****–****10****10**^{3}^{3}**. ****. **
**Imaginary**

**Imaginary** **time****time** **steps:8****steps:****8**^{3 }^{3 }**x****x****300****300****(high Ts) to 8****(high Ts) to ****8**^{3 }^{3 }**x 1800****x 1****800** **(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 di****agonal (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 larg****e matrices.****e matrices****.**

•• **Update field configurations using the Metropolis importance sampling algorithm****Update field configurations using the Metropolis importance samp****ling 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 s****igns the auxiliary **
**fields **

**fields ****σ****σ****(r****(****r,****,τ****τ****) so as to maintain a running average of the acceptance rate bet****) so as to maintain a running average of the acceptance rate bet****ween****ween**
**0.4 and 0.6 **

**0.4 and 0.6 ****.****.**

•• **Thermalize****Thermalize** **for 50,000 –****for 50,000 ****–** **100,000 MC steps or/and use as a start****100,000 MC steps or/and use as a start****-up****-****up** **field ****field **
**configuration a **

**configuration a σ****σ(x,****(x,τ****τ)****)-****-field configuration from a different T****field 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 evo****lution operator **
**U({σ****U({****σ****}) ****}) ****to stabilize the numerics****to stabilize the ****numerics****.****.**

•• **Use 2****Use ****200,000****00,000-****-2,000,000 ****2,000,000 σ****σ(x,****(x,****τ****τ****)-****)****-** **field configurations for calculations****field configurations for calculations**

•• **MC correlation “****MC correlation ****“time****time”****”** **≈****≈** **250 ****2****50 –****–** **300 time steps****3****00 time steps** **at T ≈****at T ****≈** **T****T**_{c}_{c}

**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

^{F}### 16

_{s}

_{F}

^{s}*E* *T* ε *N* π *T* ξ

### ξ ε

### = ⎛ ⎞ ⎜ ⎟ ≈

### ⎝ ⎠

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

### ε ξ ε π ε

### ⎛ ⎞

### ⎜ ⎟

### ⎝ ⎠

### = =

^{3}

### = =

^{2 2}

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

### +

### =

### = ∂

### ∫ ∂

### ∫

0 0### 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

^{F}

_{F}*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*

*F*

*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

^{F}

_{F}

_{F}

^{s}

^{C}*T* *T*

*F T* *N* *E TS* *T* *T*

### µ ε ϕ ϕ ε ξ

### ε ε ε

### ⎡ ⎛ ⎞ ⎛ ⎞ ⎤

### = = ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ≈

### ⎝ ⎠ ⎝ ⎠

### ⎣ ⎦

### ( ) 2

### ( ) '

*F*

### 5

*F s*

*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 6*Li* 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

^{5 }

^{5 }

### 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 ?

### ( )

^{1/ 3}

### (3 ) ;

1/ 3### / 1

*F**ho* *x y z*

*F**ho* *B*

*N*

*k* *K*

### ε ω ω ω

### ε µ

### = Ω Ω =

### ≈

### =

0

### 10

_{F}

^{ho}*U* ≈ ε

### 2

### - virial theorem

### 3 2

### ( )

### ( ) - local density

*PV* *E* *E*

*N U* *P* *n r U*

*n r*

### = ⎫ ⎪⇒ ⎬ =

### ∇ = − G G ∇ ⎭ G ⎪

### G

Holds at unitarity and fornoninteracting 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*

*B*

*E S*

=

### ⇒

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

/ _{B}*S k*

(*E E*− (0)) /(*N*ε_{F}* ^{ho}*)

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

### 0.29(3)

*c* *F**ho*

*c* *B*

*c* *F**ho*

*E T* *E* *N*

*S* *N* *k*

*T*

### ε

### ε

### ⎧ − ≈

### ⎪ ≈

### ⎨ ⎪ ≈

### ⎩

**Theory**

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

### 3 ( )

### 5

^{F}*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** **with****with** **experiment****experiment**

John Thomas’ group at Duke University,

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

( ) * _{ho}*3

*n r a*

Superfluid

2 max

*a**ho*

*m*ω

= =

(0) - Fermi energy at the center of the trap
ε*F*

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

*F*

### 1200 1/

_{F}### 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/

_{F}### 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

**Fully non**

**Fully non**

**-perturbative calculations for a spin**

**-**

**perturbative calculations for a spin ½**

**½**

**many**

**many**

**fermion**

**fermion**

**system in the unitary regime at finite temperatures are feasible**

**system in the unitary regime at finite temperatures are feasible** **and****and**
**apparently the system undergoes a phase transition in the bulk a**
**apparently the system undergoes a phase transition in the bulk at ****t **
**T****T**_{c}_{c}**= 0.****= 0.****15****15** **(1****(****1) ****) ε****ε**_{F}_{F}**.****.**

99 **Between****Between** **T****T**_{c}_{c}**and****and****T****T**_{0 }_{0 }**=0.23(2) ε****=0.23(2) ****ε**_{F}_{F}**the****the** **system is****system ****is** **neither****neither** **superfluid****superfluid** **nor ****nor **
**follows**

**follows** **the****the****normal****normal** **Fermi****Fermi** **gas****gas** **behavior****behavior****. ****. ****Possibly****Possibly****due****due** **to ****to ****pairing****pairing** **effects****effects****.****.**
99 **Chemical****Chemical** **potential****potential****is****is****constant****constant** **up****up****to ****to ****the****the****T****T**_{0}_{0}**–****–****note****note**

**similarity**

**similarity** **with****with****Bose systems****Bose ****systems!****!**

99 **Below the transition temperature,****Below the transition temperature****,** **both phonons and fermioni****both phonons and ****fermionic****c**
**quasiparticles**

**quasiparticles** **contribute almost equaly****contribute almost ****equaly** **to the specific heat. In mor****to the specific heat. In ****more ****e **
**than****than****one way the system is at crossover between a Bose and Fermi****one way the system is at crossover between a Bose and Fermi**
**systems**

**systems.****.**

99 **Results****Results** **(energy****(****energy, ****, entropy****entropy** **vs****vs** **temperature) ****temperature****) agree****agree** **with****with** **recent****recent****measurments****measurments****: ****: **
**L. ****L. ****Luo****Luo** **et al., PRL 98, 080402 (2007)****et al., PRL 98, 080402 (2007)**

99 **There****There** **is****is** **an****an****evidence****evidence** **for the****for ****the** **existence****existence****of ****of ****pseudogap****pseudogap****at****at** **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* *F**ho*

*c* *B*

*c* *F**ho*

*E T* *E* *N*

*S N* *k*

*T*

### ε

### ε

### ⎧ − ≈

### ⎪ ≈

### ⎨ ⎪ ≈

### ⎩

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

### 0.27(3)

*c* *F**ho*

*c* *B*

*c* *F**ho*

*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*