Partial dynamical symmetries in Bose-Fermi systems* Jan Jolie, Institute for Nuclear Physics, University of Cologne

What are dynamical symmetries?Illustration with the interacting boson model.

What are partial dynamical symmetries?Illustration with the interacting boson model.

What are dynamical Bose-Fermi symmetries?Illustration with the interacting boson fermion model.

How to extend partial dynamical symmetries to Bose-Fermi systems.

Application to 195Pt.

Work done with Piet Van Isacker, Tim Thomas and Ami Leviatan,

*: work supported by BriX during a one month sabbatical stay at GANIL in June 2013

Dynamical symmetries

Hamiltonian EH

Operators {Gi}

Lie Algebra G

[gi,gj]=kcijkgk.

{gm} {gl} {gk} {gi}

Casimir Operator

[Cn[G] ,gk]= gk ij

jii

i gggH

H = i ai Cn[Gi]E = a f() =>Ek = a f() kwithk {gk} E( = i ai f(n,Gi)

Gm Gl Gk Gi

H = a Cn[G]

Example: Angular momentum algebra

Hamiltonian EH

Operators {Gi}

Lie Algebra G

Casimir Operator ij

jii

i GGGH

H = a L2 + bLz

E = a f(L) =>Ek = a L(L+1) kwithk {L+L-Lz}

E(LM = a L(L+1) + b M

O(2) O(3)

H = a L2

zLLL ,, O(3)

)3(,0,2 OLLL ii

}{ iz LL

L

-J

0

+J

M

fermions cj

N bosons

A nucleons

valence nucleons

N nucleon pairs

L = 0 and 2 pairss,d

even-even nuclei

The Interacting Boson Model

A. Arima, F. Iachello, T. Otsuka, I Talmi

Schrödinger equation in second quantisation

N s,d boson system

with

N=cte

U(5)

SU(3)

SO(6)196Pt

156Gd

110Cd

U(5): Vibrational nuclei

SO(6): -unstable nuclei

SU(3): Rotational nuclei (prolate)

Dynamical symmetries of a N s,d boson system

U(5) SO(5) SO(3) SO(2) {nd} () L M

U(6) SO(6) SO(5) SO(3) SO(2)[N] <> () L M SU(3) SO(3) SO(2)

() L M

Partial dynamical symmetries

Dynamical symmetries lead to:

1.Solvability of the complete spectrum.2.Existence of exact quantum numbers for all states.3.Pre-determined structure of the wavefunctions independent of the used parameters.

H = (1-)1C1[U(5)]+3C2[SO(6) ] +4C2[SO(5) ] + 5C2[ SO(3)]

Example of case 2:

SO(5) SO(3) SO(2)() L M

All states still have:

We want to relax these conditions such that:1. Only some states keep all quantum numbers.2. All eigenstates keep some quantum numbers.3. Some eigenstates keep some quantum numbers.

System exhibiting one of the three conditions have a partial dynamical symmetry.

Projection of a O(6) and SU(3) wavefunction on a U(5) basis

Due to SO(5) only even or odd d-bosonsin a given state with v is even (odd) occur

Example of case 1: Partial dynamical symmetry within the SO(6) limit

SO(6) limit for 6 bosons

}

=n=6=4

=2

Step 1: construct operatorshaving a definite tensorial character under all groups

Ex: n=2

Step 2 chose an interaction Vpds such that :

for certain states because the by the irrep coupling allowed final states do not exist.

Example:

with the boson pairing operator

but

since

and does not exist.

Note: and this is true for

all other states with <n

2

..

ˆ3

1~ˆ 0)0}(0]{2[0)0}(0]{2[

ddssP

PnPBnBV sspds

0)}(2]{[~

)}(]{2[

}{}0{}{

0)}(]{[~

0)0}(0]{2[

0)0}(0]{2[

LvnnB

Lvnn

nn

LvnnB

0)}(]{[ LvnVpds

H = aP+nsP-+3C2[SO(6) ] +4C2[SO(5) ] + 5C2[ SO(3)]

