Cooper pairs and BCS (1956-1957)
Richardson exact solution (1963).
Ultrasmall superconducting grains (1999).
SU(2) Richarson-Gaudin models (2000).
Cooper pairs and pairing correlations from the exact solution. BCSBEC crossover in cold atoms (2005) and in atomic nuclei (2007).
Generalized Richardson-Gaudin Models for r>1 (2006-2009). Exact
solution of the T=0,1 p-n pairing model. 3-color pairing.
The Cooper Problem
Problem : A pair of electrons with an attractive interaction on top of
an inert Fermi sea.
 

k  kF
1
1
1
ckck  FS ,

2 k  E
G k kF 2 k  E
“Bound” pair for arbitrary small attractive interaction. The FS is unstable
against the formation of these pairs
Bardeen-Cooper-Schrieffer
 e

vk  
0 ,    ck  c k 
k uk

BCS in Nuclear Structure
Richardson’s Exact Solution
Exact Solution of the BCS Model
H P    k nk  g  c † k  c† k  c k ' ck '
k
Eigenvalue equation:
k ,k '
HP   E 
Ansatz for the eigenstates (generalized Cooper ansatz)
M
1
    0 ,   
ckck 
k 2 k  E
 1
†
†
Richardson equations
M
1
1
1 g
 2g 
 0,
k  0 2 k  E
    1 E  E
M
E   E
 1
Properties:
This is a set of M nonlinear coupled equations with M unknowns (E).
The first and second terms correspond to the equations for the one pair
system. The third term contains the many body correlations and the
exchange symmetry.
The pair energies are either real or complex conjugated pairs.
There are as many independent solutions as states in the Hilbert space.
The solutions can be classified in the weak coupling limit (g0).
154Sm
0
C5
C4
Real Part
-20
-40
-60
-80
-20
C3 C2 C1
C1 C2 C
3
-60
-80
G=0.106
-100
-100
-120
-120
-1,0
-0,5
0,0
0,5
1,0
-20
Real Part
-40
G=0.2
-20 -15 -10 -5
0
5 10 15 20
0
20 40 60 80
-50
-40
-60
-60
-70
-80
-80
-90
G=0.3
-100
-100
G=0.4
-110
-120
-40
-20
0
20
Imaginary Part
40
-120
-80 -60 -40 -20
Imaginary Part
Pair energies E for a system of 200 equidistant levels at half filling
Recovery of the Richardson solution: Ultrasmall
superconducting grains
•A fundamental question posed by P.W. Anderson in J. Phys. Chem.
Solids 11 (1959) 26 :
“at what particle size will superconductivity actually disappear?”
• Since d~Vol-1 Anderson argued that for a sufficiently small metallic
particle, there will be a critical size d ~bulk at which superconductivity
must disappear.
• This condition arises for grains at the nanometer scale.
• Main motivation from the revival of this old question came from the
works:
• D.C. Ralph, C. T. Black y M. Tinkham,
PRL’s 74 (1995) 3421 ; 76 (1996) 688 ; 78 (1997) 4087.
The model used to study metallic grains is the reduced BCS
Hamiltonian in a discrete basis:
H    j    c j c j   d  c j  c j c j 'c j '
j
j j'
Single particles are assumed equally spaced
 j  j d ,
j  1, , 
where  is the total number of levels given by the Debye frequency D
and the level spacing d as
d   2 D
PBCS study of ultrasmall grains:
D. Braun y J. von Delft. PRL 81 (1998)47
Econd   H    0 H  0
Condensation energy for even and odd grains
PBCS versus Exact
J. Dukelsky and G. Sierra, PRL 83, 172 (1999)
Exact study of the effect of level statistics in ultrasmall
superconducting grains. (Randomly spaced levels)
G. Sierra, JD, G.G. Dussel, J. von Delft and F. Braun. PRB 61(2000) R11890
Richardson-Gaudin Models
JD, C. Esebbag and P. Schuck, PRL 87, 066403 (2001).
•Combine the Richardson’s exact solution of the Pairing Model
and the integrable Gaudin Magnet
•Based on the rank 1 pair algebra of su(2) for fermions or su(1,1)
for bosons
 J 0 , J     J  ,  J  , J    2 J 0
Pair realization
J
0
k
1



