Virial expansion
in the unitary limit
L.P. Pitaevskii and S. Stringari
1
Basic conditions
Let us the gas is weakly degenerated:
n
1/ 3


 T
mT
1
but cold enough:
T 

 r0
mT
2
r0is the size of an atom. Gas is classical from
the thermodynamic point of view, but
atom-atom collisions are quantum.
2
Large scattering length
Due to the condition (2) only s-scattering is
important. Near the Feshbach resonance swave scattering length can be large with
compare to T :

a 
 r0
mT
(3)
3
Diluteness
Due to the unitarity condition the s-wave scattering
amplitude is restricted

| f | ~ T .
p
Then the diluteness condition follows from (1)
n | f |  1.
3
4
Virial coefficient
The equation of state of the gas accounting for the
second “virial” correction is
PV  nT1  nBT 
In the absence of interaction:
BT 
B,F
id
3/ 2
1   
    , g  2s  1
2 g  mT 
2
g=1 for spinless bosons (upper sign) and g=2
for s=1/2 fermions
5
Interaction
Contribution of the interaction in s-channel
to the virial coefficient (E. Beth, G. Uhlenbeck, 1937):
  

Bint T   G8
 mT 
2
3/ 2
 | | 1    k d 
dk 
e T   e mT
0
dk 

2 2
G  (s  1) /(2s  1) for bosons
andG  s / ( 2s  1 ) for fe rmions
.
 0
 is the s-wave scattering phase shift,
is the bound state energy (if the state
exists), k is the wave vector of relative
motion.
6
Scattering near Feshbach
resonance
We consider vicinity of the Feshbach resonance
where the s-wave scattering length is large
comparing to the thermal wave-length 
T

a 
mT
(4)
This is the unitarity limit.
7
Positive a means existence of a bound state
with small negative energy
2

 
,
2
ma
  T
The phase of s-wave scattering is
tg  ka, (5)
The scattering amplitude is
a
f 
1  ika
8
Calculation of the virial coefficient
Equation (5) gives
d
a

2
dk
1  ka
and using condition (4)
1

e


0
 2k 2

mT
d
1 a
dk  
.
dk
2|a|
Contribution of the bound state is
 0 if a  0
 | |
e T  1 if a  0
9
Results
Finally
  

Bint T   G 4
 mT 
2
3/ 2
independently of value and sign of the scattering
length. The derivation assumes that bound state is in
equilibrium with continuum.
The total (statistic+interaction) virial coefficient for
spinless bosons is
3/ 2
9  
B
BT       9 BT id
2  mT 
2
B
10
and for s=1/2 fermions
BT 
F
3  

  
4  mT 
2
3/ 2
 3BT 
F
id
In both cases the sign of the coefficient corresponds to
effective attraction.
11