Transcript hard core

Unitarity potentials and
neutron matter at unitary limit
T.T.S. Kuo (Stony Brook)
Collaborators:
H. Dong (Stony Brook), R. Machleidt (Idaho)

Atom-Atom interaction VAA for trapped
cold fermionic gases can be experimentally
tuned by external magnetic field,
giving many-body problems with
tunable interactions:
(H0  VAA )0 (A)  E00 (A), A  


By tuning VAA to Feshbach resonance,
scattering length as
 .
At this limit (unitary limit),
interesting physics observed.


this is BCS-BEC cross-over


1
Near the Feshbach resonance (
0 )
as
1
 0
gas in BEC
as
1
gas in BCS
 0
as


1
At
 0 , the equation of state (EOS)
as
E 0free 3 2 kF2 

has an ‘universal’ form: E 0  E 0free 

10m 
 A
with ξ=0.44 for ‘all’ gases.

E 0 depends only
 on kF


Above often known as ‘Bertsch challenge problems’
Experimental values for ξof atomic gases
Authors
ξ
0.39(15)
Bourdel et al., PRL (2004)
0.51(4)
Kinast et al., Science (2005)
0.46(5)
Partridge et al., Science (2006)
0.46 +0.05
Stewart et al., PRL (2006)
-0.12


Neutron matter is a two-species fermionic system,
should have same unitary-limit properties as cold
fermi gas, and neutron-neutron as  19 fm, it is
rather long.

We study the EOS of neutron
matter at and near

the unitary limit, using different unitarity potentials
If E 0  E 0free is universal, then results
should be independent of the potentials
as long as their as 

How to obtain unitarity potentials with as  ?
tuned CDBonn meson-exchange potential
tuned square-well ‘box’ potentials


How to calculate ground state energy E 0 ?
ring-diagram and model-space HF methods

Results and discussions


Atom-Atom interactions VAA can be
experimentally varied by tuning
external magnetic field.

How to vary the NN interaction VNN ?
Can we tune VNN experimentally?
May use Brown-Rho scaling to tune VNN ,
namely slightly changing
 its meson masses.


Ask Machleidt to help!


CD-Bonn VNN ( 1 S 0 ) of different as
m [MeV]
a s [fm]
original
452.0
-18.98
tunned

475.0
σ


-5.0
447.0
-42.0
442.850
-∞(-12070)
442.800
+∞
434.0
+21
We tuned only m σ , as attraction in S 0
mainly from σ-exchange. asdepends
sensitively on m σ .
1

We have also used hard-core square-well
(HCSW) potentials
V(r) Vc ;
r  rc
Vb ; rc  r  rb


0;
r  rb
 scattering length ( a ) and effective range
 Their
s
( re) can be obtained analytically.
 
We can have many HSCW unitarity potentials
Phase shiftsδfor HCSW potentials:
K3
tan(  K 3 rb ) 
tan(K 2 rb   )
K2
K2
tan(  K 2rc ) 
tanh( K1rc )
K1

with


K1  (Vc  E)
m
K 2  (E  Vb )
m
2
,
,
2
K3  E
where E is the scattering energy.


m
2
From phase shift δ, the scattering length is
B
as  
A
with
A  K10K 20  K 202 tanh( K10rc )tan[K 20 (rb  rc )],

B  K20 tanh( K10rc )  K10 tan[K20 (rb  rc )]
2
rb K10K 20  rb K 20
tanh( K10rc )tan[K 20 (rb  rc )],

where


K10  Vc
m
2
, K 20  Vb
m
2
.
The effective range also analytically given.
Condition for unitarity potential is
1
K10
1
rb  rc 
tan [
]
K 20
K 20 tanh( K10rc )
c

c
b
c
e
s
Three different HCSW unitarity potentials
Potentials
Vc
/MeV
rc /fm
Vb /MeV
rb
/fm
6
as /x10
fm
reff
/fm
HCSW01
3000
0.15
-20
2.31
15.2
2.36
HCSW02
3000
0.30
-30
2.03
3.38
2.21
HCSW03
3000
0.50
-50
1.81
-4.58
2.20

Ground state energy shift E 0  E 0  E 0free
E 0 
 0 d
1
*
Y
(ij,

)Y
m
m (kl,  ) ij Vlowk kl
m ijkl
Ym* (kl, )  m (, A  2) al ak 0 (, A)
By summing the pphh ring diagrams to all

orders,
the transition amplitudes Y are given by the

RPA equations: AX  BY  X
A*Y  B* X  Y
 Above is quasi-boson RPA




Model-space approach:
 Space (k > Λ) integrated out:
Vbare renormalized to Vlowk
Vbare has strong short range repulsion
Vlowk is smooth and energy independent
 Space (k ≤ Λ)

use Vlowk to calculate all-order sum of ring
diagrams
Vlowk of specific scattering length
Note
we
need

including as  



a
of specific scattering length
Vlowk
Starting from a bare CD-Bonn potential of
a
scattering length a, Vlowk
given by
T(k ' ,k,k 2 )  V a (k ' ,k) 


