Advanced condensed matter physics
Keio Univ.
BCS-BEC Crossover in a Superfluid Fermi Gas
A unified description of Fermi and Bose superfluids
and
New Material Science
Yoji Ohashi
Department of Physics, Keio University, Japan
Ultracold Atom Physics
Atoms are trapped in a magnetic/optical potential,
and are cooled down to <O(mK). Then, one can
observe quantum phenomena, such as superfluidity.
History of Fermi and Bose superfluids
Fermi superfluids
liquid 4He (K. Onnes)
1908
Superconductivity in Hg (K. Onnes) 1911
1924~1925
Bose superfluids
Theoretical prediction by Bose and Einstein
1932
Superfluid 4He (W. Keesom)
1938
“Superfluidity” (Kapitza)
BCS theory (Bardeen, Cooper, Schrieffer) 1957
Superfluid 3He (Osheroff) 1972
High-Tc superconductor (Bednorz and Muller) 1986
1995
BEC in ultracold Bose gas
2004
BCS-BEC crossover in 40K and 6Li Fermi gases (Jin, Ketterle, et al)
History of Fermi and Bose superfluids
Fermi superfluids
liquid 4He (K. Onnes)
1908
Superconductivity in Hg (K. Onnes) 1911
1924~1925
Bose superfluids
Theoretical prediction by Bose and Einstein
1932
Superfluid 4He (W. Keesom)
1938
“Superfluidity” (Kapitza)
BCS theory (Bardeen, Cooper, Schrieffer) 1957
Superfluid 3He (Osheroff) 1972
High-Tc superconductor (Bednorz and Muller) 1986
1995
BEC in ultracold Bose gas
2004
BCS-BEC crossover in 40K and 6Li Fermi gases (Jin, Ketterle, et al)
BCS-BEC crossover
Unified phenomenon to describe Fermi and Bose superfluids
which have been studied independently in 20th century.
Character of fermion superfluidity continuously changes from the weak-coupling
BCS type (such as superconductivity) to the Bose-Einstein condensation (such as
superfluid 4-He) of tightly bound molecular bosons, as one increases the strength
of a pairing interaction between Fermi atoms.
In cold atom physics, we can easily tune the interaction between
particles by using a Feshbach resonance.
interaction
40K
|9/2,-9/2>
|9/2,-5/2>
(JILA)
BCS-BEC crossover in a superfluid Fermi gas
Is this class useful for your work?
Related research fields:
cold atom physics
laser physics (quantum electronics)
condensed matter physics, especially, superconductivity
and strongly correlated electron systems
superfluidity, such as liquid 4-He
nuclear physics (neutron rich nucleus)
elementary particle physics (color superconducvity; di-quark state)
astrophysics (BCS-BEC crossover in neutron star)
quantum chemistry (ultracold molecules)
The cold Fermi gas system is expected as a useful quantum simulator
Impoprtant material
parameters in material science
interaction
band structure
(crystal lattice)
particle (electron)
density
“obstacles”
in material science
defect, impurity
High controllability of cold atom gas
Impoprtant material
parameters in material science
interaction
cold atom gases
tunable by Feshbach resonance
band structure
(crystal lattice)
particle (electron)
density
(JILA)
“obstacles”
in material science
defect, impurity
High controllability of cold atom gas
Impoprtant material
parameters in material science
interaction
band structure
(crystal lattice)
particle (electron)
density
“obstacles”
in material science
defect, impurity
cold atom gases
tunable by Feshbach resonance
various crystal lattice by optical lattice
High controllability of cold atom gas
Impoprtant material
parameters in material science
interaction
cold atom gases
tunable by Feshbach resonance
band structure
(crystal lattice)
various crystal lattice by optical lattice
particle (electron)
density
tunable
Particle statistics （fermion、boson、mixture）
“obstacles”
in material science
defect, impurity
High controllability of cold atom gas
Impoprtant material
parameters in material science
interaction
cold atom gases
tunable by Feshbach resonance
band structure
(crystal lattice)
various crystal lattice by optical lattice
particle (electron)
density
tunable
Particle statistics （fermion、boson、mixture）
“obstacles”
in material science
defect, impurity
ideal crystal without defects and
impurities
High controllability of cold atom gas
search for new material
accidental discovery!
（electron system）
High controllability of cold atom gas
theoretical prediction
atom
electron
search for new material
We can confirm the prediction
by using cold atom gas loaded
on an artificial optical lattice.
accidental discovery!
＋
（electron system）
optimal condition can
be also examined.
electron
atom
Search for a material to
realize the prediction
Advanced condensed matter physics
Starting from the review of statistical mechanics and quantum
physics, I will explain the physics of BCS-BEC crossover
phenomenon.
The real BCS-BEC crossover has been observed in cold Fermi
gases, but this phenomenon itself is very general, being widely
applicable to various fields. In this lecture, I will explain this
interesting physics from general theoretical viewpoint so that
you can get something useful from this lecture series.
Advanced condensed matter physics (2014)
Grading A,B,C,D
I will check your attendance at the beginning of each lecture. You need
to attend more than 70% of lectures.*
At the end of this course, you need to submit a report, where you solve
some problems related to the BCS-BEC crossover. I will show the
problems during lectures. At the submission, we are required to give me a
brief explanation about your solutions (oral interview!).
*When you cannot attend a lecture due to an inevitable reason (eg., illness),
please let me know after you come back. Then I will treat the absence as
“attendance”.
Chapter 1.
Experimental Overview
Recent Development in cold atom physics
Trap potential
1995
BEC in Bose
atom gases
JILA
Recent Development in cold atom physics
Trap potential
1995
Optical lattice
BEC in Bose
atom gases
2002
Mott Transition in
Superfluid Bose gases
Greiner, Nature 2002
Recent Development in cold atom physics
Trap potential
Feshbach resonance
1995
Optical lattice
BEC in Bose
atom gases
2002
Mott Transition in
Superfluid Bose gases
2004
Superfluid Fermi gas and
BCS-BEC crossover
JILA, MIT (2004)
Recent Development in cold atom physics
Trap potential
Feshbach resonance
1995
Optical lattice
BEC in Bose
atom gases
2002
Mott Transition in
Superfluid Bose gases
2004
2006
Superfluid Fermi gas and
BCS-BEC crossover
Superfluid Fermi gas
in optical lattice
Ketterle (2006)
Recent Development in cold atom physics
Trap potential
Feshbach resonance
1995
Lin Nature 471 (2011) 83
Optical lattice
BEC in Bose
atom gases
2002
Mott Transition in
Superfluid Bose gases
2004
2006 Artificial Gauge field
2008
Superfluid Fermi gas and
BCS-BEC crossover
Superfluid Fermi gas
in optical lattice
Synthetic spin-orbit
interaction
Recent Development in cold atom physics
Trap potential
Feshbach resonance
1995
Optical lattice
BEC in Bose
atom gases
2002
Mott Transition in
Superfluid Bose gases
2004
2006 Artificial Gauge field
2008
2014
Superfluid Fermi gas and
BCS-BEC crossover
Superfluid Fermi gas
in optical lattice
Synthetic spin-orbit
interaction
2D-Fermi superfluid
(KT-transition)
Jochim arXiv14095373 (2014)
Fermion Superfluidity in 40K Fermi gas
| 9 / 2, 7 / 2   | 9 / 2, 9 / 2 
TF  0.35m K
N  105
C. A. Regal, et al. PRL 92 (2004) 040403.
Tc / TF ~ 0.08  0.2  104  102 (metal )
Condensate fraction (=the number of
Bose condensed Cooper pair bosons)
Cooper pair
| 9 / 2, 7 / 2 
| 9 / 2, 9 / 2 
superconductivity
Superfluid Fermi gas
Fermion superfluidity in 6Li Fermi gas
|1/ 2, 1/ 2   |1/ 2, 1/ 2 
M. Zwierlein, et al. PRL 92 (2004) 120403.
Single-particle excitations in a superfluid 6Li Fermi gas
rf-tunneling current spectroscopy
BEC
unitarity limit
BCS
C. Chin , et al. Science 305 (2004) 1128.
photon

a
I ( )
Binding energy of a
Cooper pair boson
Vortex phase in a 6Li superfluid Fermi gas
M. Zwierlein, et al., PRL 2005
(This figure shows molecular density profiles.)
The observation of vortices is a clear evidence of
“superfluidity” in superfluid Fermi gases.
Collective (surface) mode in a 6Li Fermi gas
M. Bartenstein, et al., PRL 92 (2004) 203201
The gas cloud behaves like a
macroscopic single wavefunction.
optical lattice
artifical lattice produced by standing wave of laser light
standing wave
of laser light
E( x)  E0 sin(kx)
atom gas
Stark shift by s-p dipole transition
periodic potential
V ( x)  
p
s
H  H 0  eEx
alkali metal atom (Li, Na, K,…)
E 

2
| E ( x) |2  sin 2 (kx)
| s | eEx | p |2
Es  E p
 E2
optical lattice
artifical lattice produced by standing wave of laser light
standing wave
of laser light
E( x)  E0 sin(kx)
atom gas
Stark shift by s-p dipole transition
periodic potential
V ( x)  
40K
(Fermion)
py
2
| E ( x) |2  sin 2 (kx)
observation of the
Brillouin zone
px
V / Er  5

