Exactly-solvable Richardson-Gaudin models and their applications *

Stuart Pittel

Bartol Research Institute, University of Delaware, Newark, Delaware 19716, USA * Work carried out in collaboration with J. Dukelsky (CSIC, Madrid), G.G. Dussel (CNEA, Buenos Aires) and C. Esebbag (Alcala).

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Introductory Remarks and Outline

Shown by Richardson in the 60s that the pure pairing model with constant

g

and non-degenerate single-particle energies is exactly solvable. Recently, a revival of work on exactly-solvable pairing models building on work of Richardson and related work of Gaudin. - Will summarize recent advances - Will then discuss one particular example of relevance to nuclear structure.

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Summary of Recent Developments

2000 -

Richardson’s exact solution revived and used to provide new insight into transition from superconducting regime to fluctuation dominated regime in small metallic grains. [J. Dukelsky and G. Sierra, Phys. Rev. B61 (2000) 12302].

2001 -

Richardson’s solution of pure pairing model generalized to a much wider variety of exactly-solvable pairing hamiltonians, relevant to both fermion and bosons systems. [J. Dukelsky, C. Esebbag and P. Schuck, PRL 87 (2001) 066403]

2001 -

Extended models applied to system of bosons confined to an oscillator trap and interacting via a repulsive interaction. Showed that fragmentation of the ground condensate possible. [J. Dukelsky and P. Schuck, PRL 86 (2001) 4207]

2001 -

Models used to identify a new mechanism for enhancing s-d boson dominance in interacting boson models of nuclei, arising from repulsive interaction due to Pauli exchange of constituent nucleons. [J. Dukelsky and S. Pittel, PRL 86 (2001) 4791] 14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

2002 -

Exactly-solvable nature of pure pairing model used to find a new pictorial representation of how superconductivity arises in a finite fermi system like the nucleus. [J. Dukelsky, C. Esebbag and S. Pittel, PRL 88 (2002) 062501.]

2004 -

Review article on Richardson-Gaudin exactly-solvable models. [J. Dukelsky, S. Pittel and G. Sierra, RMP 76 (2004) 643.]

2004 -

Exactly-solvable models extended to describe coupling between an atomic system governed by pairing correlations and another bosonic mode. Used to model a system of bosonic atoms coupled to a molecular dimer. [ J. Dukelsky, G. G. Dussel, S. Pittel and C. Esebbag, PRL 93 (2004) 050403.] 14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Richardson’s solution of pure pairing model (fermions)

Standard pure pairing hamiltonian:

H l

   

l l



m l



g a lm

2

A l

'

ll

' and

A l

 

m a



l m

Richardson ansatz for ground state (N pairs): |     

i N

  1

B

  | 0  ,

B

   

l

2 

l

1 

e



A l

  | Ψ> is an exact eigenstate of H if

pair energies e α

satisfy 1  2

g



l

2 

l

 

l e

  4

g

     

l



l

 1 / 2

e

  

e

  0 , 14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

These coupled eqns., one for each Cooper pair, called the Richardson equations.

The ground state energy is a sum of the resulting pair energies,

E

α = Σ α

e

α Method can be used to get all eigenstates of H and all eigen energies. 14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Electrostatic analogy for pairing models

There is an electrostatic analogy for such pairing models that emerges from the Richardson equations. Will focus on pure pairing for fermion systems.

In this case, ground state solution governed by pair energies obtained from set of coupled Richardson equations: 1  2

g



l

2 

l



l



e

  4

g

    

e

  

e

  0 14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Consider

energy functional U

  1 4

g

[  

e

  

j



j



j

]  1 2 

j

 (  

j

) ln | 2 

j



e

 |  1 2     ln |

e

 

e

 |  1 8

i

 

j

  

j

ln | 2    2 

j

| If we differentiate

U

with respect to the

e α

and equate to zero, we recover precisely the Richardson equations.

Question:

What is the physical meaning of

U

?

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

U

  1 4

g

[  

e

  

j



j



j

]  1 2

j

  (  

j

) ln | 2 

j



e

 |  1 2     ln |

e

 

e

 |  1 8

i

 

j

  

j

ln | 2    2 

j

|

Reminder:

The Coulomb interaction between two point charges in 2D is:

v

 (

r

1 , 

r

2 )  

q

1

q

2 ln | 

r

1  

r

2 |

Thus: U

represent

s

the physics of a classical 2D electrostatic problem with the following ingredients: – A number of fixed charges (one for each active orbit) located at 2ε i and with charges Ω i /2. Called

orbitons.

– N free charges located at e α and with unit charge. Called – A Coulomb interaction between all charges.

pairons

.

– A uniform electric field with strength 1/4g.

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

For fermion systems, can show that - Orbitons are c

onstrained to real axis, since s.p. energies all real.

- Pairons

lie either on real axis or in complex conjugate pairs.

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Application to Nuclear Pairing

Will use electrostatic analogy to obtain pictorial representation of how “superconductivity” develops in nuclei.

Typically hard to see effects of transition to superconductivity because of limited number of nucleons involved.

Will use info on classical positions of pairons (from analogous 2D problem) to provide insight into quantum problem which otherwise was not readily evident.

Will focus on even Sn isotopes, with closed Z=50 proton shell and N-50 active (valence) neutrons. Will do calculations as function of

g.

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

The tin isotopes

14-18 June 2005

Orbiton d

5/2

g

7/2

s

1/2

d

3/2

h

11/2

Position 0 0.44

3.8

4.4

5.6

International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

Charge -1.5

-2.0

-0.5

-1.0

-3.0

14-18 June 2005 114

Sn , 7 pairons

Lines drawn to connect each pairon with its nearest neighbor.

For weak pairing, pairons organize themselves as

artificial atoms

around associated orbitons, subject to Pauli principle.

Note:

Physical g ≈-0.095 MeV International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

114

Sn, Evolution with

g

As pairing strength grows, a transition takes place from a set of isolated “atoms” to a “cluster”, in which pairons have lost memory of the orbitons from which they came.

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

116

Sn , 8 pairons

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

116

Sn , stronger pairing

Two-stage transition to full superconductivity.

14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden