**Equation**

**Equation**

**Equation** **of ** **of ** **state** **state** **for ** **for ** **dilute** **dilute** **and** **and** **strongly**

**Equation**

**of**

**of**

**state**

**state**

**for**

**for**

**dilute**

**dilute**

**and**

**and**

**strongly**

**strongly** **interacting** **interacting** **Fermi** **Fermi** **gas** **gas**

**strongly**

**interacting**

**interacting**

**Fermi**

**Fermi**

**gas**

**gas**

**Piotr Magierski (Warsaw University of Technology)** **In collaboration** **In ** **collaboration** **with:** **with** **: Aurel Bulgac, Joaquin E. Drut **

**(University of Washington, Seattle)**

**Outline** **Outline**

### ¾ ¾ **General** **General** **remarks** **remarks**

### ¾ ¾ **Path** **Path** **integral** **integral** **Monte Carlo** **Monte ** **Carlo** **description** **description** **of ** **of ** **strongly** **strongly** **interacting** **interacting** **Fermi**

**Fermi** **gases.** **gases** **.**

### ¾ ¾ **Equation** **Equation** **of ** **of ** **state** **state** **for ** **for ** **the** **the** **Fermi** **Fermi** **gas** **gas** **in** **in** **the** **the** **unitary** **unitary** **regime** **regime** **. ** **. ** **Thermodynamic properties**

**Thermodynamic properties** **.** **.** **Critical** **Critical** **temperature** **temperature** **.** **.**

### ¾ ¾ **Conclusions.** **Conclusions** **.**

### Superconductivity and

### Superconductivity and superfluidity superfluidity in Fermi systems

### in Fermi systems

**20 orders of magnitude over a century of (low temperature) physi** **20 orders of magnitude over a century of (low temperature) physics** **cs**

### 9 **Dilute atomic Fermi gases** **Dilute atomic Fermi gases** **T** **T**

_{c}

_{c}**≈** **≈** **10** **10**

^{-12}

^{-}

^{12}**–** **–** **10** **10**

^{-9}**eV** **eV** 9 9 **Liquid ** **Liquid **

^{3}

^{3}**He** **He** **T** **T**

_{c}

_{c}**≈** **10** **10**

^{-7}

^{-}

^{7}**eV** **eV**

### 9 9 **Metals, composite materials** **Metals, composite materials** **T** **T**

_{c}

_{c}**≈** **10** **10**

^{-}

^{-}

^{3 }

^{3 }**–** **–** **10** **10**

^{-2}

^{-}

^{2}**eV** **eV** 9 9 **Nuclei, neutron stars** **Nuclei, neutron stars** **T** **T**

_{c}

_{c}**≈** **10** **10**

^{5}

^{5}**–** **–** **10** **10**

^{6}

^{6}**eV** **eV**

### • • **QCD color superconductivity** **QCD color superconductivity** **T** **T**

_{c}

_{c}**≈** **10** **10**

^{7 }

^{7 }**–** **–** **10** **10**

^{8 }

^{8 }**eV** **eV** **units (1 **

**units (1**

**units (1 ** **eV** **eV** **≈** **≈** **10** **10**

**units (1**

**eV**

**eV**

**10**

**10**

^{4}

^{4}**K)** **K)**

**K)**

**K)**

**Fermi**

**Fermi** **gas:** **gas** **:** *n* - number density, - scattering length *a*

**What is the energy of the dilute**

**What is the energy of the dilute** **Fermi gas?** **Fermi gas?** *E k a* ( _{F} ) ? =

_{F}

2 2 3

; 2

2 3

*F* *F*

*F*

*k* *k*

*m* *n*

ε ^{=} ^{=} π

=

**- particle density**

(*k r*

*0 <<1)*

_{F}### ( ) ( )( )