7
Er  (2 / laser )2 / 2m
8
12
Kohl et al., PRL 94 (2005) 080403.
Superfluid-Mott insulator transition in 87Rb Bose gas
87Rb-Bose
Bose condensate at q=0
gas in 3D optical lattice
1st Brillouin zone
U/t =0
py
px
87
Rb :| F  2, mz  2 
N  2 105
optical  852[nm](65  65  65)
V  20Er
Er  (2 /  )2 / 2m
large U/t
Greiner et al., Nature 415 (2002) 39
h
superfluid phase
Mott phase
U / zt ~ 6 (z=6: coordination number)
Theory: U / zt  5.83
Sheshadri et al, Europhys. Lett. 22, 257 (1993)
Superfluid 6Li gas in an optical lattice
Bose condensed Cooper pairs
0
laser
atom gas
py
px
B[G] 
Laser intensity
-12
33
83
J. K. Chin, et. al. (MIT),
Nature 443 (2006) 961.
6Li
|1/ 2, 1/ 2 
|1/ 2, 1/ 2   |1/ 2, 1/ 2 
TF  1.4m K
N  4 105
|1/ 2, 1/ 2 
Chapter 2.
Quantum statistics of particle:
Difference and similarity between
Fermion and Boson
Fermi atom and Bose atom
atom = proton(F) + neutron(F) + electron (F) =NF
Fermi atom: NF= odd
Bose atom : NF= even
In cold atom physics, in most experiments, alkali atoms are used. In
particular, the Fermi superfluid has been realized in K and Li gases.
isotope proton neutron electron sum
6Li
7Li
39K
40K
41K
3
3
19
19
19
3
4
20
21
22
3
3
19
19
19
9
10
58
59
60
statistics
Fermi
Bose
Bose
Fermi
Bose
Alkali atoms zA with z being even are Fermions, and those with z
being odd are Bosons (which generally holds!).
Atomic hyperfine state
Atomic spin state (=hyperfine state) = nuclear spin +electron spin
I
S
F
alkali atom: S=1/2
F  I  1/ 2
In metallic superconductivity, an up-spin electron and down-spin
electron form a Cooper pair. In cold Fermi gases, two atomic hyperfine
states work as pseudospin-up and –down.
6Li
(I=1): F=3/2, 1/2
| F , Fz
 1/ 2, 1/ 2
40K
(I=4): F=9/2, 7/2
| F , Fz
 9 / 2, 9 / 2  9 / 2, 7 / 2
Fermi and Bose statistics 1
While the boson wave function is symmetric with respect to the
exchange of two particles, the fermion wave function is antisymmetric.
x  (r ,  )
n,m: quantum labels
fermions
( x1 , x2 )  n ( x1 )m ( x2 )  n ( x2 )m ( x1 )
bosons
( x1 , x2 )  n ( x1 )m ( x2 )  n ( x2 )m ( x1 )
( x2 , x1 )  ( x1 , x2 )
( x2 , x1 )  ( x1 , x2 )
More than two fermions cannot occupy the same quantum state.
Pauli’s exclusion principle
( x1 , x2 )  n ( x1 )n ( x2 )  n ( x2 )n ( x1 )  0
Thus, in a two-component Fermi gas (consisting of atoms with two
hyperfine states), the maximum occupation number of each momentum
state is two.
Fermi and Bose statistics 2
Even when two fermions are in different quantum states, they
cannot locate at the same (generalized) position x=(r,).
( x1  x2  x)  n ( x1  x)m ( x2  x)  n ( x2  x)m ( x1  x)  0
In particular, when their spin states are the same, they cannot
meet each other at the same spatial position r.
statistical repulsion
(This kind of repulsive effect is absent in the boson case.)
Bosons tend to occupy the same quantum state (q=0) at low
Temperatures (statistical attraction).
Occupation of atoms in the ground state (T=0)
Free Fermi gas with
two hyperfine states
Free Bose gas
 F ( TF ) (kB  1)
Fermi energy
Fermi temperature
py
py
pF
px
Pauli’s exclusion principle
(= statistical repulsion)
px
Bose-Einstein condensation
(= statistical attraction)
Occupation of atoms at finite temperautres
Fermions: Fermi distribution function
f (  m ) 
1
e
 (  m )
1
(  1/ T )
The Fermi chemical potential m can take positive and negative
values.
Bosons: Bose distribution function
nB (  m ) 
1
e  (  m )  1
Since the distribution function must be positive, the Bose
chemical potential have to be larger than the lowest energy
of atomic states (m >0). (Otherwise, the Bose distribution
Becomes unphysically negative!) For a free Bose gas with
=p2/2m, we find m<0.
Finite temperatures 1: Fermion
The occupation of Fermi atoms obeys the Fermi distribution
function.
T
T=0
1
1
T >0
f (  m )   (  m )
e
1
 F ( TF )
0
The chemical potential is determined from the equation for the
number n of Fermi atoms:
n  2 f ( p  m )
(V = 1)
p
At T=0, we find m   F , and
n  2
F
0
pF3
d  ( )  2
3
pF2
F 
2m
(  1)
pF: Fermi momentum
inter-particle distance
d ~ n 1/ 3 ~ 1 / pF
Finite temperatures 1: Fermion
In a bulk system, we can replace the momentum sum with the
momentum/energy integration (useful!).
n  2 f ( p  m )
(V = L3=1)
p
When we impose the periodic boundary condition, the momenta
px,py, and pz are discretized as
2
px 
nx , nx  0, 1, 2,......
L
p 
1
1
2
L
L
1D:  g ( p )  p  g ( p )p  p  dpg ( p)  2  dpg ( p)
x
x
px
px
L
V
1
3D:  g ( p ) ( 2 )  dp g ( p)  8  dp g ( p)  8  dp g ( p) (V  L3  1)
3
3
3
3
3
3
p
p2

2m
1
8 3
3
 dp g ( ) 
4
8 3
2
 p dp g ( ) 
m 2m
d   g ( )
2

2
Finite temperatures 1: Fermion
At finite temperatures, the Fermi edge at  F is smeared in the
region ~[F-T, F+T]. Thus, effects of the Fermi statistics (giving
the step function at T=0), become weak when T ~ TF .
At finite temperatures, m deviates from the Fermi energy. At low
temperatures, we obtain (Sommerfeld expansion)
2
2
 T 
m / F
m (T )   F [1    ]
12  TF 
When m<0 (which occurs around
T~TF), f(-m) reduces to the
classical Boltzman factor.
f (  m ) ~ e  |m|e 
T / TF
Namely, the Fermi energy is the characteristic temperature where
effects of the quantum Fermi statistics become important.
Finite temperatures 2: Boson
The occupation of Bose atoms is described by the Bose distribution
function.
nB (  m ) 
1
e  (  m )  1
nB
f
0
The chemical potential is determined from the equation for the
number of Bose atoms:
n   nB ( p  m )
(V = 1)
p
where m must be lower than the lowest energy, m   0 . This
equation is not satisfied below a certain temperature TBEC, where
TBEC is the phase transition temperature of the Bose-Einstein
condensation . At TBEC, we find that m=0, and
TBEC
2
n2 / 3

( (3/ 2)) 2 / 3 m
 (3/ 2)  2.612
Finite temperatures 2: Boson
Below Tc, the condensate fraction n0 , which describes the
number of Bose-condensed bosons, is obtained from
n  n0   nB ( p )
p
p2
p 
,
2m
m / Tc
m 0
At T=0, the second term vanishes,
so that all the atoms are Bosecondensed.
T / Tc
In the high temperature region, m is negatively large, and we
obtain the classical Boltzmann distribution function:
nB (  m ) ~ e  |m|e 
Similarity between Fermion and Boson
characteristic
temperature
Fermion
Boson
TF
thermal de Broglie length
TBEC
quantum mechanical
size of a particle
p2
T
2m
, p
h


2

(  1)
  2 / 2mT
(T  2 / mT )
Fermion: T=TF:  ~ 1/ pF ~ 1/ n1/ 3 ~ d F
Boson: T=TBEC:  ~ 1/ n1/ 3 ~ d B
TF and TBEC have the same physics that, around these temperatures,
quantum mechanical size of a particle is comparable to the interparticle distance, and the ‘classical particle picture’ is no longer
valid.
From classical to quantum regime
T  Tc
T  TF or TBEC
T  d ~ 1/ n1/ 3
T ~ d ~ 1/ n1/ 3
Boson/Fermion
thermal de Broglie length
T  h / 2 mT
The classical particle picture
breaks down and the quantum
wave function picture becomes
necessary.
Thermal de Bloglie length
T  2 / mT
Thermal de Broglie length：quantum-statistical size of a particle
p
h


2

2

p
(  1)
（de Bloglie length）
Including thermal fluctuations, we take the statistical average of
the de Bloglie length, which gives
T 
2
d
p
 pe
 dpe
2

mT


2
p
2m
p2
2m
 2


0


0
dppe

dpp 2e
p2
2m

2
p
2m

2
2mT



0

0
dxxe  x
2
2  x2
dxx e
Thermal de Bloglie length
Thermal de Broglie length：quantum-statistical size of a particle
T  2 / mT
ipr
wavefunction：  (r)  e
Including thermal fluctuations, we take the statistical average of this
Wavefunction. Then one has,
eipr
T

ipr
d
p
e
e

 dpe
波動関数の広がり
2
T 
mT

p2
2m
p2

2m
e

mT 2
x
2
~ T
Chapter 3.
Pair formation and
Fermi surface effect
Bose condensation in Fermi gases
Boson
BEC
Fermi degeneracy
Fermion
To form molecules, a pairing
interaction is necessary.
BEC of molecules (=bosons)
Pair formation 1: two Fermi particle ( ,  ) case
Assume two fermions interacting with each other. The system is described
by the Hamiltonian ( x  (r,  ))

1
[12  22 ]( x1 , x2 )  U (r1  r2 )( x1 , x2 )  E( x1 , x2 )
2m
When the interaction is spin-independent, the wavefunction has the form
(r1 , r2 , 1 ,  2 )  (r1 , r2 ) 1 ,  2
(r1 , r2 )   g (k )eik (r1 r2 ) (center of mass momentum =0)
k
Spin singlet:   
Spin triplet :   , , 
g (k )  g (k )
g (k )   g (k )
(The assumed wavefuntction must be antisymmetric!)
k2
g (k )   Vkk ' g (k ')  Eg (k )
m
k'
Vkk '   d 3rV (r)ei (k k ')r
Pair formation 1: two Fermi particle ( ,  ) case
For a contact-type interaction (U (r)  U (r) ), we obtain
k2
(  E ) g (k )  U  g (k ')
m
k'
=
k2
E
 2 k
m
0
spin-triplet
C
spin-singlet
In the spin-triplet case, two fermions never meet at the same spatial position due to
the statistical repulsion, so that the contact interaction does not work at all.
singlet case:
g (k ) 
U
2
k
E
m

g (k ')
k'

k
g (k )  
k
U
k2
E
m

k
g (k ')
'

