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. BCS-

BEC 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.

1 1 1,

2 2F F

k kk k k kk k

c c FSE G E

“Bound” pair for arbitrary small attractive interaction. The FS is unstable against the formation of these pairs

0 , kk k

k k

ve c c

u

Bardeen-Cooper-Schrieffer

BCS in Nuclear Structure

Richardson’s Exact Solution

Exact Solution of the BCS Model

Eigenvalue equation:

Ansatz for the eigenstates (generalized Cooper ansatz)

PH E

† †

1

10 ,

2

M

k kk k

c cE

† †' '

, 'P kk k kk k

k k k

H n g c c c c

Richardson equations

0 1 1

1 11 2 0,

2

M M

k k

g g E EE E E

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 (g0).

-120

-110

-100

-90

-80

-70

-60

-50

-80 -60 -40 -20 0 20 40 60 80

-120

-100

-80

-60

-40

-20

0

-1,0 -0,5 0,0 0,5 1,0

-120

-100

-80

-60

-40

-20

-40 -20 0 20 40

-120

-100

-80

-60

-40

-20

-20 -15 -10 -5 0 5 10 15 20

Imaginary Part

G=0.4

C3

C3

C2C

2C1

C1

R

eal P

art

G=0.106

C4

C5

Imaginary Part

Rea

l Par

t

G=0.3

G=0.2

154Sm

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:

' ''

j j j j j j jj j j

H c c d c c c c

Single particles are assumed equally spaced

,,1, jdjj

where is the total number of levels given by the Debye frequency D and the level spacing d as

2 Dd

PBCS study of ultrasmall grains:D. Braun y J. von Delft. PRL 81 (1998)47

0 0condE H H

Condensation energy for even and odd grains

PBCS versus ExactJ. Dukelsky and G. Sierra, PRL 83, 172 (1999)

G. Sierra, JD, G.G. Dussel, J. von Delft and F. Braun. PRB 61(2000) R11890

Exact study of the effect of level statistics in ultrasmall superconducting grains. (Randomly spaced levels)

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

0 0, , , 2J J J J J J

Pair realization

Two-level realization [Only su(2)]

0 1 ( ) ,

2k k m k m k m k m k k m k mm m

J a a a a J a a

Spin realization[Only su(2)]. Bosonization for large S

Finite center of mass momentum realization(FFLO)

0 1 1 ,

2k k k k kk k kJ a a a a J a a

0, ,

11 ,

2k Q k Q k Q k k k Q k Q kJ a a b b J a b

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

0 0 0'' ' ' '

'

22ll

l l l l l l ll l ll l

XR J g J J J J Y J J

•The integrability condition leads to, 0i jR R

0ij jk jk ki ki ijY X X Y X X

•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 1ij ij

i j

X Y

•Richardson equations

11 4 0

2j

j j

g ge e e

•Eigenvalues

1 1 11 4

4 2 2i

ij i i j i

r ge

•Hamiltonianos

, 1l ll

H R g C

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.

rs : Interparticle distance , : Size of the “Cooper” pair

The Structure of the Cooper pairs in BCS-BEC

Quasibound molecules Pair resonaces + Free fermions+

Quasibound molecules Pair resonances

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

The equation for the arc is,

22 2

2 20

ln

0 Re ln

Ez

E zid

i E

2 2z

Thermodynamic limit of the Richardson equations J.M. Roman, G. Sierra and JD, Nucl. Phys. B 634 (2002) 483.

2 2

1 10

2

gd

G

2 2

1d g

Leggett model (1980) for a uniform 3D system. The divergency of the gap equation can be cured with

2

1 1

4 2s

gmd

a G

Scattering length

The Leggett model describes the BCS-BEC crossover in terms of a single parameter . The resulting equations can be integrated (Papenbrock and Bertsch PRC 59, 2052 (1999))

1/ F sk a

2 241/ 2 2 2

3/ 42 23/ 2 2 2

4

3

P

P

-2 -1 0 1 2-4

-3

-2

-1

0

1

Evolution of the chemical potential and the gap along the crossover

-2 -1 0 1 20.0

0.5

1.0

1.5

2.0

What is a Cooper pair in the superfluid is medium?G. Ortiz and JD, Phys. Rev. A 72, 043611 (2005)

1 ik rk

k

r eV

“Cooper” pair wavefunction

From MF BCS:

From pair correlations:

From Exact wavefuction:

BCS kk BCS

k

vC

u

Pk P k kk k

BCS c c BCS C u v

2

E Ek

k

CE

E

/ 2r E

E E

er C

r

• 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.