### 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

### ⎡ ⎤

### = + ⎢ ⎣ + − + ⎥ ⎦

### =

2 2 1 1

iff | | 1 and - size of the Cooper pair

2

### 8 exp ,

### 2 2

*F* *k* *a* *F*

*BCS* *F* *F* *k**F* *k**F*

*k*

*m* *k a* *e*

η ε

### π

<< << =∆

### ⎛ ⎞

### ∆ = ⎜ ⎟

### ⎝ ⎠

### =

### BCS pairing gap

2 HF+BCS

4 FG

10 5 10

1 ( ) ... 1 ( ) ... exp

9 8 9

### E = 40

### E

^{π}

^{k a}

^{F}^{ε}

^{BCS}

_{F}^{π}

^{k a}

^{F}*e*

^{k a}^{π}

_{F}⎛ ∆ ⎞ ⎛ ⎞

+ + − ⎜ ⎟ = + + ⎜ ⎟

⎝ ⎠

### −

⎝ ⎠**Mean-field term** **BCS term**

### 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.

### The The only only scale scale : : ^{3}

### 5

*FG* *F*

*E* *N* = ε

### 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* →±∞

**UNIVERSALITY:**

**UNIVERSALITY:** ^{( )} ( )

*F*

*T* *FG*

*E T* = ξ _{ε} *E*

**QUESTIONS:**

**QUESTIONS:** **What is the shape of ?**

**What is the critical temperature for** **the superfluid-to-normal transition?**

**...**

### ( )

*F*

*T*

### ε

### ξ

**NONPERTURBATIVE**

**NONPERTURBATIVE**

**REGIME**

**REGIME**

**System **

**System ** **is** **is** **dilute** **dilute** **but ** **but ** **strongly**

**strongly** **interacting!** **interacting** **!**

### 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 interaction**

**weak interaction** **weak interactions** **weak interactions**

**Strong interaction** **Strong interaction** **UNITARY REGIME** **UNITARY REGIME**

**?**

**?**

**Bose**
**molecule**

**EASY!**

**EASY!** **EASY!** **EASY!**

**A little bit of history** **A little bit of history**

**Bertsch**

**Bertsch** **Many-** **Many** **-Body X challenge, Seattle, 1999** **Body X challenge, Seattle, 1999**

**What are the ground state properties of the many****What are the ground state properties of the many****-****-****body system composed of ****body system composed of ****spin ½ fermions interacting via a zero**

**spin ½ fermions interacting via a zero****-****-****range, infinite scattering****range, infinite scattering****-****-****length contact****length contact****interaction****i****nteraction****. ****. **

### Why? Besides pure theoretical curiosity, this problem is relevan

### Why? 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 experiment**

**In 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 that**
**under 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 Carlo**

**and 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 systems**
**are (meta)stable. **

### ( *T* 0) 0.44

### ξ = ≈

**Neutron **

**Neutron ** **matter** **matter** **:** **:**

### Effective

### Effective range range : : r r _{0 } _{0 } ≈ ≈ 2.8 2.8 fm fm Scattering

### Scattering length length : : a a ≈ ≈ - - 18.5 18.5 fm fm r r _{0 } _{0 } n n ^{-} ^{-} ^{1/3 } ^{1/3 } ≈ ≈ λ λ _{F } _{F } /2 /2 |a| |a|

**Density range**

**corresponds to**

### n n ≈ ≈ 0.001 0.001 - - 0.01 0.01 fm fm ^{-} ^{-} ^{3 } ^{3 }

### k k _{F } _{F } ≈ ≈ 0.3 0.3 - - 0.7 0.7 fm fm ^{-} ^{-} ^{1} ^{1}

**Neutron **

**Neutron ** **matter** **matter**

_{≈ −}

0 ≈

N e u tro n -n e u tro n s c a tte rin g S c a tte rin g le n g th : 1 8 .5 E ffe c tiv e ra n g e : 2 .8

*a* *fm*

*r* *fm*

**ss--wave pairing gap in infinitewave pairing gap in infinite** **neutron matter with realisticneutron matter with realistic** **NNNN--interactionsinteractions**

**BCS**

Dilute matter: only , matter*a r*0 **Details of the n-n potential matter**

**Nuclear density**

### n n ≈ ≈ 0.001 0.001 - - 0.01 0.01 fm fm

^{-3 }

^{-}

^{3 }

### k k

_{F }

_{F }

### ≈ ≈ 0.3 0.3 - - 0.7 0.7 fm fm

^{-1}

^{-}

^{1}

_{the} _{the} ^{Close} ^{Close} _{unitary} _{unitary} ^{to } ^{to } _{limit} _{limit}