1
1
1U
 U  d  ( )
0
2  E
k 2 k  E
 ( ) 
m 2m

2 2
density of states (DOS)
A bound state is obtained when this equation has a solution with E<0. We note that
the RHS has a ultraviolet divergence (    ).
Pair formation 1: two Fermi particle ( ,  ) case
Introducing a cutoff, we solve the following equation:
1U
k

1
1
 U  d  ( )
0
2 k  E
2  E
1U
k
c
1
1
 U  d  ( )
0
2 k  E
2  E
For E<0, we obtain


1
|E|
1 2c
 1 
Tan

U  (c ) 
2c
|E|
1
solution!
1/ U  (c )
|E|
There is a threshold value of the interaction U (lower limit) to form a molecule in
this two particle case.
U  (c )  1
 (c ) 
m 2m
c
2
2
Discussion: what is c?
U  (c )  1
 (c ) 
m 2m
c
2
2
This comes from introducing the contact interaction U (r)  U (r)
U
assumed contact interacion
|U()|
real interaction
c

To evaluate the value of the cutoff, we need to know the details of
the interaction in the high-energy region.
However, we usually do not know what happens in the highenergy region, and we do not want to know such a high-energy
physics in considering the interesting low energy physics.
Renormalization (Regularization)
To avoid introducing the unknown cutoff c, we introduce the two-particle
s-wave scattering length as.
Following the standard scattering theory, we calculate the scattering matrix
(t-matrix) in terms of the contact interaction. The scattering length is then
obtained as the low-energy limit of the t-matrix.
4 as
 (0, 0)
m
=
+
U
c
U
U
+ U
1
 U  (U )
(U )  .....  
p 0  2
U U
+…….
U
c
1U 
p
1
2
Since the scattering length is observable experimentally, we use this equation
to determining the value of the cutoff c.
Renormalization (Regularization)
c
1
1
1U
 U  d  ( )
0
2  E
k 2 k  E
c
c
c
 1
 1 
1
1 
1U
U


U




2


E
2


E
2

k
k 
k  2 k 
k
k
k 
c
c
 1 
 1
1 
1U  

U

k  2  E 2 

k  2 k 
 k
k 
 1
1  4 as c
1
 



c
m k
 1  k  2 k  E 2 k 
1U   
k  2 k 
c
U
4 as

m
U
c
1U 
p
1
2
 1
1 
 

 2 k  E 2 k 
This summation
converges without
the cutoff!

~  d 
0
|E|

 (  | E |)
Renormalized theory for the two-body bound state problem
4 as c
1 

m k
 1
 1
4 as 
1 
1 



d

(

)






0
2


E
2

m
2


E
2

 k
k


This renormalized equation is written by using only physical parameters.
Molecular formation:
as  0.
weak U
as 1  0
Bound state energy
E
1
mas2
Bound state wavefunction
E
0
strong U
molecular size= as
a 1s  0
E
molecular size
1
mas2
Renormalized theory for the two-body bound state problem
In a two-fermion system, a bound state is not always formed even when
they are interacting attractively.
weak U
as 1  0
strong U
0
a 1s  0
E
molecular size
1
mas2
Pair formation 2: Cooper problem
py
U
px
many free Fermions (
Fermi surface)
+ two attractively interacting Fermions
1

[12  22 ]( x1 , x2 )  U (r1  r2 )( x1 , x2 )  ( E  2 F )( x1, x2 )
2m
The energy is measured from the Fermi level.
Pair formation 2: Cooper problem
py
U
px
many free Fermions (
Fermi surface)
+ two attractively interacting Fermions
(r1 , r2 ) 

k  kF
g (k )eik (r1 r2 )
Fermion states inside the Fermi
surface are occupied.
U (r1  r2 )  U (r1  r2 )
A bound state is obtained in the spin-singlet state.
Pair formation 2: Cooper problem
In the singlet case, the bound state equation is found to be
1U 
k  kF
1
2 k  2 F  E
As in the two-fermion problem, this equation involves the unphysical ultraviolet
divergence. To avoid this, we introduce the scattering length for the interaction
renormalized down to the Fermi level:
4 as ( F )

m
U
1U
c

k  kF
1
2
,
4 as
4 as ( F )
V
m


c
F
m
4 as  F 1
1
1
1V  V 
1

m k 0 2
k  0 2
k  0 2
The renormalized (regularized) equation for the bound state is then given by

4a s ( F ) 
1
1
1 
d


(

)




m
2


2


E
2

F


F
The important thing is that this equation always has a bound state solution (E<0),
Irrespective of the value of the scattering length as.
Pair formation 2: Cooper problem
weak-coupling case (|E|<< F)
strong-coupling case (|E|>> F)

E  8 F e
1
1
E
~ 2
2
mas ( F )
mas
k F as ( F )

1  2 kF as
 
e
kF
 ( R) 
1 e R / 
2 R
  as
molecular size
Fermi surface
+two Fermions
Two Fermions
0
(kF as )1
Fermi surface effect
(Cooper instability)
Bound state energy
Pair formation 2: Cooper problem
weak U
strong U
molecular size
Fermi surface
+two Fermions
Two Fermions
0
(kF as )1
Fermi surface effect
(Cooper instability)
many-body bound state
Bound state energy
two-body bound state
Essence of the BCS-BEC crosover phenomenon
T  Tc
T  Tc
T  d ~ 1/ n1/ 3
T ~ d ~ 1/ n1/ 3
boson
thermal de Broglie length
T  h / 2 mT
Phase diagram in the BCS-BEC crossover
Eg
T
Fermi gas
Molecular Bose gas
Tc
Superfluid phase
strong U
weak U
BCS
BEC
Tc
strong U
weak U
JILA 2004
Phase diagram in the BCS-BEC crossover
Eg
unitarity limit
T
Fermi gas
crossover
region
(kF as )1  1
1
0
+1
(kF as )1
Molecular Bose gas
Tc
Superfluid phase
(kF as )1  1
Phase diagram in the BCS-BEC crossover
NOTE: interatomic distance ~ kF-1 (kF: Fermi momentum)
unitarity limit
T
Fermi gas
Molecular Bose gas
Tc
Superfluid phase
crossover
region
(kF as )1  1
1
0
+1
(kF as )1
(kF as )1  1
two-body bound state
A molecule is formed
in a two fermion system.
molecular size:
 ~ as
Phase diagram in the BCS-BEC crossover
NOTE: interatomic distance ~ kF-1 (kF: Fermi momentum)
T
crossover
region
Fermi gas
Molecular Bose gas
(molecular size)
 ~ kF1e


Tc
2 kF as
Superfluid phase
 ~ as
 ~ kF1
(kF as )1  1
1
many-body bound state
A Fermi surface is necessary
to form a Cooper pair.
0
+1
(kF as )1
(kF as )1  1
two-body bound state
A molecule is formed
in a two fermion system.
Chapter 4.
tunable interaction associated with
a Feshbach resonance
pairing interaction mediated by boson
phonon
‘phonon’ mechanism
superconductivity
Phonon, AF spin fluctuations
superfluid 3He
Ferromagnetic spin
fluctuations
molecule
Feshbach mechanism
superfluid Fermi gas
40K, 6Li
Conventional phonon mechanism in superconductivity
 p 'q
 p q
D
effective interaction
mediated by phonon
Veff 
 p 'q
 p q
J

q
J phonon
phonon
 p'
p
p
 p'
J is an electron-phonon coupling constant. Evaluating these scattering processes, we
obtain
( p, p '  p  q, p ' q)  J 2
1
 J2
1
( p   p ' )  ( p q   p '  D )
( p   p ' )  ( p   p ' q  D )
1
1
 J2
 J2
.
( p   p q )  D
( p '   p 'q )  D
For electrons in the low energy region (near the Fermi surface), satisfying |  p   pq | D
|  
|  , we obtain the phonon-mediated attractive pairing interaction
p
p 'q
D
Veff  2
J2
D
0
Feshbach resonance
open channel
Zeeman energy
B
closed channel
Atomic hyperfine states in the closed channel are different from
those in the open channel.
open
F , Fz  9 / 2, 9 / 2  9 / 2, 7 / 2
close
F , Fz  9 / 2, 9 / 2  7 / 2, 7 / 2
40K
The Zeeman energy of the open-channel is different from that of the
closed channel. This is because of the different magnitude of the
electron Bohr magneton and the nuclear one.
Tunable interaction associated with a Feshbach resonance
Fermi atom
g
2n
g
1
Veff   g
2n
2
molecule
2n is referred to as the threshold
energy of the Feshbach resonance.
(JILA)
Near the resonance field, 2n=(B-B0). Including
a residual weak interaction part U, we obtain the
Feshbach mediated pairing interaction, given by
40K
|9/2,-9/2>
|9/2,-5/2>
Veff  U  g 2
(2n)
1
1
 U  g 2
2n
 ( B  B0 )
Tunable interaction associated with a Feshbach resonance
Fermi atom
g
2n
g
1
Veff   g
2n
2
molecule
This interaction is obtained within the second order perturbation theory, but
the value of Veff becomes very large when 2n~0.
question
Don’t we need to consider higher order corrections beyond the second
order perturbative calculation?
Tunable interaction associated with a Feshbach resonance
Fermi atom
g
2n
g
1
Veff   g
2n
2
molecule
This interaction is obtained within the second order perturbation theory, but
the value of Veff becomes very large when 2n~0.
question
Don’t we need to consider higher order corrections beyond the second
order perturbative calculation?
answer
Higher order corrections are formally included when parameters are replaced by
renormalized ones.
Renormalized Feshbach induced pairing interaction
4 as
 (0, 0) 
m
+
U
+
U
U
+
+ U
U U
+…….
+
+
g2
~
U

