Integrable pairing models in nuclear and mesoscopic physics
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Transcript of Integrable pairing models in nuclear and mesoscopic physics
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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.
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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
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“Bound” pair for arbitrary small attractive interaction. The FS is unstable against the formation of these pairs
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0 , kk k
k k
ve c c
u
Bardeen-Cooper-Schrieffer
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BCS in Nuclear Structure
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Richardson’s Exact Solution
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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
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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).
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-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
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Pair energies E for a system of 200 equidistant levels at half filling
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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.
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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
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PBCS study of ultrasmall grains:D. Braun y J. von Delft. PRL 81 (1998)47
0 0condE H H
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Condensation energy for even and odd grains
PBCS versus ExactJ. Dukelsky and G. Sierra, PRL 83, 172 (1999)
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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)
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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
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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.
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•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
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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.
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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
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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
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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
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-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
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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
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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)
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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
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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
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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
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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.
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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
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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.
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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
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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
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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
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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)
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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
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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
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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
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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
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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?