Profesor Stefan Ćwiok1933-2003

ur. 1933r. – Przeryty Bór k. Tarnowa.Studia: Akademia Górniczo-Hutnicza w Krakowie,

Wydział Fizyki UniwersytetuMoskiewskiego im .Łomonosowa.

Doktorat: Wydział Fizyki Uniwersytetu Warszawskiego (1969)Adiunkt: Wydział Fizyki Uniwersytetu Warszawskiego (1969-1973)Od 1973: Instytut Fizyki Politechniki Warszawskiej1998: tytuł Profesora Fizyki

Wieloletni z-ca Dyrektora Instytutu Fizyki PW i Kierownik Zakładu Fizyki Jądrowej

Badania naukowe: struktura jądra atomowegostruktura jądra atomowego

Neutron star

JĄDRA ATOMOWEJĄDRA ATOMOWE

Jądra stabilne

Jądra znane

neutrony

??

prot

ony

Gwiazdyneutronowe

Jądra ciężkie i superciężkie

Profesor Stefan Ćwiok1933-2003

ur. 1933r. – Przeryty Bór k. Tarnowa.Studia: Akademia Górniczo-Hutnicza w Krakowie,

Wydział Fizyki UniwersytetuMoskiewskiego im .Łomonosowa.

Doktorat: Wydział Fizyki Uniwersytetu Warszawskiego (1969)Adiunkt: Wydział Fizyki Uniwersytetu Warszawskiego (1969-1973)Od 1973: Instytut Fizyki Politechniki Warszawskiej1998: tytuł Profesora Fizyki

Wieloletni z-ca Dyrektora Instytutu Fizyki PW i Kierownik Zakładu Fizyki Jądrowej

Badania naukowe: struktura jądra atomowegostruktura jądra atomowego

Osiągnięcia:Osiągnięcia:-hipoteza stabilności bardzo ciężkich jąder o N=162 – potwierdzonaprzez doświadczenia w GSI Darmstadt i Berkeley (USA).-wyjaśnienie zagadki istnienia dwóch kanałów rozszczepienia w izotopach fermu.-hipoteza istnienia (metastabilnych) stanów hiperzdeformowanych w aktynowcach- potwierdzona eksperymentalnie.-systematyczne obliczenia dla nieparzystych jąder superciężkich (do dziś interpretacja wielu eksperymentów bazuje na tych wynikach).-hipoteza istnienia liczby magicznej Z=126, a nie Z=114 jak dotychczas sądzono.-hipoteza współistnienia kształtów w jądrach superciężkich (Nature, 433(2005)705)

www.if.pw.edu.pl/~zak7www/stefan/index_stefan.htm

O pewnych własnościach O pewnych własnościach kwantowych gazów atomowychkwantowych gazów atomowych

Piotr Magierski (WF PW)

Współpracownicy:Współpracownicy: Aurel Bulgac, Joaquin E. Drut (University of Washington, Seattle)

OutlineOutline

ParticleParticle scatteringscattering atat lowlow energiesenergies. .

BCSBCS--BEC BEC crossovercrossover. . What What is the unitary regime?is the unitary regime?

HHowow one can manipulateone can manipulate the twothe two--body body interactiointeraction n ininexperimentsexperiments withwith atomicatomic gasesgases??

TheoreticalTheoretical approachapproach: : pathpath integralintegral descriptiondescription of of stronglystronglyinteractinginteracting FermiFermi gasesgases..

EquationEquation of of statestate for for thethe FermiFermi gasgas inin thethe unitaryunitary regimeregime. . CriticalCritical temperaturetemperature..

ConclusionsConclusions..

SScatteringcattering atat lowlow energiesenergies((ss--wavewave scatteringscattering))

2

- radius of the interaction potential

Rk

R

πλ = >>

scattering amplitude( ) ; - ikr

ik r er e f fr

ψ ⋅= +

02

0

1 , - scattering length, r - effective range112

f aik r ka

=− − +

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

SScatteringcattering atat lowlow energiesenergies: : attractiveattractive interactioninteraction

V(r)( )rψ

0 a bound state existsa >0 there is no bound statea < a = ±∞

1) 2) 3)

What is the energy of the diluteWhat is the energy of the dilute Fermi gas?Fermi gas? ( ) ?FE k a =2 2 3

2; 2 3

F FF

k knm

επ

= = - particle density0( 1)Fk r <<

( ) ( )( )10 61 1 11 2 2 ...9 35

3 - Energy of the noninteracting Fermi gas5

F FFG

FG F

E k a k a lnE

E N

π π

ε

⎡ ⎤= + + − +⎢ ⎥⎣ ⎦

=

PPerturbationerturbationseriesseries::

PAIRING NOT INCLUDED YET!