 

a
a

a
a

1
,
J

a


k k
k
k ak
k k
2
Two-level realization [Only su(2)]
J
0
k
1
  (ak m ak  m  akm ak m ) , J k 
2 m

a
 k mak m
Finite center of mass momentum realization(FFLO)
J k0,Q 
m
1 
ak Q ak Q  bk b k  1 , J k,Q  akQ bk

2
Spin realization[Only su(2)]. Bosonization for large S
Construction of the Integrals of Motion
•The most general combination of linear and quadratic generators, with
the restriction of being hermitian and number conserving, is
Rl  J  2 g 
0
l
l '  l 
 X ll '  
 
0 0
 2  J l J l '  J l J l '  Yll ' J l J l ' 


•The integrability condition
 Ri , R j   0
leads to
Yij X jk  X jkYki  X ki X ij  0
•These are the same conditions encountered by Gaudin (J. de Phys.
37 (1976) 1087) in a spin model known as the Gaudin magnet.
•Gaudin (1976) found three solutions
•Rational Model
1
X ij  Yij 
i   j
•Richardson equations
1 g
j
j
2 j  e
1
4g 
0
    e  e
•Eigenvalues
i
ri  
4
 1
1 
 2
1

j   i  i   j
1 
4g 

 2i  e 

•Hamiltonianos
H    l Rl  , g   C
l
1
Some models derived from RG
BCS Hamiltonian (Fermion and Boson)
Generalized Pairing Hamiltonians (Fermion and Bosons)
Central Spin Model
Gaudin magnets (spin glass models)
Lipkin Model
Two-level boson models (IBM, molecular, Josephson, etc..)
Atom-molecule Hamiltonians (Feshbach resonances)
Generalized Jaynes-Cummings models,
Breached superconductivity. LOFF and breached LOFF states.
Px + i Py pairing.
Review:
J.D., S. Pittel and G. Sierra, Rev. Mod. Phys. 76, 643 (2004).
G. Ortiz, R. Somma, JD, S. Rombouts, Nucl. Phys. B 707 (2005) 421.
The Structure of the Cooper pairs in BCS-BEC
rs : Interparticle distance ,  : Size of the “Cooper” pair
Quasibound molecules
Pair resonaces +
Quasibound molecules
Free fermions+
Pair resonances
Thermodynamic limit of the Richardson equations
J.M. Roman, G. Sierra and JD, Nucl. Phys. B 634 (2002) 483.
V, N, G=g / V and =N / V
Using an electrostatic analogy and assuming that extremes of the arc are
2+2 i , the Richardson equations transform into the BCS equations
Gap
Number
1
d

2
g  
     2
2

   d  g   1 


The equation for the arc  is,
z

1
0
G


2
      2 
 
   
2
 2



 E    

 z       ln 

  2   E         z     2   2   
i


  ln 
 
0  Re   d  

2


2


i

E




0










  


 
Leggett model (1980) for a uniform 3D system. The divergency
of the gap equation can be cured with
g  
1 1
   d
2
4 as G 2

m
Scattering length
The Leggett model describes the BCS-BEC crossover in terms of a
single parameter
 1/ kF as . The resulting equations can be
integrated (Papenbrock and Bertsch PRC 59, 2052 (1999))



2
2
4
     P1/ 2  
2
2










3/
4
4

2
2

        P3/ 2  
2
2

3








Evolution of the chemical potential and the gap along the crossover
1
2.0
0
1.5
-1


1.0
-2
0.5
-3
-4
-2
-1
0

1
2
0.0
-2
-1
0

1
2
What is a Cooper pair in the superfluid is medium?
G. Ortiz and JD, Phys. Rev. A 72, 043611 (2005)
  A 1  r1  2  r2 
“Cooper” pair wavefunction
 N / 2  rN / 2  
 r  
vk
uk
1
ik r