_{the}

_{the}

^{Close}

^{Close}

_{unitary}

_{unitary}

^{to }

^{to }

_{limit}

_{limit}

**Theoretical approach: Fermions on 3D lattice**

**- Spin up fermion:**

**- Spin down fermion:**

**External conditions:**

### - temperature

### - chemical potential *T*

### µ

*cut*

### ;

*k* *x*

*x*

### = π ∆

### ∆

**L –limit for the****spatial** **correlat****ions****in****the****syst****em**

**Coordinate**

**Coordinate** **space** **space**

*Volume* *L*

3
*lattice spacing* *x*

### =

### = ∆

### Periodic boundary conditions imposed

### ( )

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

**Hamiltonian**

**Hamiltonian**

**Theoretical approach: Fermions on 3D lattice** **Momentum space**

**Momentum space**

### π π

### ε

### Λ =

### ∆

### Λ =

### Λ Λ

### << ∆ <<

### = =

U V

IR

2 2 2 2

### U V m om en tu m cu toff IR m om en tu m cu toff 2

### ,

### 2 2

*IR* *U V*

*F*

*x* *L*

*m* *m*

**k** **k**

_{y}

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

_{x}

_{x}### 2π/L

*x*
π

∆

*x*
π

∆

**k** **k**

**kk**_{cut}_{cut}**==π/π/**'x'**x**

**2**

**2**

**π/L**

**π**

**/L**

**n(k)n(k)**

**REAL SPACE**

**REAL SPACE**

*FFTFFT*

**MOMENTUM SPACE** **MOMENTUM SPACE**

### ( ) { ( ) }

2

3 † 3

3 †

### ˆ ˆ ˆ ( ) ( ) ( ) ( ) ˆ ˆ

### 2

### ˆ ˆ ( ) ˆ ( ) , ( ) ˆ ( ) ( ), ,

### ˆ ˆ ˆ ˆ 1

### ( ) Tr exp Tr exp ,

### ( )

*s* *s*

*s*

*s* *s* *s*

*N*

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

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

*Z* *H* *N* *H* *N* *N*

*T* *E T*

τ

τ

### ψ ψ

### ψ ψ

### β β µ τ µ β τ

↑ ↓

=↑↓

↑ ↓

### ⎛ ∆ ⎞

### = + = ⎜ − ⎟ −

### ⎝ ⎠

### = ⎡ ⎣ + ⎤ ⎦ = =↑ ↓

### ⎡ ⎤ ⎡ ⎤

### = ⎣ − − ⎦ = ⎣ − − ⎦ = =

### =

### ∫ ∑ ∫

### ∫

### =

### G G G G

### G G G G G

### ( )

### ( )

### 1 Tr exp ˆ ˆ ˆ

### ( )

### 1 ˆ ˆ ˆ

### ( ) Tr exp

### ( )

*H* *H* *N*

*Z T*

*N T* *N* *H* *N*

*Z T*

### β µ

### β µ

### ⎡ − − ⎤

### ⎣ ⎦

### ⎡ ⎤

### = ⎣ − − ⎦

**Grand Canonical Path**

**Grand Canonical Path** **-** **-** **Integral Monte Carlo** **Integral Monte Carlo**

### Trotter expansion

### Trotter expansion

### ( ) ( ) ( )

3

( ) 1

### ˆ ˆ ˆ ˆ ˆ ˆ ˆ

### exp exp /2 exp( )exp /2

### ( )

### ˆ 1 ˆ ˆ

### exp( ) 1 ( ) ( ) 1 ( ) ( ) , exp( ) 1

*r*

### 2

*r*

*H* *N* *T* *N* *V* *T* *N*

*O*

*V* *r An r* *r An r* *A* *g*

σ

### τ µ τ µ τ τ µ

### τ

### τ σ

_{↑}

### σ

_{↓}

### τ

=±

### ⎡ − − ⎤ ≈ ⎡ − − ⎤ − ⎡ − − ⎤

### ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

### +

### − = ∏

_{G}

### ∑

_{G}

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

### Discrete Hubbard

### Discrete Hubbard- -Stratonovich Stratonovich transformation transformation

### σ σ -fields fluctuate both in space and imaginary time - fields fluctuate both in space and imaginary time

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*

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 x* *U*

*D r* *N*

*T*

*U* *T* *d h*

*D x* *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!**

**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 matrices** **particle density matrices**

### 1

*kT*

β

### σ

τ### τ σ µ σ

### σ ψ σ ψ ψ

### = − − −

### =

### ∫

0### ˆ ({ }) exp{ [ ({ }) ˆ ]}; ({ }) one-body operator ˆ ({ })

