Exactly solvable Richardson-Gaudin models in nuclear structure
Jorge Dukelsky
In collaboration with people in audience:
S. Pittel, P. Schuck, P. Van Isacker.
And many others
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 kk 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 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).
Exact solvability reduces an exponential complex problem to an
algebraic problem.
Evolution of the real and imaginary part of the pair energies with g. L=16,
M=8. R. W. Richardson, Phys. Rev. 141 (1966) 949. Solved numerical systems up to L=32,
dim=108
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.0
0.2
0.4
0.6
0.8
1.0
E=1.7+0.0i
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.0
0.2
0.4
0.6
0.8
1.0
E=12.0+4.0i
|i|2
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.0
0.2
0.4
0.6
0.8
1.0
E=0.0+2.0i
2i
†
1
1
2
L
k kk k
c cE
The SU(2) Algebra
, , , , , 2z z zS S S S S S S S S
Rank 1 and 1 quantum degree of freedom
The pair realizations is:
1 1 , S
2 4 2
jz
j jm jm j jm jm
m m
S a a a a
Other realizations like, two level atoms, spin, finite center of mass
pairs, Holstein-Primakoff or Schwinger, give rise to different physical
Hamiltonians
•The most general combination of linear and quadratic generators, with the
restriction of being hermitian and number conserving, is
22
ijz z z
i i i j i j ij i j
j i
XR S g S S S S Y S S
•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.
Richardson-Gaudin Models:
Construction of the Integrals of Motion
J. D., C. Esebbag and P. Schuck, Phys. Rev. Lett. 87, 066403 (2001).
Gaudin (1976) found three solutions
1ij ij
i j
X Y
XXX (Rational)
XXZ (Hyperbolic Trigonometric)
12 ,
i j i j
ij ij i j
i j i ji j
X Z Coth x xSinh x x
i iR r Exact solution
Eigenstates of the Rational and Hyperbolic Models
10 , 0i
XXX i XXZ i
i ii i
S SE E
Richardson ansatz
Any function of the R operators defines a valid integrable Hamiltonian. The
Hamiltonian is diagonal in the basis of common eigenstates of the R operators.
•Within the pair representation two body Hamiltonians can be obtain by a
linear combination of R operators:
•The parameters g, ´s and ´s are arbitrary. There are 2 L+1 free parameters
to define an integrable Hamiltonian in each of the families. (L number of single
particle levels)
• The constant PM or reduced BCS Hamiltonian solved by Richardson can be
obtained by from the XXX family by choosing = .
•For the same linear combination in the Hyperbolic family:
,l l
l
H R g
2 z
BCS ï i i j
i ij
H S g S S
2 z
Hyper ï i ï j i j
i ij
H S g S S
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. Dim. of
the pairing Hamiltonian matrix ~ 1049 to 1053.
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=gn/A is assumed
for the isotope chain.
-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
Re
al P
art
G=0.3
G=0.2
154Sm
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
The Hyperbolic Model in Nuclear Structure
,
z
i i i j i j
i i j
H S G S S
Redefining the 0 of energy , absorbing the constant in
the chemical potential μ i i
,
i i i i j i ji ji i j
H c c G c c c c
The separable integrable Hyperbolic Hamiltonian
α is a new parameter that serves as an energy cutoff.
In BCS approximation:
The BCS Hamiltonian has ' ' '
'
i i i i i i
i
G u v
Exactly solvable H with non-
constant matrix elements
J. Dukelsky, S. Lerma H., L. M. Robledo, R. Rodriguez-Guzman, S. Rombouts, Phys. Rev. C 84, 061301(R) (2011)
i
unphysical
Mapping of the Gogny force in the Canonical Basis
We fit the pairing strength G and the interaction cutoff to the paring
tensor uivi and the pairing gaps i of the Gogny HFB eigenstate in the
Hartree-Fock basis.
Protons
o Gogny
_ Hyperbolico
' ' '
'
2 22
i i i i i i
i
i
i i
i i
G u v
u v
M L D G EBCScorr EExa
corr
154Sm 31 95 9.9x1024 2.2x10-3 32.7 0.158 1.0164 2.9247
238U 46 148 4.8x1038 2.0x10-3 25.3 0.159 0.503 2.651
Models derived from r = 1 RG [SU(2) and SU(1,1)]
BCS or constant pairing Hamiltonian
Generalized Pairing Hamiltonians (Fermion and Bosons)
Central Spin Model (Quantum dot)
Gaudin magnets (Quantum magnetism)
Lipkin Model
Two-level boson models (IBM, molecular, etc..)
Atom-molecule Hamiltonians (Feshbach resonances in cold atoms)
Generalized Jaynes-Cummings models.
