2009/7/23 1

BCS-BEC Crossover

Yoji Ohashi

July 22, 2009, Colloquium, KEK, Tsukuba, Japan

Faculty of Science and Technology

Keio University, Japan

Allan Griffin (Toronto, Canada)

N. Fukushima (Tsukuba (Canon))

S. Tsuchiya (Keio, Japan)

collaborators

A Unified Description of Fermi and Bose superfluid Phenomena

R. Watanabe (Keio, Japan)

2009/7/23 2

BCS-BEC crossover

A unified Description of Fermi and Bose Superfluid Phenomena

Introduction

Feshbach resonance and BCS-BEC crossover in cold Fermi gases

BCS-BEC crossover theory and crossover behavior of Tc

Pairing fluctuations and pseudogap effect above Tc

NSR-Gaussian fluctuation theory

Single-particle excitations and Goldstone mode

Extension to the superfluid phase below Tc

Phase diagram of Fermi superfluids in the BCS-BEC crossover region

Summary

2009/7/23 3

Recent progress in cold atom physics

1995

2002BEC in Bose

atom gasesMott Transition in

Superfluid Bose gases

Superfluid Fermi gas and

BCS-BEC crossover

Optical lattice Feshbach resonance

Trap potential

Superfluid Fermi gas

in optical lattice

2004

2006

2009/7/23 4

Fermion Superfluidity in 40K Fermi gas

C. A. Regal, et al. PRL 92 (2004) 040403.

4 2/ ~ 0.08 0.2 10 10 ( )c FT T metal

| 9 / 2, 7 / 2 | 9 / 2, 9 / 2

0.35FT K510N

superconductivity

| 9 / 2, 7 / 2

| 9 / 2, 9 / 2

Superfluid Fermi gas

Cooper pair

2009/7/23 5

Fermion Superfluidity in 40K and 6Li Fermi gases

C. A. Regal, et al. PRL 92 (2004) 040403.

4 2/ ~ 0.08 0.2 10 10 ( )c FT T metal

| 9 / 2, 7 / 2 | 9 / 2, 9 / 2

0.35FT K510N

Cooper-pair formation Superfluidity

single-particle excitation gap Goldstone sound mode

C. Chin , et al. Science 305 (2004) 1128.

gapE

vortex lattice formation

M. Zwierlein, et al., PRL 2005

v

Thomas et al., PRL 98 (2007) 170401

6Li

6Li

6Li

2009/7/23 6

Fermion Superfluidity in 40K and 6Li Fermi gases

C. A. Regal, et al. PRL 92 (2004) 040403.

Key words to understand this new superfluidity

Feshbach resonance

BCS-BEC crossover

4 2/ ~ 0.08 0.2 10 10 ( )c FT T metal

| 9 / 2, 7 / 2 | 9 / 2, 9 / 2

0.35FT K510N

2009/7/23 7

Feshbach resonance

Fermi atom

tunable by magnetic field

2 1

2effV g

molecule

BCS-BEC crossover tuned by F.R.

(Timmermans (2001), Holland (2001))

(Ohashi and Griffin, PRL 89, 130402 (2002))

2009/7/23 8

Bose-Einstein Condensation (BEC)

thermal de Broglie length

boson

/ 2T h mT

cT T

1/3~1/T d n

cT T

1/3~ ~1/T d n

2009/7/23 9

Essence of the BCS-BEC crossover phenomenon

weak U strong U

Ebind (binding energy)

T

Tc

1/3~ ~ ( / 2)MOLT MOLd n

BCS BEC

~c FT

cT

weak U

strong U

2009/7/23 10

T

Superfluid phase

Fermi gas

Molecular Bose gas

Tc

Phase diagram of Fermi superfluids

BCS regime BEC regime

Ebind

crossover region

weak U strong U

2009/7/23 11

Model superfluid Fermi gas with a Feshbach resonance

Coupled Fermion-Boson (CFB) model

two atomic hyper-fine states:

2

Fermion Boson

2

22

q

qE

M Feshbach resonance

2Fermion BosonN N N H N

2

2eff

gV

† † † † †

/ 2 / 2 / 2 / 2 ' / 2 ' / 2, , , ',

. .q bgp q p qH c c E b b g b c c h c U c c c c

p p p q q q p q p q p q p q

p q p q p p q

2009/7/23 12

Model superfluid Fermi gas with a Feshbach resonance

Coupled Fermion-Boson (CFB) model

two atomic hyper-fine states:

† † † † †

/ 2 / 2 / 2 / 2 ' / 2 ' / 2, , , ',

. .q bgp q p qH c c E b b g b c c h c U c c c c

p p p q q q p q p q p q p q

p q p q p p q

2

2bg

gU U

† † †

/ 2 / 2 ' / 2 ' / 2, , ',

H c c c c cU c

p p p p q p q p q p qp p p q

broad Feshbach resonance ( )Fg n

In a broad FR, one can safely study the BCS-BEC crossover by using the

ordinary BCS model, without explicitly considering Feshbach molecules.