2
~
Veff
g
2n


 U  ~
c
2 c
1
g
1
2n
1  Veff 
1  (U  )
2n p 2
p 2
renormalized parameters
c
~
U
U
c
1
1U 
p 2
~
g
g
c
1
1U 
p 2
~
2n  2n  g 2
1
p 2
c
1U 
p
1
2
Chapter 5.
Many-body Hamiltonian in the second
quantization representation
second quantization
The wavefunction of a many-fermion system can be written as, by using the Slater
determinant,
1 ( x1 ) 1 ( x2 ) 1 ( x3 )
1
   (1) 1 ( x1 )2 ( x2 )3 ( x3 ) 
2 ( x1 ) 2 ( x2 ) 2 ( x3 )
3!
P
3 ( x1 ) 3 ( x2 ) 3 ( x3 )
P
This kind of wavefunctions can be written more simply when one uses the number
representation.
１
２
３
４
５
６
７
  n1 , n2 , n3 , n4 , n5 , n6 , n7 .......  0110101...
To include the Fermi statistics, we write this wavefunction in the second quantization
form:
  1110000  c1†c2†c3† 0
The creation operators c †j satisfy the (Fermi) anticommutation relation
[c , c ]  c c  c c  0
†
j
†
i 
† †
j i
† †
i j
c†j c†j 0  0 (exclusion principle)
second quantization
†
c †j is called the creation operator. From the hermite conjugate of 1  c1 0 ,
We obtain 1  (c1† 0 )†  0 c1. When 1 is normalized, they satisty
1  1 1  0 c1 c1† | 0
From this, we can easily understand why c is called the annihilation operator.
(The vacuum state |0> is (exactly speaking) defined as the state which vanishes
when any cj acts on it. c j 0  0 )
[ci , c†j ]   ij
[c†j , ci† ]  [c j , ci ]  0
In the bosonic case, the creation and annihilation operators satisfy the following
commutation relation, refecting the that the wavefunction is symmetric in terms
of particle exchange.
[bi , b†j ]   ij
[b†j , bi† ]  [b j , bi ]  0
second quantization
  c1†c2†c3† 0
When we use the second quantized wavefunction, we need to rewrite
the Hamiltonian in the form appropriate to this scheme.
one-particle part (kinetic term, potential)
F   fi
F   f nm cn†cm
i ,
i
2m
i ,
i
2m
V  Vi  V (ri )
mn
i
H 0  
H 0  
i
H 0    p c†p c p
i
p ,
two-particle part (interaction)
1
G
gij

2 i , j (i  j )
Vint 
1
U  (ri  rj )
2 i j
1
G   mn g ij cm† cn†c j ci
2 i , j ,m,n
Vint  U
c
p , p ', q
†
†
p  q  p '  q  p ' p 
c
c c
Vint 
U
2
 u(r  r )
i j
(volume=1)
i
j
second quantization
In the second quantization scheme, the “particle picture” revives.
H 0    p c†p c p
p ,
number operator

1 
n p   c†p c p   

 0
N   c†p c p
p
Vint  U
c
( p   )
( p   )
total number operator
†
†
p  q p '  q p ' p
c
c c
p , p ', q
U
p'
p
Microscopic models to describe the Fermi superfluids
BCS model (single-channel)
H   ( p  m )cp† cp  U
p
c
†
†
p q p ' q p ' p
c
c c
p ,p ',p
In this model, the pairing interaction is simply treated as the constant
parameter U, without specifying its origin.
Coupled fermion-boson model (two-channel)
.
H   ( p  m )cp† cp U
p

p ,p ',p
cp†qcp†'qcp 'cp   ( qB  2n  2m )bq†bq
q
 g  bq†cpq / 2cpq / 2  c†pq / 2c†pq / 2bq 
p ,q
Fermion
Feshbach resonance
Boson
2n
N  NF  2NB  H  m N
1
U eff n  Un  ( g n )
2n  2m
2
crossover region
narrow F.R.
g n  F
U eff n ~  F
broad F.R.
g n   F
Boson
Fermion
n ~ F
2n
Feshbach molecules appear as
real particles in the crossover
region.
n   F
Feshbach molecules only
appear in the virtual process
to produce Ueff.
(40K, 6Li)
BCS  CFB
BCS  CFB
Chapter 6.
Ground state of a superfluid Fermi gas
and BCS-BEC crossover
From the Cooper problem to the BCS state
Cooper problem
(r1 , r2 ) 
F 
g
k k F

ik ( r1 r2 )
†
†
e
F

g
c
c

k
k k  k  F
†
p
c
p  pF ,
k
~
0  ( g p1 c c
p1
†
p1 
†
 p1 
~
)( g p2 c
†
p2 
p2
†
p2 
c
~
)....(  g p N / 2 cp†N / 2  c†p N / 2  ) 0
pN / 2
~
g p   ( pF  p)
  ( g k c c
†
†
k  k 
k
~
)( g p1 c c
p1
†
†
p1   p1 
~
)( g p2 c
p2
†
†
p2  p2 
c
~
)....(  g p N / 2 cp†N / 2  c†p N / 2  ) 0
pN / 2
many Cooper pairs
  ( g p1 cp†1  c†p1  )( g p2 cp†2  c†p2  )....(  g p N / 2 cp†N / 2  c†p N / 2  ) 0
p1
p2
pN / 2
BCS ground state


BCS  C  1  g k ck†c†k   0

k 
| > is obtained from ‘N-particle terms’ in |BCS >. Setting g=v /u and C   uk vk ,
k
we obtain the famous expression,


BCS   uk  vk ck†c†k   0

k 
BCS BCS  1 
uk2  vk2  1
The BCS state involves terms with the different numbers of fermions, which looks
strange. However, it can be shown that the fluctuation around the mean-value <N>
is very small when N is large (=106~108 >>1 in cold atom physics).
N
N

N
2
 N2
N

1
N
BCS ground state


BCS   uk  vk ck†c†k   0

k 
uk2  vk2  1
u and v are determined by minimizing the ground state energy Eg  BCS H BCS BCS
The resulting “optimized u and v” are given by ( p   p  m )
p
1
u p  (1 
),
2
2
2
p  
2
p
1
v p  (1 
)
2
2
2
p  
2
Here,  is the (Fermi) superfluid order parameter, having the form
  U  BCS c c
† †
k k
p

BCS  U  u p v p  U 
p
p 2E p
Thus, the order parameter is determined by the BCS gap equation
1
1U
p 2E p
E p   p2   2
.
Regularization of the BCS gap equation
The BCS gap equation involves the ultraviolet divergence. In super conductivity,
the phonon-mediated pairing interacrtion has a physical cutoff associated with the
upper limit of the phonon frequency (=Debye frequency D ~ 1000[ K ] ) .
In cold Fermi gases, in contrast, there is no physical cutoff like the Debye cutoff,
so that we have to regularize the theory to eliminate the divergence.
c 
1
1
1
1U
U

2 p
p 2E p
p 
 2E p
c
4 as
1 
m
c 

1
 U  
p 


 2 p
 1
1



p 
 2 E p 2 p



This summation now converges.




BCS superfluid theory at T=0
Gap equation
number equation
(equation of state)
4 as
1 
m
 1
1



p 
 2 E p 2 p
N   BCS c c p
p
†
p



(cutoff-free)

p 
BCS  2 v   1 

E
p
p 
p 


2
p
 and m are determined self-consistently from these coupled equations.
NOTE
In the usual (weak-coupling) BCS theory, the chemical potential can be taken to be
equal to F. However, from general point of view, it should be determined by the
equation for the number of fermions. Indeed, the chemical potential is found to
remarkably deviates from the Fermi energy when the pairing interaction is strong.
BCS-BEC crossover theory at T=0 (Leggett theory)
4 as
1 
m
 1
1



p 
 2 E p 2 p
Weak coupling BCS limit:

8
Fe
2
e
m  F
,
Strong-coupling BEC limit:

2
p
(kF as )1  

2 k F as

p 
N  2 v   1 

E p 
p
p 




16
| m |1/ 4  F3/ 4 ,
3
The chemical potential is at
The Fermi level, as expected.
(kF as )1  
m
1
0
2
2mas
Binding energy of a Cooper pair molecule obtain from the Cooper problem:
(BCS)
Ebind 
1
mas2 (BEC)
BCS-BEC crossover theory at T=0 (Leggett theory)
4 as
1 
m
 1
1



p 
 2 E p 2 p
Weak coupling BCS limit:
8
  2 Fe
e
m  F
,
Strong-coupling BEC limit:

2
p
(kF as )1  

2 k F as

p 
N  2 v   1 

E p 
p
p 




16
| m |1/ 4  F3/ 4 ,
3
(kF as )1  
m
1
0
2
2mas
Binding energy of a Cooper pair molecule obtain from the Cooper problem:

Ebind  8 F e kF as ( F )
(BCS)
Ebind 
1
mas2 (BEC)
BCS-BEC crossover at T=0 (self-consistent solution)
BCS
 / F
BEC
The chemical potential
gives the size of the
Fermi surface.
The chemical potential
gives the value of the
binding energyof a
molecule.
m / F
BEC
BCS
(kF as )1
BEC
Fermi superfluid = molecular BEC ?
(1) superfluid order parameter
  U  BCS ck†c† k  BCS
p
(2) ground state energy
BEC regime: EG  BCS H BCS   | m | N   1 2 N   Ebind N B
mas 2
All atoms form Cooper pairs with Ebind.