e
 k
V k
From MF BCS:
 kBCS  CBCS
From pair correlations:
kP  BCS ck ck  BCS  CPuk vk
From Exact wavefuction:
 E  r   CE
e
r  E / 2
r
kE  E  
CE
2 k  E
• E real and <0, bound eigenstate of a zero range
interaction parametrized by a.
• E complex and R (E) < 0, quasibound molecule.
• E complex and R (E) > 0, molecular resonance.
• E Real and >0 free two particle state.
BCS-BEC Crossover diagram
f pairs with Re(E) >0
f=1 Re(E)<0
1-f unpaired, E real >0
= -1,
f = 0.35 (BCS)
 = 0,
f = 0.87 (BCS)
 = 0.37, f = 1 (BCS-P)
 = 0.55, f = 1 (P-BEC)
 = 1,2, f=1 (BEC)
f=1 some Re(E)>0
others Re(E) <0
“Cooper” pair wave function
2
r |(r)|
2
Weak coupling BCS
Strong coupling BCS
x 10
6
5
4
3
2
1
0
10
-2

0
2
4
6
8
E
6
P
4
2
8
BCS

3
4
10
0
0
1
2
5
25
20
BEC
15
10

5
0
0.0
0.5
1.0
r
1.5
2.0
Sizes and Fraction of the condensate

  E   0 / 2 

 0  2 / 

  P   0 / 2 2 
 
3
r

9 / 4

 2
  BCS  21/ 2 
 r2 
 E  1/ Im
 E
2
2
   dr1dr2  P  r1 , r2 
N
3
2

16 Im   i


Nature 454, 739-743 (2008)
Cooper wavefunction in the BEC region
  r   e r / b , Eb 
2
/ mb2
A spectroscopic pair size can be defined from the
threshold energy of the pair dissociation spectrum as
 
2
th
2
/ 2mEth
Application to Samarium isotopes
G.G. Dussel, S. Pittel, J. Dukelsky and P. Sarriguren, PRC 76, 011302 (2007)
Z = 62 , 80  N 96
Selfconsistent Skyrme (SLy4) Hartree-Fock plus BCS in 11 harmonic
oscillator shells (40 to 48 pairs in 286 double degenerate levels).
The strength of the pairing force is chosen to reproduce the
experimental pairing gaps in 154Sm (n=0.98 MeV, p= 0.94 MeV)
gn=0.106 MeV and gp=0.117 MeV. A dependence g=G0/A is assumed
for the isotope chain.
Correlations Energies
Mass
Ec(Exact)
Ec(PBCS)
Ec(BCS+H) Ec(BCS)
142
144
146
148
150
152
154
156
158
-4.146
-2.960
-4.340
-4.221
-3.761
-3.922
-3.678
-3.716
-3.832
-3.096
-2.677
-3.140
-3.014
-2.932
-2.957
-2.859
-2.832
-3.014
-1.214
0.0
-1.444
-1.165
-0.471
-0.750
-0.479
-0.605
-1.181
-1.107
0.0
-1.384
-1.075
-0.386
-0.637
-0.390
-0.515
-1.075
Fraction of the condensate in mesoscopic systems
1

 


c
c
c
c

c

   
 c

M 1  M / L   
BCS

c c 


1
2 2

u

 v
M 1  M / L  
From the exact solution f is the fraction of pair energies whose
distance in the complex plane to nearest single particle energy is
larger than the mean level spacing.
1,0
Exact
Fraction
0,8
BCS
BEC
0,6
154
Sm
0,4
0,2
0,0
0,0
1
g0
0,2
0,4
g
0,6
0,8
1,0
0
C5
C4
Real Part
-20
-40
-60
-80
-20
C3 C2 C1
C1 C2 C
3
-80
-100
-100
-0,5
0,0
0,5
1,0
|(k)|
2
Real Part
-20
C5
40
42
44
46
k
48
50
-60
-60
-70
-80
-80
-90
G=0.3
-100
-100
52
G=0.4
-110
-20
0
20
Imaginary Part
C4
38
-50
-40
C3
-20 -15 -10 -5 0 5 10 15 20
-40
-120
C2
G=0.2
-120
-1,0
BCS
C1
-60
G=0.106
-120
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
-40
40
-120
-80 -60 -40 -20 0 20 40 60 80
Imaginary Part
Exactly Solvable RG models for simple Lie algebras
Cartan classification of Lie algebras
rank
An su(n+1)
Bn so(2n+1)
Cn sp(2n)
Dn so(2n)
1
su(2), su(1,1)
pairing
so(3)~su(2)
sp(2) ~su(2)
so(2) ~u(1)
2
su(3) Three
level Lipkins
so(5), so(3,2)
pn-pairing
sp(4) ~so(5)
so(4) ~su(2)xsu(2)
su(4) Wigner
so(7)
FDSM
sp(6) FDSM
so(6)~su(4)
color
superconductivity
sp(8)
so(8) pairing
T=0,1.
Ginnocchio. 3/2
fermions
3
4
su(5)
so(9)
Exactly Solvable Pairing Hamiltonians
1) SU(2), Rank 1 algebra
H    i ni  g  Pi  Pj
i
ij
2) SO(5), Rank 2 algebra
H    i ni  g  Pi Pj
i
ij
J. Dukelsky, V. G. Gueorguiev, P. Van Isacker, S. Dimitrova, B. Errea y S. Lerma H. PRL 96 (2006) 072503.
3) SO(6), Rank 3 algebra
H    i ni  g  Pi Pj
i
ij
B. Errea, J. Dukelsky and G. Ortiz, PRA 79 05160 (2009)
4) SO(8), Rank 4 algebra
H ST    i ni  g  Pi Pj  g  Di D j
i
H 3/ 2
ij
ij
2
 