_{kl}

_{k}### ˆ ({ })

_{l}### ;

_{l}### - single-particle wave function

*U* *T* *d h* *h*

*U* *U*

### σ τ

σ### σ

### σ

### = = ∫ ^{G}

− ^{[ ]}

### energy associated with a given sigma field

### [ ( , )]

### ( ) ˆ [ ({ })]

### ( ) [ ({ })]-

*D* *r* *e*

*S*

*E T* *H* *E U*

*Z T* *E U*

### Sigma space sampling

### σ •

### ( )

*[ ]*

^{S}*P* σ ∝ *e*

^{−}

^{σ}

**Quantum Monte-Carlo:**

### ( )

1

### ( ) 1 ({ })

*N*

*k*
*k*

*E T* *E U*

*N*

σ

σ

### σ

=

### = ∑

2 2

( ) - stochastic variable

( ) ( )

( ) ( ) 1

- number of

uncorrelated samples
*E T*

*E T* *E T*

*E T* *E T*

*N*
*N*

σ σ

=

− ∝

β

### σ

τ### τ σ µ σ

### σ ψ σ ψ ψ

### = − − −

### =

### ∫

0### ˆ ({ }) exp{ [ ({ }) ˆ ]}; ({ }) one-body operator ˆ ({ })

_{kl}

_{k}### ˆ ({ })

_{l}### ;

_{l}### - single-particle wave function

*U* *T* *d h* *h*

*U* *U*

**Quantum Monte-Carlo: parallel computing**

**For each sigma ****n single particle states have to be evolved.****n **

### ψ

1### ψ

2### ψ

3### ψ

*n*

σ = ψ ˆ σ ψ
({ })_{kl}* _{k}* ({ })

_{l}*U* *U*

### ˆ({ }) σ

*U* *U* ˆ({ }) σ ^{U} ^{ˆ({ })} ^{σ} . . . ^{U} ^{ˆ({ })} ^{σ}

^{U}

^{U}

### ...

**More details of the calculations:**

**More details of the calculations:**

•• **Lattice sizes used from 8Lattice sizes used from 8**^{3 }^{3 }**xx257257(high Ts) to 8(high Ts) to 8**^{3 }^{3 }**x 1732x 1732** **(low Ts), <N>=50,(low Ts), <N>=50,**
**andand66**^{3 }^{3 }**xx257 (high Ts) to 257 (high Ts) to 66**^{3}^{3}**x 1361x 1361** **(low Ts), <N>=30.(low Ts), <N>=30.**

•• **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 matricese 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 ..**

•• **ThermalizeThermalizefor 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-upup** **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 evoAt low temperatures use Singular Value Decomposition of the evolution operator lution operator **
**U({σU({σ}) to stabilize the }) to stabilize the numericsnumerics..**

•• **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””** **≈≈** **150 150 ––** **200 time steps200 time steps** **at T ≈at T ≈** **TT**_{c}_{c}

**Superfluid**

**Superfluid** **to Normal Fermi Liquid Transition** **to Normal Fermi Liquid Transition**

**Bogoliubov**

**Bogoliubov--Anderson phononsAnderson phonons**
**and quasiparticle**

**and **

**quasiparticle**

**contribution**

**contribution**

(dashed### (d

ashed line### lin

e )### )

### Bogoliubov

Bogoliubov--Anderson phonons Anderson phonons

### contribution only (

contribution only (dotteddotted line)

### line

)**Quasi**

**Quasi--particle contribution onlyparticle contribution only**
(dotted

### (d

otted 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**

Phase transition

### ξ

### +

### =

### = ∂

### ∫ ∂

### ∫

0 0### 3

/### '( )