BCS regime: EG  EN   1  ( F ) 2   3  N   
2
8
F 
Only a small fraction ( N  ( /  F ) ) of atoms form Cooper
pairs with Ebind=.
The molecular picture is not good in the BCS regime.
Fermi superfluid = molecular BEC ?


BCS   uk  v k ck†c† k   0

k 
v
u
~
uk ~ 0, vk ~ 1   p   F  

F
n p  v 2p
Energy levels deep inside the Fermi level are fully occupied (as
in the case of a free Fermi gas F   c†p 0 ). These occupying
p ,
atoms do not contribute to the condensation energy.
Cooper pairs near the Fermi surface (|  p |  )only contribute to the
condensation energy.
N pair ~ N 

F
3

EG  EN    N
8  F



Fermi superfluid = molecular BEC ?
(3) molecular picture in the BCS regime

p
n p  BCS c c p
p
1 
BCS  v  1 
2
2
2
(



)


p
F

2
p





1
np
F
0
size of a Cooper pair 
 p ~ 1/ 
 F ~ ( pF   p)2 / 2m
uncertainty
principle

kF
1  2 kF as
~

e

m k F
Cooper problem
Fermi superfluid = molecular BEC ?
(4) condensate fraction
= the number of Bose-condensed Cooper pair molecules
g k c p  c p 

BCS  e
0
 e
n0 b0
(Boson BEC ground state)
0
q2
U
H  (
 m )bq bq 
2m
2
q
Bose gas BEC


b
b
 p  q p ' q b p 'b p
p , p ', q
b0  b0
(BEC order parameter)
n0  b0
(condensate fraction in BEC)
2
b0   g c c
?
† †
k k k
k
vk † †
  ck c k 
k uk
Fermi superfluid = molecular BEC ?
[b0 , b0† ]   g k2 (1  ck†ck   c† k c k  ).
k
Within the expectation value in terms of |BCS >, this commutation
relation is evaluated as
[b0 , b ]   g (1  2v )
†
0
2
k
B0 
2
k
k
1
g
2
k
(1  2vk2 )
b0 ,
[ B0 , B0† ]  1.
k
Boson!
When RHS is positive,
This condition is always satisfied when m<0 (v2<0.5). This is
realized in the strong-coupling BEC regime, where the
molecular character would be OK. In this case, we have
BCS  e
gk c p c p
0 e
n0   g k2 (1  2vk2 )   (1 
k
k
n0 B0†
0
k
k2   2
BEC limit
)2
k
E

N
2
Fermi superfluid = molecular BEC ?
[b0 , b0† ]   gk2 (1  2vk2 )
k
B0 
1
i |  g k2 (1  2vk2 ) |
b0 , [ B0 , B0 ]  1.
†
k
In the BCS regime (m>0), RHS can be negative and one cannot
introduce the approximate boson operator.
In this regime, Bose-condensed pairs consist of particle pairs
and hole pairs due to the presence of the Fermi surface.
Fermi surface
Fermi superfluid = molecular BEC ?
BCS   uk  vk ck†c†k   0   uk c k ck   vk  uk  vk ck†c†k   F ( m )
 m
k
F ( m )   ck† c† k  0
b
 vkk c kck   ukk ck†c† k
e
 m
(‘Fermi vacuum’)
 m
†
0
u
 m
v
 m
F (m )  e
uk2
vk2
2
[b0 , b ]   2 (1  2vk )   2 (1  2vk2 )  0
  m vk
  m uk
†
0
†
B0 
n0 B0†
F (m )
The first term is set to
be equal to 0 when m<0.
1
uk2
vk2
2
 2 (1  2vk )   2 (1  2vk2 )
  m vk
  m uk
uk2
vk2
N  
2
2
n0   2 (1  2vk )   2 (1  2vk )  

2  F 
  m vk
  m uk
b0† ,
[ B0 , B0† ]  1.
BCS limit
Fermi superfluid = molecular BEC ?
condensate fraction
ODLRO
n0 / N
BCS
(kF as )1
BEC
condensed molecular bosons
Particle pairs
Hole pairs
(Fermi surface effect)
Particle pairs
Condensate fraction (ODLRO)
(1) Boson BEC
1  † (r1 ) (r2 )  † (r1 )  (r2 )  f (r1 , r2 )
This vanishes when | r1  r2 | 
In the BEC phase, the condensate fraction is given by the
maximum O(N)-eigen-value of this density matrix:
1  n0 * (r1 ) (r2 )  O(1)
normalized eigen funcion of 1
From these two expressions, it is reasonable to equate the red terms.
n0 
 dr |
2
(r ) |
Condensate fraction (ODLRO)
(2) Fermi superfluid
2  † (r1 )† (r2 ) (r3 ) (r4 )  † (r1 )† (r2 )  (r3 ) (r4 )  f
 0 when | (r1 , r2 )  (r3 , r4 ) | .
2  n0 (r1 , r2 ) (r3 , r4 )  O(1)
*
n0   dr1dr2 | † (r1 ) † (r2 ) |2
Uniform Fermi superfluid at T=0 (|BCS >)
 (r )   e ik r ck
k
BCS  (r) (r ') BCS   eik(r r ') BCS ckck  BCS   uk vk eik(r r ')
k
2
n0   u v   2
k
k 4 Ek
2 2
k k
k

uk2
vk2
2
2 
 n0   2 (1  2vk )   2 (1  2vk ) 
  m vk
  m uk


Chapter 7.
Excitations in a Fermi superfluid at T=0
Excitations in Fermi and Bose superfluids
collective modes single-particle
excitations
Bose atom
BEC
Fermi
superfluid
Cooper-pair is a molecular
Boson with a finite binding
Energy.
Mean field theory of Fermi superfluids (Bogoliubov)
In the previous discussion, we have shown that the BCS state
(describing the ground state of a Fermi superfluid) is characterized
by the finite value of the BCS order parameter
 U
ck† c† k 
p
In constructing the mean field theory of the Fermi superfluid, we
expect that the interaction term may be approximated by replacing
the operator cc by its expectation value <cc>.
Dividing the pair-operator ck† c† k  into the mean value and
fluctuations around it as
ck†c† k   ck†c† k    Ak†
we can write the interaction part of the BCS Hamiltonian as
U  c†p 'c† p 'c pc p  U  ( c†p 'c† p '   A† )( c pc p   A)
p, p '
p, p '
 U  2  U  ( c†p 'c† p '  Ap   Ap ' c pc p   Ap '  Ap )
†
p, p '

||
  (*c†pc† p  c pc p )    A† A
U
p
2
†
Mean field theory of Fermi superfluids (Bogoliubov)
H   ( p  m )c c p   ( c c
†
p
p
* †
†
p  p
p
 c pc p )  U 
|  |2
 A  A
U
†
In the mean field approximation, we ignore this
term, which describes pairing fluctuations.
Actually, we may take D to be real, because even if  |  | ei , the phase
factor can be eliminated by the U(1) gauge transformation,
c p  c p ei / 2
H   ( p  m )c c p    (c c
†
p
p
p

  c , c p
p
†
†
p  p
†
p

 p  m

 
|  |2
 c pc p ) 
U
   c p   2

 ( p  m )

( p  m )   c† p  U 
p


Mean field theory of Fermi superfluids (Bogoliubov)
This mean-field Hamiltonian can be diagonalized by the so-called
Bogoliubov transformation,
 c p   u p
 † 
 c   v p
  p 
v p    p 


u p    † p 


Here,  obeys the Fermi statistics, and u and v are the same as those
appeared in |BCS>.
H   E p †p  p  W0
p
2
W0   [( p  m )  E p ] 
U
Ep 
( p  m )2   2
0
Ep is the excitation spectrum of quasi-particle described by  (Bogolon).
E p :  †p 0
Bogolon
 †p 0
What is the vacuum state for the Bogolons?
Answer: 0  BCS
The vacuum for the Bogolons is not the vacuum for Fermi atoms!
Ep 
( p  m )2   2
What is this excitation?
Answer: single particle excitation associated with the breakup of a
Cooper pair.
Excitation gap in the Bogolon spectrum
Eg  
(m  0)
Eg  m 2   2 | m | ( m  0)
BCS regime
BEC regime
Binding energy of a
Cooper pair
Single particle excitations at T=0
[ F ]
Eg

|m|
BCS
Energy gap Eg
BEC


 2
  m  2
 |m|

BCS: m>0
BEC: m<0
BEC limit (|m|>>)
Note: The binding energy of a Cooper-pair is given by 2Eg.
Single particle excitations at T=0
Single-particle excitations can be seen in the superfluid density
of states.