How pairing emerges?0a <Cooper’s argument (1956)

Gap 2∆

Cooper pair

2 21 1

iff | | 1 and - size of the Cooper pair28 exp ,

2 2F Fk aBCS F k kF FF

km k ae

εη

π<< << =

∆

⎛ ⎞∆ = ⎜ ⎟

⎝ ⎠

Fermi gas

2

HF+BCS4

FG

10 5 101 ( ) ... 1 ( ) ... exp9 8 9

E 40 = EBCS

F FF F

k a k ak aeπ

π ε π⎛ ⎞ ⎛ ⎞∆

+ + − = + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−

Hartree-Fock term BCS term

Superconductivity and Superconductivity and superfluiditysuperfluidityin 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 gasesDilute atomic Fermi gases TTcc ≈≈ 1010--1212 –– 1010-9 eVeV

Liquid Liquid 33HeHe TTcc ≈ 1010--77 eVeV

Metals, composite materialsMetals, composite materials TTcc ≈ 1010--3 3 –– 1010--22 eVeV

Nuclei, neutron starsNuclei, neutron stars TTcc ≈ 101055 –– 101066 eVeV

•• QCD color superconductivityQCD color superconductivity TTcc ≈ 10107 7 –– 10108 8 eVeV

units (1 units (1 eVeV ≈≈ 101044 K)K)

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.

TheThe onlyonly scalescale:: 35

FGF

EN ε=

nn -- particleparticle densitydensityn |a|n |a|33 11n rn r003 3 11

rr00 -- effectiveeffective rangerangea a -- scattering lengthscattering length

0. . 0, i e r a→ →±∞

UNIVERSALITY:UNIVERSALITY: ( )( )F

TFGE T Eεξ=

QUESTIONS:QUESTIONS: What is the shape of ?What is the critical temperature forthe superfluid-to-normal transition?...

( )F

Tεξ

NONPERTURBATIVENONPERTURBATIVEREGIMEREGIME

System System isis dilutedilute but but stronglystrongly interactinginteracting!!

24 aσ π=

1/a

T

a<0no 2-body bound state

a>0shallow 2-body bound state

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

BCS BCS SuperfluidSuperfluid

Molecular BEC andMolecular BEC andAtomic+MolecularAtomic+MolecularSuperfluidsSuperfluids

weak interactionweak interaction weak interactionsweak interactions

Strong interactionStrong interactionUNITARY REGIMEUNITARY REGIME

?Bose

molecule

EASY!EASY! EASY!EASY!

A little bit of historyA little bit of historyBertschBertsch ManyMany--Body X challenge, Seattle, 1999Body X challenge, Seattle, 1999

What 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 contactiinteractionnteraction. .

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.

( 0) 0.44Tξ = ≈

In dilute atomic systems experimenters can control nowadaysIn dilute atomic systems experimenters can control nowadaysalmost anything:almost anything:•• The number of atoms in the The number of atoms in the tratrap: tp: typicallyypically about 10about 1055--10106 6 atoms atoms

divideddivided 5050--50 among50 among the lowest two hyperfine statesthe lowest two hyperfine states..•• The density of atomsThe density of atoms•• Mixtures of various atomsMixtures of various atoms•• The temperature of the atomic cloudThe 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?•• 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 fermionicfermionic atom in magnetic fieldatom in magnetic field

FF m ; F I J J L S= + = +

Nuclear spin Electronic spin

TwoTwo hypefinehypefine statesstates arearepopulatedpopulated inin thethe traptrap

CollisionCollision of of twotwo atomsatoms:: 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 hyperfinestate do not interact with one anotherstate do not interact with one another..

•• Atoms in different hyperfine states experience interactions Atoms in different hyperfine states experience interactions only in sonly in s--wave.wave.