1 1 2 2 / 2 / 2N NA r r r

BCS-BEC Crossover diagram

f pairs with Re(E) >0

1-f unpaired, E real >0f=1 Re(E)<0

f=1 some Re(E)>0

others Re(E) <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)

0 2 4 6 8 100

1

2

3

4

5

6

x 10-2

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

r

r2 |(r

)|2

0 1 2 3 4 50

2

4

6

8

10

E

P

BCS

“Cooper” pair wave function

Weak coupling BCS

Strong coupling BCS

BEC

Sizes and Fraction of the condensate

0

020 3

2

/ 22 /

/ 2 29 / 4

21/ 2

E

P

BCS

rr

1/ ImE E

2

1 2 1 2

2

2,

3

16 Im

Pdrdr r rN

i

Nature 454, 739-743 (2008)

/ 2 2, /r bbr e E mb

A spectroscopic pair size can be defined from the threshold energy of the pair dissociation spectrum as

2 2 / 2th thmE

Cooper wavefunction in the BEC region

Application to Samarium isotopesG.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.

Mass Ec(Exact) Ec(PBCS) Ec(BCS+H) Ec(BCS)

142 -4.146 -3.096 -1.214 -1.107

144 -2.960 -2.677 0.0 0.0

146 -4.340 -3.140 -1.444 -1.384

148 -4.221 -3.014 -1.165 -1.075

150 -3.761 -2.932 -0.471 -0.386

152 -3.922 -2.957 -0.750 -0.637

154 -3.678 -2.859 -0.479 -0.390

156 -3.716 -2.832 -0.605 -0.515

158 -3.832 -3.014 -1.181 -1.075

Correlations Energies

Fraction of the condensate in mesoscopic systems

2 2

1

1 /

1

1 /BCS

c c c c c c c cM M L

u vM 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.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

F

ract

ion

g

BECBCS

Exact

g0

1

154Sm

38 40 42 44 46 48 50 52

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

|(k

)|2

k

BCS C

1

C2

C3

C4

C5

-120

-110

-100

-90

-80

-70

-60

-50

-80 -60 -40 -20 0 20 40 60 80

-120

-100

-80

-60

-40

-20

0

-1,0 -0,5 0,0 0,5 1,0

-120

-100

-80

-60

-40

-20

-40 -20 0 20 40

-120

-100

-80

-60

-40

-20

-20 -15 -10 -5 0 5 10 15 20

Imaginary Part

G=0.4

C3

C3

C2C

2C1

C1

Real

Part

G=0.106

C4

C5

Imaginary Part

Real

Part

G=0.3

G=0.2

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)

1su(2), su(1,1)

pairingso(3)~su(2) sp(2) ~su(2) so(2) ~u(1)

2su(3) Three

level Lipkins so(5), so(3,2) pn-pairing

sp(4) ~so(5) so(4) ~su(2)xsu(2)

3 su(4) Wignerso(7)

FDSMsp(6) FDSM

so(6)~su(4)

color superconductivity

4 su(5) so(9) sp(8)

so(8) pairing T=0,1.

Ginnocchio. 3/2 fermions

Exactly Solvable Pairing Hamiltonians1) SU(2), Rank 1 algebra

i i i ji ij

H n g P P 2) SO(5), Rank 2 algebra

i i i ji ij

H n g P P

4) SO(8), Rank 4 algebra

J. Dukelsky, V. G. Gueorguiev, P. Van Isacker, S. Dimitrova, B. Errea y S. Lerma H. PRL 96 (2006) 072503.

S. Lerma H., B. Errea, J. Dukelsky and W. Satula. PRL 99, 032501 (2007).

3) SO(6), Rank 3 algebra

i i i ji ij

H n g P P

2

3/ 2 00 00 2 22

ST i i i j i ji ij ij

i i i j i m j mi ij m

H n g P P g D D

H n g P P P P

B. Errea, J. Dukelsky and G. Ortiz, PRA 79 05160 (2009)

0

2

4

G R

R B

Ni

G R BG R

G

A

0 20 40 60 80 1000

2G R

R B

GG R

Ni

i

G R B

B

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pg

Normal

Breached

Unbreached

Phase diagram

L=500, N=150, P=(NG-NB)/(NG+NB)

Breached, Unbreached configurations

3-color Pairing

50 100 1500.0

0.5

1.0

1.5

2.0

2.5

3.0

50 100 150

0.0

0.5

50 100 1500.0

0.5

0.0

0.5

1.0

R

B

G

< N

i >

i

Breached

50 100 1500.0

0.5

0.0

0.5

1.0

0.0

0.5

B

G

R

i

Unbreached

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

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

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

n(r/

R0)

r/R0

Breached

r/R0

Unbreached

Occupation probabilities Density profiles

3-color Pairing

1

1 1

M L

i ii

E e u

1 2

' ' '

2 12 1 10

2

M M Li

i i

l

e e e e g

The exact solution

32 1 4 1

' ' ' ' '' ' ' '

2 1 1 1 10

2

MM M M M

ie

3 2

' '' '

2 1 10

2

M M L

i i

4 2

' '' '

2 1 10

2

M M L

i i

Odd-Even Pair effect as a signal of quartet correlations

90 95 100 105 110 115

-3

-2

-1

0

1

2

2EA

+2-E

A-E

A+

4

Z=N

Exact p-n BCS

200 levels, g=-0.2

T=0,1 Pairing

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?