Exactly-solvable Richardson-Gaudin models and their applications *

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

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

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

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

5. Richardson’s solution of pure pairing model (fermions) Standard pure pairing hamiltonian: Richardson ansatz for ground state (N pairs): • |Ψ> is an exact eigenstate of H if pair energieseαsatisfy International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

6. These coupled eqns., one for each Cooper pair, called the Richardson equations. • The ground state energy is a sum of the resulting pair energies, Eα= Σαeα • Method can be used to get all eigenstates of H and all eigen-energies. International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

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

8. Consider energy functional • If we differentiate U with respect to the eα and equate to zero, we recover precisely the Richardson equations. • Question: What is the physical meaning of U? International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

9. Reminder:The Coulomb interaction between two point charges in 2D is: Thus: Urepresents thephysicsof aclassical 2D electrostaticproblem with the following ingredients: • A number of fixed charges (one for each active orbit) located at 2εi and with charges Ωi/2. Called orbitons. • N free charges located at eα and with unit charge. Called pairons. • A Coulomb interaction between all charges. • A uniform electric field with strength 1/4g. International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

10. For fermion systems, can show that - Orbitons are constrained to real axis, since s.p. energies all real. - Pairons lie either on real axis or in complex conjugate pairs. International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

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

12. The tin isotopes International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

13. 114Sn , 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

14. 114Sn, 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. International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

15. 116Sn , 8 pairons International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden

16. 116Sn , stronger pairing • Two-stage transition to full superconductivity. International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden