bcs-bec crossovern. fukushima, y. ohashi, e. taylor, and a. griffin, pra 75 (2007) 033609 c f t t...
TRANSCRIPT
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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)
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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
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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
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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
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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
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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
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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))
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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
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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
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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
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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
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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.
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2009/7/23 13
BCS-BEC crossover theory and crossover behavior at Tc
NSR-Gaussian Fluctuation theory2
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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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
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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:
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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
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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
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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)
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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.
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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.
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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
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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
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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
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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
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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
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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!
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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.
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2009/7/23 36
Pseudogap region in the BCS-BEC crossover
cT
MF
cT *T2 | |bindE
*T =“pseudogap temperature”
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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)
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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
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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,)
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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
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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