L –limit for thespatialcorrelationsinthesystem

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Equation

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

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)

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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 .

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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

cc

10 10

-12-12

10 10

-9

eV eV 9 9 Liquid Liquid

33

He He T T

cc

10 10

-7-7

eV eV

9 9 Metals, composite materials Metals, composite materials T T

cc

10 10

--3 3

10 10

-2-2

eV eV 9 9 Nuclei, neutron stars Nuclei, neutron stars T T

cc

10 10

55

10 10

66

eV eV

• • QCD color superconductivity QCD color superconductivity T T

cc

10 10

7 7

10 10

8 8

eV eV units (1

units (1 eV eV 10 10

44

K) K)

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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 ) ? =

2 2 3

; 2

2 3

F F

F

k k

m n

ε = = π

=

- particle density

(k rF 0 <<1)

( ) ( )( )

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 kF kF

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 aF εBCSF π k aF

e

k aπF

⎛ ∆ ⎞ ⎛ ⎞

+ + − ⎜ ⎟ = + + ⎜ ⎟

⎝ ⎠

⎝ ⎠

Mean-field term BCS term

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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

00

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

. . 0 0,

i e ra →±∞

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 !

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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!

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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 scatteringrange, infinite scattering--length contactlength contact interactioninteraction. .

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 = ≈

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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

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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 , mattera r0 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

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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 thespatial correlationsinthesystem

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

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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

yy

k k

xx

2π/L

x π

x π

k k

kkcutcut==π/π/'x'x

2

2

π/L

π

/L

n(k)n(k)

REAL SPACE

REAL SPACE

FFTFFT

MOMENTUM SPACE MOMENTUM SPACE

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( ) { ( ) }

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

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( ) ( ) ( )

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

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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

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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

=

− ∝

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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

...

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More details of the calculations:

More details of the calculations:

•• Lattice sizes used from 8Lattice sizes used from 83 3 xx257257(high Ts) to 8(high Ts) to 83 3 x 1732x 1732 (low Ts), <N>=50,(low Ts), <N>=50, andand663 3 xx257 (high Ts) to 257 (high Ts) to 6633 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 ≈ TTcc

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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)

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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

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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

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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.

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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?

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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

cc

= 0.23 = 0.2 3 (2 ( 2) ) ε ε

FF

( ( Exp: Exp : T T

cc

= = 0.27(2) 0.27(2) ε ε

FF

, J. Kinast , J. Kinast et al. et al. Science Science , 307, 1296 (2005): , 307, 1296 (2005):

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.

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

OURS OURS

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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

Obraz

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