2 2

0 0 1 11

2

( ) ( ) ( )2

, ( )

hf Z di i

i

hfhf Ze z n z

pH V V V r P V r P V

aV I J V J I B

µ

γ γ

=

= + + + + +

= ⋅ = −

∑

Regal and Jin, PRL 90, 230404 (2003)

Feshbach resonance

Tiesinga, Verhaar, Stoof, Phys. Rev.A47, 4114 (1993)

Channel coupling

Interatomic distance

E∆

resonance: a →±∞

Theoretical approach: Fermions on 3D lattice

- Spin up fermion:

- Spin down fermion:

External conditions: - temperature - chemical potential

Tµ

; cutk xxπ

= ∆∆

L –l

imit

for t

hesp

atia

lco

rrel

atio

nsin

the

syst

emCoordinateCoordinate spacespace

kkyy

kkxx

2π/L

Momentum spaceMomentum spaceπε

πδε

πδ

∆∆

>

>

2 2

2

2 2

2

, , << 2 ( )

2

2

F kTm x

mL

pL

xπ∆

xπ∆

kk

kkcutcut==ππ// xx

22ππ/L/Ln(kn(k))

HamiltonianHamiltonian

( )

23 † 3

3 †

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )2

ˆ ˆ ˆˆ ˆ ˆ( ) ( ) ; ( ) ( ) ( )

s ss

s s s

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

N d r n r n r n r r r

ψ ψ

ψ ψ

↑ ↓=↑↓

↑ ↓

⎛ ⎞∆= + = − −⎜ ⎟

⎝ ⎠

= + =

∑∫ ∫

∫

2 2 21

4 2cutmkm

g aπ π= − + Running coupling constant g defined by lattice Running coupling constant g defined by lattice

21 - U N IT A R Y LIM IT

2m

g xπ=

∆

{ } ( )

{ } ( ) ( )

1

11 ˆ ˆ

1 1ˆ ˆ ˆ ˆ( ) ( , , ) ( )

ˆ ˆ ˆ ˆ( ) ( , , ) ; ( , , )=

n n

n n

E NkTn

n

H NE N kTkT

n

E T H T r H H N T E eZ T Z

Z T T r H N T e H N T e

µ

µµ

ρ

ρ ρ

− −

− −− −

= = =

= =

∑

∑

EigenenergiesEigenenergies of of thethe HamiltonianHamiltonian areare unknownunknown!!

GrandGrand--canonicalcanonical ensembleensemble::

PathPath integralintegral approachapproach::SingleSingle--particleparticle quantum quantum mechanicsmechanics::

1r

1iSe 2iSe

( )

( )( )

0 0

0

( ), ( )ˆ ( )

1 1

( ), ( )[ ( )]

[ ( )]

( )( ), ( ) ( ); 2

t

t

t

t

i L r t r t dtiH t t

i L r t r t dtiS r t

r e r D r t e

mr tL r t r t V r e e

− −∫

=

∫= − =

∫

{ }

ˆ ˆ ˆ ˆ( ) exp( ( ) exp( ( )

1 ; imaginary time:

n manybody states

Z Tr H N n H N n

itkT

β β µ β µ

β τ

−

= − − = − −

= =

∑Quantum Quantum statisticalstatistical mechanicsmechanics::

[ ] σβ σ τ

σ τ σ

+=

= − +

∫ˆln{det[1 ({ })]}( ) ( , )

ˆ[ ( , )] ln{det[1 ({ })]} - action

UZ D r e

S r U

1kT

β

τσ τ σ µ σ

σ ψ σ ψ ψ

= − − −

=

∫0

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

ˆ({ }) ({ }) ; - single-particle wave function kl k l l

U T d h h

U U

σσ τ σ

σ

−

= = ∫[ ]

energy associated with a given sigma field

[ ( , )]ˆ( ) [ ({ })] ( )

[ ({ })]-

SD r eE T H E U

Z T

E U

Sigma space sampling

σ •

[ ]( ) SP e σσ −∝

Quantum Monte-Carlo:

( )1

1( ) ({ })N

kk

E T E UN

σ

σσ

=

= ∑

22

( ) - stochastic variable

( ) ( )

1( ) ( )

- number of uncorrelated samples

E T

E T E T

E T E TN

Nσ

σ

=

− ∝

β

τσ τ σ µ σ

σ ψ σ ψ ψ

= − − −

=

∫0

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

ˆ({ }) ({ }) ; - single-particle wave function kl k l l

U T d h h

U U

Quantum Monte-Carlo: parallel computing

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

1ψ2ψ3ψnψ

σ ψ σ ψ= ˆ({ }) ({ })kl k lU U

σ(̂{ })U σ(̂{ })U . . .σ(̂{ })U σ(̂{ })U

...

More details of the calculations:More details of the calculations:

•• Lattice sizes used from Lattice sizes used from 883 3 xx 257257 (high Ts) to (high Ts) to 883 3 x 1x 1732732 (low Ts)(low Ts), <N>=50,, <N>=50,andand 663 3 xx 257 257 (high Ts) to (high Ts) to 6633 x 1x 1361361 (low Ts)(low Ts), <N>=30., <N>=30.