   i ni  g   Pi 00 Pj 00   Pi 2 m Pj 2m 
i
ij 
m 2

S. Lerma H., B. Errea, J. Dukelsky and W. Satula. PRL 99, 032501 (2007).
3-color Pairing
1.0
4
A
0.9
Ni
RB
0.8
2
GRB
GR
Breached
0.7
GR
0.6
P
G
0
B
0.5
Normal
0.4
0.3
RB
2
GR
Ni
GRB
0
20
40
i
60
Unbreached
0.1
GR
G
0
0.2
80
100
0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
g
Breached, Unbreached configurations
Phase diagram
L=500, N=150, P=(NG-NB)/(NG+NB)
3-color Pairing
3.5
3.0
1.0
0.5
0.5
0.0
0.5
< Ni >
R
R
0.0
0.5
2.0
0.5
B
B
0.0
1.0
2.5
0.5
0.0
1.5
G
G
0.0
2.0
3.0
50
0.0
100 150
50
100 150
Unbreached
Breached
n(r/R0)
2.5
1.0
0.0
0.0
i
50
i
100
Occupation probabilities
150
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Breached
0.0
150
1.0
0.8
Unbreached
1.0
0.5
100
1.2
1.0
1.5
0.5
50
1.2
0.5
1.0
1.5
r/R0
2.0
2.5
0.0
0.5
1.0
1.5
2.0
r/R0
Density profiles
2.5
3.0
The exact solution
M1
L
 1
i 1
E   e    i ui
M2
L
2li  1 1

2
1


 0

g
   ' e '  e
  '  e
i 2 i  e
M1
M3
M1
M4
M1
2
1
1
1
1




0

 '    '  
 ' e '  
 '  '  
 '   '  
 ' 2 i  
M2
M3
M2
L
2
1
1


0

 '    '  
 '  '  
i 2 i  
M4
M2
L
2
1
1


0

 '     '   
 '  '   
i 2 i   
T=0,1 Pairing
Odd-Even Pair effect as a signal of quartet correlations
200 levels, g=-0.2
2
2EA+2-EA-EA+4
1
0
-1
Exact
p-n BCS
-2
-3
90
95
100
105
Z=N
110
115
Summary
• From the analysis of the exact BCS wavefunction we proposed a new view to
the nature of the Cooper pairs in the BCS-BEC crossover.
• Alternative definition of the fraction of the condensate. Consistent with the
change of sign of the chemical potential.
• For finite system, PBCS improves significantly over BCS but it is still far from
the exact solution. Typically, PBCS misses ~ 1 MeV in binding energy.
•The T=0,1 pairing model could be a benchmark model to study different
approximations dealing with alpha correlations, clusterization and condensation.
It can also describe spin 3/2 cold atom models where this physics could be
explored.
•The SO(6) pairing model describes color superconductivity and exotic phases
with two condensates.
• SP(6) RG model: A deformed-superfluid benchmark model for nuclear
structure?