0
a
'
2
V
(k
,q)T(q,k,k
)
2
q dq
k 2  q2  i0
a
p 2 )  Vlowk
Tlowk ( p' , p,
( p' , p)
 0



'
2
'
2 '
(
p
,p
)

T
(
p
,
p
,
p
)

T
(
p
,
p
,
p
);
low

k


a
'
2
V
(
p
,q)T
(q,
p,
p
)
2
lowk
lowk
q dq
p 2  q 2  i0 
a
Vlowk
obtained
from solving the above T-matrix

equivalence equations using the iteration method
of Lee-Suzuki-Andreozzi
Ring diagram unitary ratio given by
different unitarity potentials
Diagonal matrix elements of VNN
The ring-diagram unitary ratio near the
unitary limit
When choosing Λ= k F, ring-diagram method
becomes a Model-Space HF method,
1
kF
E 0 
k
k
V

1 2 lowk k1k 2
2 k1 ,k2 kF
and E 0 /A given by simple integral

E0 3
8
 F 
A 5


kF
0
3
3k
k
k dk[1
 3]
2k F 2k F
2
kF
 (2J  1) ,k Vlowk
,k


kF
V
Here lowk means Λ= k F.
kF
MSHF has simple relation between ξand Vlowk
:
3F
8
( 1) 
5


kF
0
3
3k
k
k 2 dk[1
 3]
2kF 2k F
1
kF
1
 
S0 ,k Vlowk
S0 ,k
kF
V
 lowk is highly accurately simulated by momentum
expansions:
k 2
k 4
)  V4 ( )
kF
kF
where V0 , 2V and4 V are constants. Then
k V  k  V0  V2 (
kF
lowk

3
V0 V2 3V4
( 1)  kF (  
)
10
3 10 70
Above is a strong sum-rule and scaling
kF
Vlowk
constraint
for 
at the unitary limit.
Checking
3
V V 3V
( 1)  kF ( 0  2  4 ) for four unitarity potentials
10
3 10 70
V2/fm
V4 /fm
Sum
ξ
-2.053
3.169
-1.801
-0.445
0.434
HCSW01
-2.001
2.865
-1.402
-0.441
0.439
HCSW02
-1.904
2.373
-0.999
-0.440
0.440
HCSW03
-1.893
2.261
-0.825
-0.440
0.439
VNN

CDBonn
k F /fm-1 V0 /fm
1.2
HCSW01
1.0
-2.102
2.202
-1.070
-0.526
0.442
HCSW01
1.4
-1.945
3.584
-1.983
-0.375
0.443
Comparison of V from four different
unitarity potentials (Λ= k =1.2
fm)
F
kF
low-k
Comparison of recent calculated values on ξ
ξ
0.326,0.568
Method
Ref.
Padé approximation
Baker et al., PRC(1999)
0.326
Galitskii resummation
Heiselberg et al., PRA(2001)
0.7
Ladder approximation
Bruun et al., PRA(2004)
0.455
Diagrammatic theory
Perali et al, PRL(2004)
0.42
Density functional theory
Papenbrock et al., PRA(2005)
0.401
NSR extension with
pairing fluctuations
Hu et al., EPL(2006)
0.475
εexpansion
Nishidaet al., PRL(2006)
0.360
Variational formalism
Haussmann et al., PRA(2007)
0.475
εexpansion
Chen et al., PRA(2007)
0.44(1)
Quantum Monte Carlo
Carlson et al., PRL(2003)
0.42(1)
Quantum Monte Carlo
Astrakharchik et al., PRL (2004)
0.44
Ring-diagram and MSHF
This work (Dong, et al., PRC 2010)

MSHF single-particle (s.p.) potential is
16
U(k1 )  (2J 1){


2

k1


k
k_

k
0
kF
k 2 dk ,k Vlowk
,k
kF
kdk[kF2  k12  4k(k1  k)] ,k Vlowk
,k }
with k − =(kF - k1)/2, k +=(k F + k1)/2.


2 2
1
k
(k1 ) 
 U(k1 )
2m
The MSHF s.p. spectrum is
which can be well approximated by
2
k12
(k1 ) 

*
 2m
m* is effective mass andΔis effective ‘well-depth’ .


At the unitary limit, m* and Δof MSHF should
obey the linear constraint
1
m
5
 

2 2m * 6F
We have checked this constraint.

1
m
5

Check   
at unitary limit
2 2m * 6F

Summary and outlook:

Our results have provided strong ‘numerical’
free
evidences that the ratio ξ= E /0 E 0
is a universal constant, independent of
the interacting potentials as long as
they have as .

However, it will be still challenging to prove
this universality analytically !

Thanks to organizers
R. Marotta and N. Itaco