Exactly-solvable Richardson-Gaudin models and

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Nov 15, 2013 (3 years and 1 month ago)

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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 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
their applications
*

14
-
18 June 2005




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

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.

Introductory Remarks and Outline

14
-
18 June 2005




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

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]

Summary of Recent Developments

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















m
l
a
m
lm
a
l
A
lm
a
m
lm
a
l
N
l
A
ll
l
A
g
l
l
N
l
H

and

ˆ
'
'
2
ˆ

Standard pure pairing hamiltonian:


Richardson’s solution of pure pairing model
(fermions)


Richardson ansatz for ground state (N pairs):















l
l
l
N
i
A
e
B
B




2
1

,

0

|

|
1


|
Ψ
> is an exact eigenstate of H if
pair energies

e
α

satisfy


2
/
1








,

0


4
2
2
1














l
e
e
g
e
g
l
l
l
l






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:

0


4
2
2
1

















e
e
g
e
g
l
l
l
14
-
18 June 2005




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

Consider
energy functional






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
?


|
2
2
|
ln

8
1


|
|
ln
2
1
|
2
|
ln

)
(
2
1


]
[
4
1
j
j
j
i
j
j
j
j
j
j
e
e
e
e
g
U



































|
|
ln


)
,
(
2
1
2
1
2
1
r
r
q
q
r
r
v







14
-
18 June 2005




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



A number of fixed charges (one for each active orbit) located at 2
ε
i


and with charges
Ω
i
/2. Called
orbitons.



N free charges located at
e
α

and with unit charge. Called
pairons
.



A Coulomb interaction between all charges.



A uniform electric field with strength 1/4g.

Reminder:

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



Thus: U

represent
s
the

physics

of a

classical 2D electrostatic

problem with the following ingredients:

|
2
2
|
ln

8
1


|
|
ln
2
1
|
2
|
ln

)
(
2
1


]
[
4
1
j
j
j
i
j
j
j
j
j
j
e
e
e
e
g
U



































|
|
ln


)
,
(
2
1
2
1
2
1
r
r
q
q
r
r
v







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

Orbiton

Position

Charge

d
5/2

0

-
1.5

g
7/2

0.44

-
2.0

s
1/2

3.8

-
0.5

d
3/2

4.4

-
1.0

h
11/2

5.6

-
3.0

14
-
18 June 2005




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

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


14
-
18 June 2005




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.