### ( ) 5 ( ) (0)

*F*

*T*

*T e*

*y*

*S T*

*S T* *S* *E dT*

*T*

*N* *dy*

*T*

*y*

### 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*

**From a talk of J.E. **

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

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

### ε ξ µ ξ

### ε ε

### ⎛ ⎞

### = ⎜ ⎟ ≈ ≈ <

### ⎝ ⎠

### 3 ( )

### ( ) and 0.41(2) for

### 5

^{F}

_{F}

_{F}

^{s}

^{C}*T* *T*

*E T* *N* *T* *T*

### µ ε ξ ξ ε ξ

### ε ε ε

### ⎡ ⎛ ⎞ ′ ⎛ ⎞ ⎤

### = = ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ≈

### ⎝ ⎠ ⎝ ⎠

### ⎣ ⎦

### ( ) 2

### ( )

^{F}

_{F}### 5

_{F}

_{F}

^{F s}*dE T* *T* *T* *T*

*T* *dN*

### ξ ξ ς ς

### ε ε

### ⎛ ⎞ ⎛ ⎞

### = + ≈

### ⎜ ⎟ ⎜ ⎟

### ⎝ ⎠ ⎝ ⎠

5/2

### , 11(1)

*s* *s* *s*

*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.**

### ε ξ ς

### π

### ⎛ ⎞ ⎛ ⎞ Γ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

### ≈ +

### ⇒ ≈

### =

3/2

5/2 2

1/ 2 2 3

### 3 3

### 3 2 2

### ( ) , if

### 5 2

### and fitting to lattice results 3

*B*

*F* *s* *B*

*B*

*m*

*E T* *N* *T V* *T* *m c*

*m* *m*

### • • **Why this value for the ** **Why this value for the ** **bosonic** **bosonic** **mass?** **mass?**

### • • **Why these bosons behave like ** **Why these bosons behave like ** **noninteracting** **noninteracting** **particles?** **particles?**

**Conclusions** **Conclusions**

### 9 9 **Fully non** **Fully non** **-** **-** **perturbative calculations for a spin ½ many fermion** **perturbative calculations for a spin ½ many ** **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**

**T**

**T**

_{c}

_{c}**= 0.23** **= 0.2** **3** **(2** **(** **2) ** **) ** **ε** **ε**

_{F}

_{F}**(** **(** **Exp: ** **Exp** **: ** **T** **T**

**T**

**T**

_{c}

_{c}**=** **=** **0.27(2) ** **0.27(2) ** **ε** **ε**

_{F}

_{F}**, J. Kinast** **, J. ** **Kinast** **et al.** **et al.** **Science** **Science** **, 307, 1296 (2005):** **, 307, 1296 (2005):**

**et al.**

**et al.**

**Based**

**Based** **on theoretical** **on ** **theoretical** **assumptions).** **assumptions** **).**

### 9 9 **Chemical** **Chemical** **potential** **potential** **is** **is** **constant** **constant** **up** **up** **to the** **to ** **the** **critical** **critical** **temperature** **temperature** **–** **–** **note** **note** **similarity**

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

### 9 9 **Below the transition temperature** **Below the transition temperature** **,** **,** **both phonons and ** **both phonons and ** **fermioni** **fermioni** **c** **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** **.** **.**

### There are reasons to believe that below the critical temperature this

### system is a new type of fermionic superfluid, with unusual properties.

### From: E.Burovski, N.Prokof’ev, B.Svistunov, M.Troyer, cond-mat/0602224

**OURS** **OURS**

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

*T* *c**/E* *F*

J. Kinhast, A. Turlapov, J.E. Thomas, Q. Chen, J. Stajic, K. Levin,

** Science 307, 1296 (2005)**

M. Wingate, cond-mat/0502372 A. Bulgac, J. E. Drut, P. Magierski, cond-mat/0505374

X.-J. Liu, H. Hu, cond-mat/0505572 P. Nozieres, S. Schmitt-Rink,

**J. Low. Temp. Phys 59, 195 (1985)**
M. Holland, S. J. J. M. F. Kokkelmans,
M. L. Chiofalo, and R. Walser,
** PRL 87, 120406 (2001)**

Analytics Numerics Experiment + assumptionns

This work

**Quest**

**Quest** **for unitary**

**for u**

**nitary**

**point critical temperature**

**point critical temperature**

Boris Svistunov’s talk (updated), Seattle 2005

E. Burovski, N. Prokofev, B. Svistunov, M. Troyer cond-mat/0602224

**Ours**

T.Lee, D. Schafer, nucl-th/0509018