Current experiments on 40 K and 6Li are all using a broad Feshbach resonance.

2009/7/23 13

BCS-BEC crossover theory and crossover behavior at Tc

NSR-Gaussian Fluctuation theory2

2009/7/23 14

qp,

ppqpqpp

pp ccccUccHp†††

,

)(

Formulation (broad Feshbach resonance: 40K, 6Li)

：two atomic hyperfine states = pseudospin ↑↓

U : effective pairing interaction associated with the F.R.

U

uniform gas is assumed.

Feshbach resonance We treat U as a tunable parameter.

2009/7/23 15

BCS-BEC crossover theory at Tc

Equation for Tc: Thouless criterion

( , ) q

Fermion

U

1( , ) 0 q

tanh ( )21

2( )

p

p

U

p

...

0 qat cT

Nozieres and Schmitt-Rink, JLTP (1985)

1G

p

The chemical potential remarkably deviates from the

Fermi energy F in the BCS-BEC crossover.

2009/7/23 16

BCS-BEC crossover theory at Tc

is determined from the number of Fermi atoms.

0,

log 1 ( , )n

F nN T U iN

q

q

MF

pair correlation function

U

.......

/ 2 / 2

/ 2 / 2

1( , )

2

p q p q

n

p q p q n

f fi

i

p

q

0 2 ( )FN f pp

pairing fluctuations

2009/7/23 17

Reguralized Nozieres-Schmitt-Rink (NSR) theory (Tc)

4 1 11 tanh ( )

2( ) 2 2

sp

p pm

a

p

0

,

41 1ln 1 ( , )

2 2n

F n

q p

sN N q im

a

as: two-body s-wave scattering length

4

1 1/ 2

sa U

m U

pp

The NSR theory is known to include pairing fluctuations within the Gaussian

fluctuation level around the mean-field (saddle point) solution.

Gaussian fluctuation theory

weak-U strong-U

1

sa 1sa

The well-known ultraviolet divergence

in the BCS theory is absorbed into as.

2009/7/23 18

Weak-coupling BCS limit

4 1 11 tanh ( )

2( ) 2 2

sp

p pm

a

p

0

,

41 1ln 1 ( , )

2 2n

F n

q p

sN N q im

a

1

sa

( )F c FT T

The Tc-equation with =F reproduces the

ordinary BCS gap equation at Tc.

2009/7/23 19

Strong-coupling BEC limit

4 1 11 tanh ( )

2( ) 2 2

sp

p pm

a

p

,

41 12 ( ) ln 1 ( , )

2 2n

n

q p

sN f q im

a

pp

1

sa

220

1

sma

2

2

1bbind

s

Ema

2-body bound state energy

2

2(2 )

1

21m

q

N

e

q

BEC of (N/2,2m)

molecular Bose gas

2009/7/23 20

BCS-BEC crossover at Tc

c

F

T

TF

1( )F sp a

1( )F sp a

2

1

2BEC

sma 0.218 FT

BCS

cT

BCS BECBEC BCS

cT

40K

Jin, 2004

experimental result

2009/7/23 21

Superfluid properties in the crossover region

Single-particle and collective excitations

N. Fukushima, Y. Ohashi, E. Taylor, and A. Griffin, PRA 75 (2007) 033609

c

F

T

T

1( )F sp a

BCS BEC

superfluid

phase

3

2009/7/23 22

Extension to the superfluid phase below Tc

pairing fluctuations

( , )ni q

ie

cT T cT T

phase fluctuations

amplidude fluctuations

( , )ni

q

4 1 11 tanh ,

2 2 2

ps

p p p

Ea

m E

2 2( )p pE

phason, ampliton, and their coupled fluctuations

0

,

41 1ln det 1 ,

2 2( , )

n p

ns

F

q

aN N

mi

q

2009/7/23 23

BCS-BEC crossover below Tc

F F

FT FT

)/(1 sFak)/(1 sFak

0

2 2( )p pE

Energy gap

22 | |

(BCS)

(BEC)

single-particle excitations:

2009/7/23 24

Goldstone collective mode in the BCS-BEC crossover

/ Fv v

/ FT T 1( )F sk a

Anderson-Bogoliubov

mode

Bogoliubov phonon

of a molecular BEC

/ 3 ( 0)Fv v T

experiment on 6Li

theory

F

v

v

Thomas et al., PRL 98 (2007) 170401

v q

2009/7/23 25

E

2

q0

single-particle

excitations

E

q0

single-particle excitations

Collective excitations are important

only in a small momentum region.

BCS region BEC region

Large single-particle excitation gap

reflecting a strong binding energy.

From single-particle excitations to collective excitations

2009/7/23 26

Superfluid density rs in the BCS-BEC crossover

( )s nnr r F

n

B

n nr r r Normal fluid density

22 ( )3

n p

p

F

p

p f Em E

r

mean-field part fluctuation part

2 2( )p pE

This part is dominant in the BCS regime.

