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?