Breached superconductivity. LOFF and breached LOFF states.
p-wave pairing in 2D lattices.
Richardson-Gaudin-Kitaev model of topological supeconductivity.
Reviews: J.Dukelsky, S. Pittel and G. Sierra, Rev. Mod. Phys. 76, 643 (2004);
G. Ortiz, R. Somma, J. Dukelsky y S. Rombouts. Nucl. Phys. B 7070 (2005) 401
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)
3 su(4) Wigner so(7)
FDSM sp(6) FDSM
so(6)~su(4)
color
superconductivity
4 su(5) so(9) sp(8)
so(8) pairing
T=0,1.
Ginnocchio. S=3/2
fermions
Exactly Solvable Pairing Hamiltonians
1) SU(2), Rank 1 algebra
i i i j
i ij
H n g P P 2) SO(5), Rank 2 algebra
i i i j
i 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 j
i ij
H n g P P
01 10
0 0
1 1,
2 2
ST i i i j i j
i ij ij
i i i i i i
H n g S S g D D
S a a D a a
B. Errea, J. Dukelsky and G. Ortiz, PRA 79, 051603(R) (2009).
1
1 1
M L
i i
i
E e u
1 2
' ' '
2 12 1 10
2
M M Li
i i
l
e e e e g
Exact solution of the SO(8) model
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
80 Nucleons in L=50 equidistant levels
Quartet: 1e, 1, 1, 1
n-n Cooper pair: 1e
p-p Cooper pair: 1e, 2, 1, 1
0 2 4 6 8 100
10
20
30
40
50
1 3 5 7 90
10
20
30
40
T=0
T=1
T=0,1
ET
T
Even T
T=0
T=1
T=0,1
T
Odd TG=0.22
1 1
,2 2
e o
T T
T T
E T T E T T EJ J
JT: iso-MoI, : Linear enhancement factor (Wigner energy),
E: 2qp excitation (=2)
Analysis of the nuclear symmetry energy vs T in terms of the Isocranking model (W.
Satula and R. Wyss, PRL 86, 4488 (2001) and 87, 052504 (2001).
Linear enhancement factor λ Inverse of the Iso-MoI
G=0.16
G=.22
T=0 circles, T=1diamonds, T=0,1 triangles. Solid (open) -> even (odd) T
Wigner limit
Picket-Fence model and the thermodynamic limit of p-n BCS
Equidistant single particle levels , 1, ,2
i
ii
ST i i i j i j
i ij ij
H n g S S D D
SU(4) symmetric pairing Hamiltonian
Quarter filling , with 0.15 0.54N g f
Thermodynamic limit 1
, ,4 4
NN
BCS equations:
1/2 1/2
2 20 02 2
14 1 1,d d
g
G. F. Bertsch, J. Dukelsky, B. Errea, C. Esebbag, Ann. Phys. 325 (2019) 1340
Unlike the SU(2) RG model, we cannot derive analytically the continuous limit. Proceed
numerically by expanding the GS and quasiparticle energies as
4
2 3
1,
1/
4 4 1 4
14 1 2 4 1 4 4 2
2
18
GS
q GS GS
o e GS GS GS
c i i
i
E b c da N
N N N N
E n E n E n
n E n E n E n
gn n
160 1000, 40 250N n
0 20 40 60 80 100
70
80
90
100
110
120
130
140E
corr
ela
cio
n
T=(N-Z)/2
exact
BCS
200 levels, 200 particles
=0.5, g=-0.2
Odd-Even Pair effect as a signal of quartet correlations
90 95 100 105 110 115
-3
-2
-1
0
1
2
2E
A+
2-E
A-E
A+
4
Z=N
Exact
p-n BCS
200 levels, g=-0.2
T=0,1 Pairing
Summary
• For finite systems, PBCS improves significantly over BCS but it is still far from
the exact solution. Typically, PBCS misses ~ 1 MeV in binding energy.
•The Isovector SO(5) and the SO(8) pairing models are excellent benchmark
models to study different approximations dealing with quartet correlations,
clusterization and condensation. The SO(8) model can also describe spin 3/2
cold atoms where nuclear physics could be explored in the lab.
•SO(5) has been used to test the QCM approximation in: N. Sandulescu, D. Negrea,
J. Dukelsky, and C. W. Johnson Phys. Rev. C 85, 061303(R) (2012)
•The exact GS energy of the T=0,1 pairing Hamiltonian goes to p-n BCS energy
in the thermodynamic limit. However, quartet correlations are important for finite
systems.
•Alpha phases in nuclear matter require more realistic interactions: contact,
schematic or realistic nuclear forces. Could they be explore with cold atoms in
optical lattices?