N ( )    (  E p )   d  ( ) (  E )
0
coherence
peak at 
N ( )   (m )

 2  2
(
 (  )
)
m  0 : BCS
m  0 : BEC
The coherecne peak is absent in the excitation spectrum in the
BEC regime.
1. Single particle excitations at T=0 (BEC regime)

Fermion band
0
m
Excitation spectrum in the BEC regime
is simply given by the density of states
of a free Fermi gas.
N ()   (  | m |)
Observation of single particle excitations in a superfluid Fermi
gas (BCS-BEC crossover region)
Single-particle excitations can be observed by using the
rf-tunneling current spectroscopy.
superfluid 6Li

Fermion band
, 
0
light
Eg
C. Chin , et al. Science 305 (2004) 1128.
2. collective excitations at T=0 (1. Boson BEC)
Bogoliubov mode in a boson BEC
q2
H  (
 m )bq bq  U  b p q b p' q b p ' b p
2m
q
p , p ', q
b0  b0 
n0 ,
+ Bogoliubov approximation
Un0
q2

H  Eg   (
 m  2Un0 )bq bq 
2m
2
q
 
[
b
 p b p b pbp ]
p
m is chosen so that the ground state energy E g
can be minimum, which gives m  Un0 .
Un
q2
H  Eg   (
 Un0 )bqbq  0
2m
2
q

 
[
b
 p b p b pbp ]
p
Bogoliubov transformation
U 2
n0  mn0
2
2. collective excitations at T=0 (1. Boson BEC)
Bogoliubov mode in a boson BEC
H 
E
q
 q  q
q
boson
Eq   q ( q  2Un0 )

Un0
q  v q
m
(q  0)
gapless Bogoliubov phonon
This Goldstone mode is associated with the spontaneous
breakdown of the continuous gauge symmetry in the
BEC phase.
b0  b0  n0 ei
The boson BEC is dominated by collective excitations only.
2. collective excitations at T=0 (1. Boson BEC)
E
b0
Realized ground state
| b0 | ei1
Another possible ground state
| b0 | ei2
Gapless Goldstone mode
2. collective excitations at T=0 (2. Superfluid Fermi gas)
BCS   (uk  vk ckck  ) 0
i
  e
i
(b0  b0  n0 e )
BCS '   (uk  vk ckck ei ) 0 .
different state with
the same energy
(degenerate!)
When the order parameter oscillates with q, the additional
†
†
†
pair amplitude, q  c pq / 2c pq / 2 and q  c pq / 2c pq / 2
are induced. The resulting mean-field Hamiltonian is
H  H BCS  U  ( q† c p  q / 2c p q / 2  q c†p q / 2c† p q / 2 )
p ,q
This perturbation again generates the oscillation of the
order parameter. In the linear response theory, we obtain
2. collective excitations at T=0 (2. Superfluid Fermi gas)

q (t )  U  dt ' q (t ); (t ')
†
q
0

q† (t )  U  dt ' q† (t );q† (t ')
0
Here,
.....

q (t ')  U  dt ' q (t ); q (t ')
q† (t ')
0

q (t ')  U  dt ' q† (t ); q (t ')
q† (t ')
0
are the double-time Green’s functions, e.g.,
q (t );q† (t ')
 i (t  t ') BCS | q (t '), q† (t ')  | BCS
In the frequency space W,
q (W)  U q ;q†
q† (W)  U q† ;q†
W
W
q (W)  U q ;q
q (W)  U q† ;q
W
W
q† (W)
q† (W)
2. collective excitations at T=0 (2. Superfluid Fermi gas)
Mode equation:
0
1  U q ; q†
U q† ; q†
W
W
U q ;q
W
1  U q† ; q
W
2
U
1
U 
1   22 (q, W)     21 (q, W)
12 (q, W)
U
2
2
 
1  11 (q, W)
2
      2 
E  E
 22 (q, W)   1 

E E  ( E  E )2  W2
p 
phason
      2 
E  E
11 (q, W)   1 

E E  ( E  E )2  W2
p 
ampliton

 
iW
12 (q, W)   21 (q, W)       
E  ( E  E )2  W2
p  E
coupling
mode
2. collective excitations at T=0 (2. Superfluid Fermi gas)
In the long wave length limit, taking W = v q, we obtain
the sound velocity in the entire BCS-BEC crossover at T=0,
v 
1
m
BCS regime: v 
p
2
 E 5  p   2E 3
p
p
1  p
 E 3    E 3
p
p

1
vF
3
U B n0
BEC regime: v 
M
UB 
2

2
 /  3
Ep

Anderson-Bogoliubov mode
Bogoliubov phonon
4 aB 4 (2as )

0
M
M
(M  2m)
Effective repulsive interaction, given by aB=2as, looks working
between Cooper-pair molecules in the BEC regime.
2. collective excitations at T=0 (2. Superfluid Fermi gas)
v 
sound velocity
1
vF
3
v 
U B n0
M
v / vF
BCS
(kF as )1
BEC
NOTE:
1. In a charged Fermi superfluid, this sound mode remains only just
below Tc (Carlson-Goldman mode). At T=0, the plasma only exists.
2. In a more sophisticated theory, it has been pointed out that the effective
interaction between molecules is given by aB=0.6as.
Chapter 8.
BCS-BEC Crossover theory based on the
Coupled Fermion-Boson Model (T=0)
BCS-BEC crossover tuned by a Feshbach resonance
H   ( p  m )c†p c p  U
p

c†p  qc†p 'qc p 'c p
p , p ', q
†
†
†
 ( qB  2n  2m )bq†bq  g  bq c p  q / 2c p q / 2  c p q / 2c  p  q / 2bq 
p ,q
q
The fermion field and boson field lead to the two superfluid order
parameters:
  U  BCS ck†c† k  BCS
BCS order parameter
m  b0  b0†
BEC molecular condensate
p
However, these are NOT independent, but related to each other
due to the resonance between atoms and molecules.
1
g
m  

2n  2m U

i

0

b

0

t

b0 , H 
 (2n  2m )m 
Thus, both order parameters are finite in the superfluid phase.
g 

U 
BCS-BEC crossover tuned by a Feshbach resonance
The mean field CFB Hamiltonian has the form
~
H   ( p  m )c c p    (c†pc† p  c pc p )   ( qB  2n  2m )bq†bq
†
p
p
q 0
p
BCS-type with composite order parameter
Free Bose gas
~
    gm
Ep 
~ 2
( p  m )  
2
: Single-particle excitations
~
  U  BCS ck†c† k  BCS  U 
p
p
g2
1  (U 
)
2n  2m p

~2
2 ( p  m )  
2
~
1
~2
2 ( p  m )2  
: gap equation for

BCS-BEC crossover tuned by a Feshbach resonance
g2
1  (U 
)
2n  2m p
1
~2
2 ( p  m )  
2
The equation indicates that the effective pairing interaction
associated with the Feshbach resonance is given by
g2
U eff  U 
2n  2m
Note that this expression is different from the two-particle result:
U
2b
eff
g2
U 
2n
At T=0, thermally excited molecules are absent, so that the
equation for the total number of Fermi atoms is given by
N  2 | m |   [1 
2
p
p m
~2
( p  m )2  
]
Broad Feshbach resonance: g n   F
2n  2m ~ 2 F
crossover region
Boson
U eff
g2
g2
U 
~U 
2n  2m
2n
Fermion
2n
m ~ 0
1 
4 as
m
1
[
p
~2

2 ( p  m )  
2
N  2 | m |   [1 
2
p
p m
~2
1
]
2

g2

U
4

a
2n
s


 m
g 2 c 1
1  (U  )

2n p 2

]
( p  m )2  
Single-channel
BCS
~
model with 






Narrow Feshbach resonance: g n   F
crossover region
2n ~ 2m
Boson
Fermion
In this case, one cannot eliminate the
cutoff by using the two-body scattering
length, because Ueff is different from Ueff2b.
2n
The cutoff can be formally eliminated by introducing the
‘generalized scattering length,’ defined by
g2
U
4 as
2n  2m

c
m
g2
1
1  (U 
)
2n  2m p 2
1 
4 as
m
1
[
p
~2

2 ( p  m )  
2
N  2 | m |   [1 
2
p
p m
~2
( p  m )2  
1
]
2
]
Chemical potential in the crossover region
Boson
m n
Fermion
m / F
2n
U eff
n / F
g2
U 
2n  2m
The pairing interaction Ueff is always attractive and become strong
as one decreases the threshold energy 2n.
In the strong-coupling regime, 2m=2n is obtained. This means that
the molecular chemical potential 2m is at the lowest boson energy
(= BEC condition).
Composite order parameter in the crossover region
~
/  F
 / F
~
/  F
 / F
m /  F
m /  F
n / F
n / F
narrow resonance
broad resonance
Un  0.02 F , g n  0.17 F
Un  0.02 F , g n  5 F
~
/  F
 / F
narrow
m /  F
~
/  F
 / F
broad
m /  F
 / F
single-channel
BCS
(kF as )1
BEC
[N ]
NF
n0  m2
narrow
m2
n0
NF
n0  m2
n0
broad
m2
n0
single-channel
BCS
(kF as )1
BEC
broad
m / F
narrow
(kF as )1
As far as we consider quantities independent of the character of
molecules (Cooper pairs or Feshbach molecules), the narrow FR
and broad FR almost give the same results when scaled by the
(generalized) scattering length.
Chapter 9.
BCS-BEC crossover at finite temperatures
1. Superfluid phase transition
Breakdown of the mean-field theory at T>0
1. Single-channal BCS model
The previous mean-field theory can be immediately extended to
the case at T>0.
A  BCS A BCS 
 †p p  f ( E p ),
1
tr e  H A
Z
 p †p  1  f ( E p ),
 †p †p   p p  0

E   2  2
 Ep
1
1U
tanh
2
p 2E p
N   c c p
p
†
p
 p
 Ep 
  1 
tanh

E
2
p 


p
(  1/ T ,
kB  1)

Breakdown of the mean-field theory at T>0
1. Single-channal BCS model
The previous mean-field theory can be immediately extended to
the case at T>0.
A  BCS A BCS 
 †p p  f ( E p ),
1
tr e  H A
Z
 p †p  1  f ( E p ),
 †p †p   p p  0

E   2  2

  U  c†p c† p  U  (u p †p  v p  p )(v p p  u p † p )
p
p
 U   u p v p  †p p  u 2p  †p † p  v 2p   p p  u p v p   p † p 


p
 U  u p v p 1  2 f ( E p ) 
p
Breakdown of the mean-field theory at T>0
1. Single-channal BCS model
At Tc, these equations reduce to
4 as
1 
m

(Tc    0)
 1
 (  m )
1
 
 2(  m ) tanh
2
2 

N  2 f ( p  m )
Free Fermi gas!
p
m ~  F (T  TF )
Breakdown of the mean-field theory at T>0
1. Single-channel BCS model
In the weak-coupling BCS limit, Tc<< TF, and the equation of states
gives m=F, as expected. In this case, the gap equation gives

Tc 
8
2 kF as
2 kF as

e

0.61

e
F
F
 e2

Note:

8
(T  0)  2  F e 2 kF as
e
 ~ 1.78
2 / Tc  2 /   3.54
(BCS universal constant)
The fact that /Tc=O(1) means that, as one increases temperatures,
the superfluid phase transition occurs when Cooper pairs are
completely destroyed thermally.
Breakdown of the mean-field theory at T>0
1. Single-channel BCS model
In the strong-coupling limit, the gap equation gives
1
m
2mas2
This equals the binding energy of a Cooper-pair obtained at T=0. This
means that molecules does not dissociate even at Tc. Thus, one expects
that Tc is essentially the same as that for an ideal molecular Bose gas
(NB=N/2, M=2m).
2
(n / 2) 2 / 3
Tc 
 0.218 F
2/3
( (3 / 2))
(2m)
N  2 f ( p  m )
p
The free Fermi gas expression cannot describe BEC transition.
Breakdown of the mean-field theory at T>0
2. Coupled Fermion-Boson model
 p
g2
1
1  (U 
)
tanh
2n  2m p 2 p
2
N  2 nB ( qB  2n  2m ) 2 f ( p  m )  2 N B 0  N F 0
q
 qB  q 2 / 2M
p
Weak-coupling regime (2n>>2m):
N B 0  0  m ~  F  BCS
Strong-coupling regime (2n~ 2m0): Gap eq.: m  0  N F 0  0
N
  nB ( bB )
2
q
ideal molecular
Bose gas!!
2
(n / 2) 2 / 3
Tc 
 0.218 F
( (3 / 2)) 2 / 3 (2m)
OK!
Crossover region:
Gap eq.:
2n  2m
A finite energy gap exists in the molecular excitations even at Tc.
Why does the mean field theory break down at T>0?
The mean field theory ignores fluctuation effects. This approximation
is valid at T=0, or at finite temperature in the weak interaction regime
(weak-coupling BCS regime). However, it does not work when the
interaction is strong (=BEC regime) at T>0, where thermally excited
pairing fluctuations play crucial roles.
We need to improve the BCS theory at finite temperaures going
past the mean-field approximation.
strong-coupling theory = mean field theory + fluctuation effects
Gaussian fluctuation theory
(Nozières and Schmitt-Rink: NSR)
How to improve the theory (BCS model)
In the strong-coupling limit, the gap equation gives
1
m
2mas2
This equals the binding energy of a Cooper-pair obtained at T=0. This
means that molecules does not dissociate even at Tc. Thus, one expects
that Tc is essentially the same as that for an ideal molecular Bose gas
(NB=N/2, M=2m).
2
(n / 2) 2 / 3
Tc 
 0.218 F
2/3
( (3 / 2))
(2m)
N  2 f ( p  m )
p
The free Fermi gas expression cannot describe BEC transition.
How to improve the theory (BCS model)
In the strong-coupling limit, the gap equation gives
1
m
2mas2
This equals the binding energy of a Cooper-pair obtained at T=0. This
means that molecules does not dissociate even at Tc. Thus, one expects
that Tc is essentially the same as that for an ideal molecular Bose gas
(NB=N/2, M=2m).
2
(n / 2) 2 / 3
Tc 
 0.218 F
2/3
( (3 / 2))
(2m)
N  2 f ( p  m )
p
The free Fermi gas expression cannot describe BEC transition.
Strong-coupling theory at Tc (Nozières and Schmitt-Rink: NSR)
Importance of thermal pairing fluctuations to consistently
describe the BCS-BEC crossover at Tc.
We calculate the thermodynamic potential W including pairing fluctuations.
Then we derive the number equation using the identity
W
N 
m
BCS model
H   ( p  m )c†p c p U
p

c†p  qc†p 'q c p 'c p  H 0  H1
p , p ', q
We treat the interaction perturbatively
Perturbation theory in the quantum statistical mechanics
   d H1 ( ) 

R(  )  T e 0





H  H 0  H1
H1 ( )  e H0 H1e H0
e   H  e   H 0 R(  )
†

c
(

)
c
( 2 )  1   2
†
1
T c( 1 )c ( 2 )    †
c ( 2 )c( 1 )  1   2
time-ordered product
(+: boson, -: fermion)
Z  tre  H
W  T log Z

 T log tr e
Z0  tre  H0
  H0

 tr e  H0 R(  )
R(  )   W0  T log 
Z0


   W

0
 T log R(  )
0
Perturbation theory in the quantum statistical mechanics
non-perturbative part
     p c†p c p
W0  T log Z 0  T log tr  e p
 
NF 0  

 
   2T  log(1  e p )
p
 
W0
 2 f ( p )
m
p
( p   p  m )
(free Fermi gas)
perturbative part
W  T log R( ) 0  T  R( ) c  1
Linked cluster theorem: We may only take into
account connected diagrams in this calculation.
N 
W
 2
m
p
n






1 
f ( p  m )  T
R(  ) c  N F 0  T 
T    d H1 ( )  
m
 0
n 1 n ! m
 

c
Perturbation theory in the quantum statistical mechanics
N 
W
 2
m
p
n






1 
f ( p  m )  T
R(  ) c  N F 0  T 
T    d H1 ( )  
m
 0
n 1 n ! m
 

c
causality (     i )
(1) n=1
W1  U

  d
p , p ', q


T c†p  q / 2 (  )c† p  q / 2 (  )c p ' q / 2 ( )c p '  q / 2 ( )
0
c
Wick’s theorem: One can decompose <cccc> into the sum of all
possible combinations in terms of <cc><cc>



T c†p  q / 2 (  )c† p  q / 2 (  )c p ' q / 2 ( )c p ' q / 2 ( )
p , p ', q








T c†p  q / 2 (  )c† p  q / 2 (  )

T c†p  q / 2 (  )c p ' q / 2 ( )

T c p ' q / 2 ( )c†p  q / 2 (  )
0
p , p ', q

0
p , p ', q

p , p ', q
0




T c p ' q / 2 ( )c p ' q / 2 ( )
0
T c† p  q / 2 (  )c p ' q / 2 ( )


T c p  q / 2 ( )c †p ' q / 2 (  )
0
.
0
Perturbation theory in the quantum statistical mechanics
N 
W
 2
m
p
(1) n=1
W1  U
n






1 
f ( p  m )  T
R(  ) c  N F 0  T 
T    d H1 ( )  
m
 0
n 1 n ! m
 


  d
p , p ', q
U
p ,q


T c†p  q / 2 (  )c† p  q / 2 (  )c p ' q / 2 ( )c p ' q / 2 ( )
0

 d G
p  q / 2
(    )G p  q / 2 (    )
c
antisymmetric condition
0
Gp (   ')   T c p ( )c†p ( ')
c
0
Single-particle thermal Green’s function
This is one of the most important functions in many-body quantum field theory.
Perturbation theory in the quantum statistical mechanics
Gp (   ')   T c p ( )c ( ')
†
p
0
Fermi dolphin
c p ( )
c ( ')
†
p
Fermi sea (ground state)
Perturbation theory in the quantum statistical mechanics
Gp ( )   T c p ( )c†p (0)
antisymmetric condition
G p ( ) 
1
e

 
 in
0
 e
 p
G (   )   c(   )c † (0)
G p (in )
G p (in )   d e
0
in
0

1
tr e   H 0 e(   ) H 0 ce  (   ) H 0 c † 
Z0

1
tr e H 0 ce  H 0 e   H 0 c † 
Z0

1
tr e   H 0 c†e H 0 ce  H 0 
Z0

1
tr e   H 0 c( )c †   G ( )
Z0
n   T (2n  1)
n

  [0,  ]
(1  f ( p ))
n  0, 1, 2,......
Fermion Matsubara frequency
1
G p ( ) 
in   p
Perturbation theory in the quantum statistical mechanics
N 
W
 2
m
p
n






1 
f ( p  m )  T
R(  ) c  N F 0  T 
T    d H1 ( )  
m
 0
n 1 n ! m
 

c
(1) n=1
W1  U 
p ,q

 d G
p  q / 2
(    )G p  q / 2 (    )
0
 U  ein n
q ,n n
1
G
 
p,
p  q / 2
(in  in n )G p  q / 2 (in )  U  ein n  (q, in n )
n
q ,n n
n n  2n T n  0, 1, 2,......
(q, in n ) 
1

 G
p,
p  q / 2
(in  in n )G p q / 2 (in )
n
Physically, this polarization function describes pairing fluctuations. The
sum of the fermion Matsubara frequencies can be carry out using the
knowledge of the complex integration.
How to sum up Matsubara frequencies
Im[Z]
in
×
×
Re[Z]
×
×
Pole of A(z)
×
C1
C
×
×
1

 A(in )  
n
 (q, in n )  
1
dzf ( z ) A( z )

2 i C
1

2 i p
 dzf ( z)
C
f ( z) 
1
e z  1
1
1
z  in n   p  q / 2  z    p  q / 2
Taking the poles at z   pq / 2  in n , and z   pq / 2 , we obtain
Perturbation theory in the quantum statistical mechanics
N 
W
 2
m
p
n






1 
f ( p  m )  T
R(  ) c  N F 0  T 
T    d H1 ( )  
m
 0
n 1 n ! m
 

(1) n=1
W1  UT  ein (q, in n )
n n  2n T
n
q ,n n
(q, in n )  
p
n  0, 1, 2,......
1  f ( p  q / 2 )  f ( p q / 2 )
in n   p  q / 2   p q / 2
(2) n=2
(a) connected type
U 2G(1   2 )G( 2  1 )G(1   2 )G( 2  1 )
(b) disconnected type U 2G(1  1 )G(1  1 )G( 2   2 )G( 2   2 )
Gp (1   2 )   T c p (1 )c†p ( 2 )
0
1
2
c
Linked cluster theorem
(a) connected type
(b) disconnected type
G(1   2 )G( 2  1 )G(1   2 )G( 2  1 )
G(1   2 )
1
U
G(1  1 )G(1  1 )G( 2   2 )G( 2   2 )
G( 2  1 )
1
2
U
2
U
U
1
2
We may only consider the connected type (linked cluster theorem).
U2
2
W2 
T   (q, in n )
2 q ,n n
Perturbation theory in the quantum statistical mechanics
N 
W
 2
m
p
n