2

2,

0

42 1ln det 1 ( , )

2n

z

sn n

zQ

B

q p

amq i

Q m

r

22 ( )3

B q

qB q

q nM

p v q

This part is dominant in the BEC regime.

(BEC)

2009/7/23 27

Superfluid density in the BCS-BEC crossover

N

N S

FTT / 1)( sF ak

Superfluid density

BCS

BEC

/s nr

BCSF

nn r

BECB

nn r

The origin of the normal fluid density continuously changes from single-

particle excitations to collective excitations in the BCS-BEC crossover.

2009/7/23 28

Effective interaction between Cooper pairs in the BEC regime

4MOL Beff

B

aV

M

+ +…..MOL

effV 0.6B sa a

2 0B sa a

2-body renormalization

MOL

effV

Petrov et al., PRL 93 (2004) 090404.

2009/7/23 29

Effective interaction between Cooper pairs in the BEC regime

two-body renormalization

level (Petrov, et al, 2004)

Many-body-renormalized molecular

interaction vanishes at Tc.

(Renormalization Group (1-loop))

Y. Ohashi; JPSJ 74 (2005) 2659

Many-body renormalization

bare interaction / 2B Fa a

2009/7/23 30

Recovery of the second order transition in the BEC regime

1/( ) 2F sp a

Many-body renormalization effects

on molecular interaction is included.present theory

0 /n n0 /n n

/ FT T

2009/7/23 31

Normal state properties in the crossover region

pairing fluctuations and pseudogap effect

S. Tsuchiya, R. Watanabe, and Y. Ohashi, cond-mat/0906.3983

c

F

T

T

1( )F sp a

BCS BEC

normal

state

4

2009/7/23 32

Pseudogap in high-Tc cuprates

Bi2212 (under dope, Tc=83[K])

C.Renner et al., PRL 80 (1997) 149

cT T

strong pairing fluctuations above Tc

strong AF spin fluctuations

hidden order (DDW)

origin of pseudogap

We can expect pseudogap phenomenon originating from strong pairing

fluctuations in the BCS-BEC crossover regime of cold Fermi gases

2009/7/23 33

Single-particle density of states (DOS) in the normal state

1( ) Im ( , )nN G i i

p

p

G involves the self-energy correction describing pairing fluctuations

+ + + …U

( , )ni p

N()(m

pF/2

2)

N()(m

pF/2

2)

Free Fermi gas (=F) BCS state (=0.2F)

/ F / F

2

2009/7/23 34

N()(m

pF/2

2)

N()(m

pF/2

2)

Free Fermi gas (=F) BCS state (=0.2F)

/ F / F

2

Single-particle density of states (DOS) in the normal state

BCS

BEC

cT T

0!

2009/7/23 35

Single-particle density of states (DOS) in the normal state

BCS

BEC

cT T

cT T

1( ) 0.6F sp a

cT T

One can define the

pseudogap temperature

T* as the temperature

where the dip structure

vanishes.

2009/7/23 36

Pseudogap region in the BCS-BEC crossover

cT

MF

cT *T2 | |bindE

*T =“pseudogap temperature”

2009/7/23 37

“phase diagram” of cold Fermi gases

cT

*T 2 | |bindE

pseudogap

Fermi atom gas

Molecular

Bose gas

Superfluid phase

(Fermion like) (Boson like)

2009/7/23 38

Spectral weight at Tc in the BCS-BEC crossover

1( , ) Im ( , )nA G i

p p

/ Fp p

Free Fermi gasweak-coupling

BCS state at T=0

/ Fp p

/ 1F

2

/ 0.2F

( , ) ( ( ))A pp

Fp p Fp p

A(p,)

/

F

/

F

( , )A p

p

2 2( , ) ( ) ( )p p p pEA u v E p

( , )A p

pE

pE

2

pu2

pv

2 2( )p pE 2 / 2p p m

( ) ( , )Ar p

p

2009/7/23 39

Single-particle excitations in the BCS state

/ Fp p

Fp p

:p

/ Fp p

particle band

( ) :p hole band

2 2( )pE

2 2( )pE

† †

p ppp pc v cu

Bogoliubov excitation = particle + hole:

/ F

A(p,)

2009/7/23 40

Spectral weight at Tc in the BCS-BEC crossover

1( ) 0.6F sp a

/ Fp p

1( , ) Im ( , )nA G i

p p

/

F

A(p,)

Pairing fluctuations induce a particle-hole coupling in the normal

state above Tc, leading to the pseudogap in DOS.

( ) ( , )Ar p

p

BCS state (T

2009/7/23 41

Summary

We have discussed the BCS-BEC crossover in ultra-cold Fermi gases.

superfluid phase

Molecular

Bose gas

Fermi gas

pseudogap

*T

BCS region BEC region

Fermion SF Boson SF

cT

Tunable pairing interaction associated with a Feshbach resonance

Unified description of Fermi and (molecular) Bose superfluids

Importance of pairing fluctuations in the crossover theory