Partial dynamical symmetries in Bose-Fermi systems*

1. Partial dynamical symmetries in Bose-Fermi systems* Jan Jolie, Institute for Nuclear Physics, University of Cologne Work done with Piet Van Isacker, Tim Thomas and Ami Leviatan, 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 supported by BriX during a one month sabbatical stay at GANIL in June 2013

2. Dynamicalsymmetries Hamiltonian Lie AlgebraG [gi,gj]=Skcijkgk. Operators {Gi} Casimir Operator [Cn[G] ,gk]=0gk H = a Cn[G] {gm} {gl}  {gk}  {gi} EY = a f(a) Y => EYk = a f(a) Yk withYk {gkY} GmGlGkGi H = Si aiCn[Gi] E(abg) = Si aifabg(n,Gi)

3. Example: Angular momentum algebra +J L 0 -J M Hamiltonian Lie AlgebraG O(3) Operators {Gi} Casimir Operator H = a L2 EY = a f(L) Y => EYk = a L(L+1) Yk withYk {L+Y, L-Y LzY} O(2)O(3) H = a L2 + bLz E(LM) = a L(L+1) + b M

4. fermions cj N nucleon pairs valence nucleons IBA s,d 1974-1979 A. Arima, F. Iachello, T. Otsuka, I Talmi The Interacting Boson Model even-even nuclei A nucleons N bosons L = 0 and 2 pairs

5. Schrödinger equation in second quantisation N=cte N s,d boson system with

6. Dynamical symmetries of a N s,d boson system  U(5) SO(5)  SO(3)  SO(2) {nd} (t) L M U(6)  SO(6)  SO(5)  SO(3)  SO(2) [N] <s> (t) L M  SU(3)  SO(3)  SO(2) (l,m) L M 110Cd U(5) SU(3) 156Gd 196Pt SO(6) U(5): Vibrational nuclei SO(6): g-unstable nuclei SU(3): Rotational nuclei (prolate)

7. Partial dynamical symmetries • Dynamicalsymmetriesleadto: • Solvabilityofthecompletespectrum. • Existenceofexactquantumnumbersfor all states. • Pre-determinedstructureofthewavefunctionsindependentoftheusedparameters. We want to relax these conditions such that: • Only some states keep all quantum numbers. • All eigenstates keep some quantum numbers. • Some eigenstates keep some quantum numbers. System exhibiting one of the three conditions have a partial dynamical symmetry. Example of case 2: H =(1-a)k1C1[U(5)]+a k3C2[SO(6) ] + k4C2[SO(5) ] + k5C2[ SO(3)] SO(5)  SO(3)  SO(2) (t) L M All states still have:

8. Projection of a O(6) and SU(3) wavefunction on a U(5) basis Due to SO(5) only even or odd d-bosons in a given state with v is even (odd) occur

9. Example of case 1: Partial dynamical symmetry within the SO(6) limit SO(6) limit for 6 bosons Step 1: construct operators having a definite tensorial character under all groups Ex: n=2 } s=2 s=4 s=n=6

10. 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 s<n

11. H =aP+nsP-+ k3C2[SO(6) ] + k4C2[SO(5) ] + k5C2[ SO(3)] k3= -42.25 keV k4 = 45.0 keV k5= 25.0 keV a= 0 keV k3= -29.5 keV k4 = 45.0 keV k5= 25.0 keV a= 34.9 keV J.E. Garcia-Ramos, P. van Isacker, A. Leviatan, Phys. Rev. Lett. 102 (2009) 112502

12. N bosons 1 fermion N+1 bosons s,d s,d,aj IBA IBFA e-e nucleus e-o nucleus Even-even nuclei: the interacting boson approximation Odd-A nuclei: the interacting boson-fermion approximation fermions cj odd-odd nuclei Nukleonen A nucleons M valence nucleons nucleon pairs L = 0 and 2 pairs

13. N s,d bosons+ single j fermion: UB(6)xUF(2j+1) 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 groups Spin(3): SOB(3) ~ SUF(2) Spin(5): SOB(5) ~ SpF(4) Spin(6): SOB(6) ~ SUF(4)

14. 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. 5/2 L=2 L=2 L´=2 3/2 x x x S´= 1/2 L=0 1/2 L=0 L´=0 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.

15. 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(s1(s1 +4)+ s2(s2 +2)) + C(t1(t1 +3)+ t2(t2 +1)) +D L(L+1) + E J(J+1)

16. 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

17. 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: wavefunction Using them we can now construct operators which have a tensorial character under all groups in the group chain

18. We now construct two body interactions: which have the property that: 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:

19. H= AC2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + DC2[SOB+F(3)] + ESpin(3) A=37.7, B=-41.5, C =49.1, D= 1.7, E= 5.6, a0=306, a11=10., a22=-97., a33= 112 P. Van Isacker, J. Jolie, T. Thomas, A. Leviatan, subm to Phys. Rev. Lett.

20. 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.