1 
f ( p  m )  T
R(  ) c  N F 0  T 
T    d H1 ( )  
m
 0
n 1 n ! m
 

(3) n=3,4,5…..
Among various diagram, we only take into account the following types,
describing pairing fluctuations.
W 
＋
＋
＋……..
1
 T  ein n  (q, in n )2  T  ein n log 1  U (q, in n )
q ,n n
n 1 n
q ,n n
N 
W

 NF 0  T
ein n log 1  U (q, in n )

m
m q ,n n
c
Gaussian fluctuation theory for Fermi superfluid at Tc (NSR)
4 as
1 
m

p

 ( p  m )
1
1 
tanh



2
2 p 
 2( p  m )
 4 a

in n
s
N  2 f ( p  m )  T
e
log
1




m
m
p
q ,n n


1 
(q, in n )  

p 2 p 

 
Tc and m are determined self-consistently for a given value
of the pairing interaction.
Gaussian fluctuation theory in the BCS regime
(BCS limit: (kF as )1  1 )
4 as
1 
m

p

 ( p  m )
1
1 
tanh



2
2 p 
 2( p  m )
 4 a

in n
s
N  2 f ( p  m )  T
e
log
1




m
m
p
q ,n n


1 
(q, in n )  

p 2 p 

 
m   F (Tc  TF )
The resulting equation is the ordinary BCS gap equation at Tc in the
weak-coupling superconductivity.
Gaussian fluctuation theory in the BEC regime
(BEC limit: (kF as )1  1 )
4 as
1 
m

p

 ( p  m )
1
1 
tanh



2
2 p 
 2( p  m )
 4 a

in n
s
N  2 f ( p  m )  T
e
log
1




m
m
p
q ,n n

1
m
2mas2

1 
(q, in n )  

p 2 p 

 
Expanding around =q=0, we obtain
condition for BEC
of N/2 Bose gas.
N 1

2 
mB 
1
q2
in n
e   nB (
 mB )

2
q
2(2m)
q ,n n
b
in n 
 mB
2(2m)
|m|
2
[1

2
ma
| m |]  0
s
2
2mas
BCS-BEC crossover behavior of Tc (Gaussian)
40
K :| 9 / 2, 7 / 2   | 9 / 2, 9 / 2 
0.218TF
BEC
BCS
C. A. Regal, et al. PRL 92 (2004) 040403.
m / F
1/ 2mas2
Extension to the coupled Fermion-Boson model
two hyper-fine states:


H   p cp c p  Eqbqbq  g bq†c pq / 2c pq / 2  h.c.  U c pc pc p 'c p '
q2
Eq 
 2n
2M
Feshbach resonance
Fermion Boson
2n
NFermi+2NBoson= N
H-mN
Extension to the coupled Fermion-Boson model
m:fluctuation contribution to thermodynamic potential
W  W MF 
D0
U
(q)
+
Nozieres and
Schmitt-Rink
Feshbach
resonance
W
N 
 N F0  N B0  T  log 1  U eff (q)(q) 
m
q
2
g
U eff (0)  U 
2n  2m
U eff (q)  U  g 2 D0 (q)
Extension to the coupled Fermion-Boson model
 p
g2
1
1  (U 
)
tanh
2n  2m p 2 p
2
 



g2
in n
N  2NB0  NF 0  T
e log 1  U 
 (q, in n ) 

B
m q ,n n
in n  ( q  2n  2m ) 
 

In the weak-coupling BCS regime, the number equation simply gives m=F, so
that the ordinary BCS theory is obtained.
In the strong-coupling BEC regime, the number equation reduces to the
equation for Tc of a N/2 ideal Bose gas:
N
q2
  nB (
)
2
2(2m)
b
BCS-BEC crossover at Tc
Theory
Experiment
(coupled fermion-boson model
for the Feshbach resonance)
40
K
Tc
0.218TF
BCS
BEC
BCS
BEC
Regal, et al. PRL 92 (2004) 040403.
Chapter 10.
BCS-BEC crossover at finite temperatures
2. Superfluid phase below Tc
Extended NSR in the superfluid phase below Tc
H   ( p  m )c†p c p   c†p  qc†p 'qc p 'c p
p
p , p ', q
  ( p  m )c†p c p  (c†pc† p  c pc p )
p
MF (BCS) part
p

 j (q)    †p q j  p
p
U
4
   (q)  (q)   (q)  (q) Fluctuation part
1
1
2
2
q
generalized density operator
1
2
 c p 
 p   †  : Nambu field
c 
  p 
: amplitude fluctuations
: phase fluctuations
j
: Pauli matrix
Extended NSR in the superfluid phase below Tc
gap equation (below Tc)
Ep
1
1U
tanh
2T
p 2E p
chemical potential
U
W 
 ij
i,j=1,2


 

 ij (q, in n )  T  tr  i G(p  q / 2, in  in n ) j G(p  q / 2, in )
p ,n
G(p, in ) 
1
in   p 3   1
T

 1 ˆ

N  NF  
tr log 1  U 
(q, in n ) 
2 q ,n n m
 2

 11 12 
ˆ
 (q)  



 21
22 

BCS-BEC crossover in the superfluid phase below Tc

m
F
F
T
TF
 (T )
 ( 0)
T / TF
(k F aS ) 1
(k F aS ) 1  2
(k F aS ) 1  2
BCS
T / Tc
(k F as ) 1
Phase diagram in the BCS-BEC crossover
Eg
T
Fermi gas
Molecular Bose gas
Tc
Superfluid phase
strong U
weak U
BCS
BEC
Tc
strong U
weak U
JILA 2004
Advanced Condensed Matter Physics
Supplementary materials
What do we learn from Tc/TF ~ 0.2 ?
Tc / TF
0.2
strong
attractive
interaction
BCS-BEC crossover
0
weak
attractive
interaction
weak
attractive
interaction
strong
attractive
interaction
What do we learn from Tc/TF ~ 0.2 ?
The Fermi superfluid phase
transiton temperature has this
upper limit (which is given by
the GOD)!
Fermi gas:
TF ~ 1μK
metal:
TF ~ 104 K
Tc / TF
0.2
BCS-BEC crossover
0
Tc ~ 2000K<<300K
weak
attractive
interaction
Room temperature superconductivity (Tc~300K)
is not prohibitted by nature! (Thanks, GOD!)
strong
attractive
interaction
BEC (Bose-Einstein condensation) = superfluid state
Condition for superflow:
There is no dissipation by elementary excitations
Landau criterion
v
superflow
Total mass of fluid: M
v'
q
superflow
elementary
excitation
Mv  Mv ' q
Total energy E’ of the fluid after the elementary excitation q is excited:
2
1
1
q
1
q
E '  Mv '2   q  M ( v  ) 2   q  Mv 2  v  q 
 q
2
2
M
2
2M
Initial energy E of the flow
BEC (Bose-Einstein condensation) = superfluid state
v
superflow
v'
q
superflow
elementary
excitation
Mv  Mv ' q
Total mass of fluid: M
Total energy E’ of the fluid after the elementary excitation q is excited:
2
1
1
q
1
q
E '  Mv '2   q  M ( v  ) 2   q  Mv 2  v  q 
 q
2
2
M
2
2M
Initial energy E of the flow
Landau criterion
Stability
condition
q2
v  q 
 q  0
2M
 q  vq
(M  )
BEC (Bose-Einstein condensation) = superfluid state
 q  vq
Stability condition =
Elementary excitation
wavenumber
Of excitation
speed of fluid
The ideal Bose gas does NOT satisfy this condition!
2
Excitation energy of an ideal Bose gas
The system energy is
lowered by excitations!
q2
q 
 q2
2m
vq
 q  vq !
0
q
How is the stable superflow realized ??
Stability condition =
 q  vq
If the excitation energy has a linear dispersion, then
 q  V q
2
q2
q 
2m
vq
0
“the stable superflow” is obtained as far as v<V.
q
How is the linear dispersion realized ??
Bogoliubov
A linear dispersion is obtained in the BEC phase when
There is a repulsive interaction between bosons.
23Na
Superfluid 4He
Bose gas BEC
 q  v q
observed velocity
Andrews et al. PRL 79 (1997) 553
How does q2 dispersion become q-linear?
m
….
0
a
2a
3a
2
4a
q2
2
q 
q
2m
5a
Ｘ
….
How does q2 dispersion become q-linear?
Natural length = a
m
….
0
a
m 2 K
L   xj
2
j 2
2a
3a
4a
5a
Ｘ
….
2
(
x

x
)
 j j 1
j

m x j   K ( x j  x j 1 )  K ( x j  x j 1 )
iq t iq ( ja )
x j  Ae
e
4K
 qa 
q 
sin   ~
m
 2 
K
4K
2  qa 
   2  2cos qa  
sin  
m
m
 2 
2
q
K
aq
m
(q ~ 0)
Linear dispersion!
How does q2 dispersion become q-linear?
m
….
0
a
2a
3a
4K
 qa 
q 
sin   ~
m
 2 
4a
K
aq
m
5a
(q ~ 0)
This is not a one-particle excitation but a collective motion.
Ｘ
….
The essesnse of the superfluid phase
The phase of the wavefunction of each boson is aligned in the BEC.
i1
| ( x1 ) | e
i
| ( x1 ) | e

| ( x2 ) | e
i2
| ( x2 ) | e


i
i3
| ( x4 ) | ei4
i
| ( x4 ) | ei
| ( x3 ) | e
| ( x3 ) | e


