14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden Stuart Pittel Bartol Research Institute, University

14-18 June 200514-18 June 2005 International Conference on Finite Fermi International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, SwedenSystems: Nilsson Model 50 years, Lund, Sweden

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).

Exactly-solvable Richardson-Gaudin models and Exactly-solvable Richardson-Gaudin models and

their applications their applications **


Shown by Richardson in the 60s that the pure pairing model with Shown by Richardson in the 60s that the pure pairing model with constant constant gg and non-degenerate single-particle energies is exactly and non-degenerate single-particle energies is exactly solvable. solvable.

Recently, a revival of work on exactly-solvable pairing models Recently, a revival of work on exactly-solvable pairing models building on work of Richardson and related work of Gaudin. building on work of Richardson and related work of Gaudin.

- Will summarize recent advances- Will summarize recent advances

- Will then discuss one particular example of relevance to nuclear - Will then discuss one particular example of relevance to nuclear structure.structure.

Introductory Remarks and OutlineIntroductory Remarks and Outline


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

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

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

2001 -2001 - Models used to identify a new mechanism for enhancing s-d Models used to identify a new mechanism for enhancing s-d boson dominance in interacting boson models of nuclei, arising boson dominance in interacting boson models of nuclei, arising from repulsive interaction due to Pauli exchange of constituent from repulsive interaction due to Pauli exchange of constituent nucleons. [J. Dukelsky and S. Pittel, PRL 86 (2001) 4791]nucleons. [J. Dukelsky and S. Pittel, PRL 86 (2001) 4791]

Summary of Recent Developments Summary of Recent Developments


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

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

2004 -2004 - Exactly-solvable models extended to describe coupling Exactly-solvable models extended to describe coupling between an atomic system governed by pairing correlations and between an atomic system governed by pairing correlations and another bosonic mode. Used to model a system of bosonic atoms another bosonic mode. Used to model a system of bosonic atoms coupled to a molecular dimer. [ J. Dukelsky, G. G. Dussel, S. coupled to a molecular dimer. [ J. Dukelsky, G. G. Dussel, S. Pittel and C. Esebbag, PRL 93 (2004) 050403.]Pittel and C. Esebbag, PRL 93 (2004) 050403.]


mla

mlm

al

Alm

am

lma

lN

lA

lll

Ag

ll

Nl

H

and ˆ

''2

ˆ

Standard pure pairing hamiltonian:Standard pure pairing hamiltonian:

Richardson’s solution of pure pairing model Richardson’s solution of pure pairing model (fermions)(fermions)

Richardson ansatz for ground state (N pairs):Richardson ansatz for ground state (N pairs):

l

l l

N

i

Ae

BB

2

1 , 0 | |

1

||ΨΨ> is an exact eigenstate of H if > is an exact eigenstate of H if pair energiespair energies eeαα satisfy satisfy

2/1

, 0 42

21

l

eeg

eg

l

l l

l


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

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

EEαα = = ΣΣαα eeαα

Method can be used to get all eigenstates of H and all eigen-Method can be used to get all eigenstates of H and all eigen-energies. energies.


Electrostatic analogy for pairing modelsElectrostatic analogy for pairing models

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

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

0 42

21

eeg

eg

l l

l


Consider Consider energy functionalenergy functional

If we differentiate If we differentiate U U with respect to with respect to the the eeαα and equate to zero, and equate to zero,

we recover precisely the Richardson equations.we recover precisely the Richardson equations.

Question:Question: What is the physical meaning of What is the physical meaning of UU??

|22|ln 8

1 ||ln

2

1

|2|ln )(2

1 ][

4

1

jjji

jj

jjj

j

ee

eeg

U

||ln ),( 212121 rrqqrrv


– A number of fixed charges (one for each active orbit) located at A number of fixed charges (one for each active orbit) located at 22εεii and with charges and with charges ΩΩii/2. Called /2. Called orbitons.orbitons.

– N free charges located at N free charges located at eeαα and with unit charge. Called and with unit charge. Called paironspairons..

– A Coulomb interaction between all charges.A Coulomb interaction between all charges.– A uniform electric field with strength 1/4g.A uniform electric field with strength 1/4g.

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

Thus: UThus: U representrepresents s thethe physicsphysics of aof a classical 2D electrostaticclassical 2D electrostatic problem with the following ingredients:problem with the following ingredients:

|22|ln 8

1 ||ln

2

1

|2|ln )(2

1 ][

4

1

jjji

jj

jjj

j

ee

eeg

U

||ln ),( 212121 rrqqrrv


For fermion systems, can show that For fermion systems, can show that

- Orbitons are c- Orbitons are constrained to real axis, since s.p. energies all onstrained to real axis, since s.p. energies all real.real.

- Pairons - Pairons lie either on real axis or in complex conjugate pairs.lie either on real axis or in complex conjugate pairs.


Application to Nuclear PairingApplication to Nuclear Pairing

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

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

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

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


The tin isotopesThe tin isotopes

OrbitonOrbiton PositionPosition ChargeCharge

dd5/25/2 00 -1.5-1.5

gg7/27/2 0.440.44 -2.0-2.0

ss1/21/2 3.83.8 -0.5-0.5

dd3/23/2 4.44.4 -1.0-1.0

hh11/211/2 5.65.6 -3.0-3.0


114114Sn , 7 paironsSn , 7 pairons

Lines drawn to connect each Lines drawn to connect each pairon with its nearest pairon with its nearest neighbor.neighbor.

For weak pairing, pairons For weak pairing, pairons organize themselves as organize themselves as artificial atomsartificial atoms around around associated orbitons, subject to associated orbitons, subject to Pauli principle.Pauli principle.

Note:Note: Physical g Physical g ≈-0.095 MeV≈-0.095 MeV


114114Sn, Evolution with Sn, Evolution with gg

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


116116Sn , 8 paironsSn , 8 pairons


116116Sn , stronger pairingSn , stronger pairing

Two-stage transition to full superconductivity.Two-stage transition to full superconductivity.

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14-18 June 2005 International Conference on Finite Fermi Systems: Nilsson Model 50 years, Lund, Sweden Stuart Pittel Bartol Research Institute, University