3= -42.25 keV4 = 45.0 keV 5= 25.0 keVa= 0 keV

3= -29.5 keV4 = 45.0 keV 5= 25.0 keVa= 34.9 keV

J.E. Garcia-Ramos, P. van Isacker, A. Leviatan, Phys. Rev. Lett. 102 (2009) 112502

odd-oddnuclei

s,d,aj

N bosons1 fermion

e-o nucleus

s,d

N+1 bosons

IBFA

e-e nucleus

fermions cj

M valence nucleons

A nucleons

L = 0 and 2 pairs

nucleon pairs

Even-even nuclei: the interacting boson approximationOdd-A nuclei: the interacting boson-fermion approximation

N s,d bosons+ single j fermion: UB(6)xUF(2j+1)

jj aa

bb

0

0

36 boson generators + (2j+1)2 fermion generators, which both couple to integer total spin and fullfil the standard Lie algebra conditions.

Bose-Fermi symmetries

Two types of Bose-Fermi symmetries: spinor and pseudo spin types

Spinor type: uses isomorphism between bosonic and fermionic groupsSpin(3): SOB(3) ~ SUF(2)Spin(5): SOB(5) ~ SpF(4)Spin(6): SOB(6) ~ SUF(4)

Pseudo Spin type: uses pseudo-spins to couple bosonic and fermionic groups

Example: j= 1/2, 3/2, 5/2 for a fermion: P. Van Isacker, A. Frank, H.Z. Sun, Ann. Phys. A370 (1981) 284.

L=2

L=0 1/2

3/25/2 L=2

L=0x x

L´=0

L´=2x S´= 1/2

UB(6) x UF(12) UB(6) x UF(6) x UF(2)

UB+F(5)xUF(2)...

UB(6)xUF(12) UB(6)xUF(6)xUF(2) UB+F(6)xUF(2) SUB+F(3)xUF(2)... SOB+F(3)xUF(2) Spin (3) SOB+F(6)xUF(2)...

H= AC2[UB+F(6)] + A C1[UB+F(5)] + A´ C2[UB+F(5)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SUB+F(3)] + E C2[SOB+F(3)] + F Spin(3)

This hamiltonian has analytic solutions, but also describes transitional situations.

Example: the SO(6) limit of UB(6)xUF(12)

H= A C2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SOB+F(3)] + E Spin(3)

E= A(N1(N1 +5)+ N2(N2 +3)) + B(1(1 +4)+ 2(2 +2)) + C(1(1 +3)+ 2(2 +1))+D L(L+1) + E J(J+1)

Result for 195Pt

A =46.7, B = -42.2 C= 52.3, D = 5.6 E = 3.4 (keV)A. Metz, Y. Eisermann, A. Gollwitzer, R. Hertenberger, B.D. Valnion, G. Graw,J. Jolie, Phys. Rev. C61 (2000) 064313

wavefunction

We consider now a system with 1 boson and 1 fermion. They can form[N1,N2] states with [2,0] and [1,1]. The [2,0]<0,0> and [1,1] states are:

Using them we can now construct operators which have a tensorialcharacter under all groups in the group chain

)0()()( )(12ˆ JLJLJLL UUJU

We now construct two body interactions:

which have the property that:

0,),0,(,0,],0,1[ˆ,),0,(,0,],0,1[ 1111 JMLNUJMLN JLL

Because in the N-1 boson the highest representation is [N-1] and

[N-1]x[1,1] = [N,1]+[N-1,1,1].

Note that:

0,),,(,,],1,[ˆ,),,(,,],1,[ 21212121 JMLNUJMLN JLL

P. Van Isacker, J. Jolie, T. Thomas, A. Leviatan, subm to Phys. Rev. Lett.

H= A C2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + DC2[SOB+F(3)] + ESpin(3)

)ˆˆ()ˆˆ()ˆˆ(ˆ 2/533

2/33333

2/522

2/32222

2/311

2/11111

2/100 UUaUUaUUaVa

A=37.7, B=-41.5, C =49.1, D= 1.7, E= 5.6, a0=306, a11=10., a22=-97., a33= 112

Conclusion

For the first time the concept of partial dynamical symmetries was applied to a mixed system of bosons and fermions.

Using the partial dynamical symmetries part of the states keep the original symmetry, while other loose it.

The description of 195Pt could be improved.

Thanks for your attention.