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

•• 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 igns the auxiliary fields fields σσ(x,(x,ττ) so as to maintain a running average of the acceptance rate bet) so as to maintain a running average of the acceptance rate betweenween0.4 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--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({U({σσ}) }) to stabilize the to stabilize the numericsnumerics..

•• Use Use 2200,00000,000--2,000,000 2,000,000 σσ(x,(x,ττ))-- field configurations for calculationsfield configurations for calculations

•• MC correlation MC correlation ““timetime”” ≈≈ 1150 50 –– 2200 time steps00 time steps at T at T ≈≈ TTcc

SuperfluidSuperfluid to Normal Fermi Liquid Transitionto Normal Fermi Liquid Transition

BogoliubovBogoliubov--Anderson phononsAnderson phononsand and quasiparticlequasiparticle contributioncontribution(d(dashedashed linlinee ))

BogoliubovBogoliubov--Anderson phonons Anderson phonons contribution only (contribution only (dotteddotted lineline))People never consider this ???People never consider this ???

QuasiQuasi--particle contribution onlyparticle contribution only(d(dottedotted line)line)

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

a = a = ±±∞∞

( 0) 0.41(2)Tξ = ≈

3

quasi-particles 4

7 /3

3 5 2( ) exp5 2

2 exp2

FF

FF

TE T NT

e k a

πεε

πε

∆ ∆⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞∆ = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

44

phonons 3/2

3 3( ) , 0.445 16F s

s F

TE T N πε ξξ ε

⎛ ⎞= ≈⎜ ⎟

⎝ ⎠

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

µ ε εε

επ

µσ

ε

⎛ ⎞= + =⎜ ⎟

⎝ ⎠

= = =

− ⎛ ⎞= = =⎜ ⎟

⎝ ⎠

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

µµ

EE

SS

Ideal Fermi gasentropy

Phase transition

LowLow temperaturetemperature behaviourbehaviour of a of a FermiFermi gasgas inin thethe unitaryunitary regimeregimeµε ξ ξ

ε ε⎛ ⎞

= ≈ ≈ <⎜ ⎟⎝ ⎠

3 ( )( ) and 0.41(2) for

5 F s CF F

T TE T N T T

µ ε ξ ξ ε ξε ε ε

⎡ ⎤⎛ ⎞ ⎛ ⎞′= = − ≈⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

( ) 2( )

5F F sF F F

dE T T T TT

dN

ξ ξ ς ςε ε⎛ ⎞ ⎛ ⎞

= + ≈⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

5/2

, 11(1)s s sF F

T T

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

ε ξ ςε

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

3( )

5

n

F s sF

TE T N

This is the same behavior as for a gas ofThis is the same behavior as for a gas ofnoninteractingnoninteracting (!) bosons below(!) bosons belowthe condensation temperature.the condensation temperature.

ConclusionsConclusions

Fully nonFully non--perturbative calculations for a spin ½ many perturbative 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 andandapparently the system undergoes a phase transition in the bulk aapparently the system undergoes a phase transition in the bulk at t TTcc = 0.2= 0.233 ((22) ) εεFF((ExpExp: : TTcc == 0.27(2) 0.27(2) εεFF , J. , J. KinastKinast et al.et al. ScienceScience, 307, 1296 (2005):, 307, 1296 (2005):BasedBased on on theoreticaltheoretical assumptionsassumptions).).

ChemicalChemical potentialpotential isis constantconstant upup to to thethe criticalcritical temperaturetemperature –– notenotesimilaritysimilarity withwith Bose Bose systemssystems!!

Below the transition temperatureBelow the transition temperature,, both phonons and both phonons and fermionifermioniccquasiparticlesquasiparticles contribute almost contribute almost equalyequaly to the specific heat. In to the specific heat. In mormore e thanthan one way the system is at crossover between a Bose and Fermione way the system is at crossover between a Bose and Fermisystemssystems..

There are reasons to believe that below the critical temperature thissystem is a new type of fermionic superfluid, with unusual properties.

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

QuestQuest for ufor unitarynitary point critical temperaturepoint critical temperature

Boris Svistunov’s talk (updated), Seattle 2005

E. Burovski, N. Prokofev, B. Svistunov, M. TroyerPrivate communication 2005

Ours

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

EvidenceEvidence for for fermionicfermionicsuperfluiditysuperfluidity: : vorticesvortices!!

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

6system of fermionic atomsLiFeshbachFeshbach resonanceresonance: :

B=834GB=834G

BEC side:a>0

BCS side:a<0

UNITARY REGIMEUNITARY REGIME