1. II. Spontaneous symmetry breaking

2. II.1 Weinberg’s chair Hamiltonian rotational invariant Why do we see the chair shape? States of different IM are so dense that the tiniest interaction With the surroundings generates a wave packet that is well oriented. Spontaneously broken symmetry

3. Tiniest external fields generate a superposition of the |JM> that is oriented in space, which is stable. Spontaneous symmetry breaking Macroscopic (“infinite”) system

4. The molecular rotor 1 2 3 Axial rotor

6. .. Born-Oppenheimer Approximation Electronic motion Vibrations Rotations CO

7. Microscopic (“finite system”) Rotational levels become observable. Spontaneous symmetry breaking = Appearance of rotational bands. Energy scale of rotational levels in

8. HCl Microwave absorption spectrum Rotational bands are the manifestation of spontaneous symmetry breaking.

9. II.2 The collective model Most nuclei have a deformed axial shape. The nucleus rotates as a whole. (collective degrees of freedom) The nucleons move independently inside the deformed potential (intrinsic degrees of freedom) The nucleonic motion is much faster than the rotation (adiabatic approximation)

10. Nucleons are indistinguishable The nucleus does not have an orientation degree of freedom with respect to the symmetry axis. Axial symmetry

11. Single particle and collective degrees of freedom become entangled at high spin and low deformation. Limitations: Rotational bands in Adiabatic regime Collective model

12. II.3 Microscopic approach: Retains the simple picture of an anisotropic object going round. Mean field theory + concept of spontaneous symmetry breaking for interpretation.

13. Rotating mean field (Cranking model): Start from the Hamiltonian in a rotating frame Mean field approximation: find state |> of (quasi) nucleons moving independently in mean field generated by all nucleons. Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) ……. Reaction of the nucleons to the inertial forces must be taken into account

14. Low spin: simple droplet. High spin: clockwork of gyroscopes. Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries Rotational response Mean field theory: Tilted Axis Cranking TAC S. Frauendorf Nuclear Physics A557, 259c (1993) Quantization of single particle motion determines relation J(  ).

15. Spontaneous symmetry breaking Symmetry operation S and Full two-body Hamiltonian H’ Mean field approximation Mean field Hamiltonian h’ and m.f. state h’|>=e’|>. Symmetry restoration Spontaneous symmetry breaking

16. Which symmetries can be broken? Combinations of discrete operations is invariant under Broken by m.f. rotational bands Obeyed by m.f. spin parity sequence broken by m.f. doubling of states

17. Rotational degree of freedom and rotational bands. Deformed charge distribution nucleons on high-j orbits specify orientation

18. Isotropy broken Isotropy conserved

19. Current in rotating Lab frameBody fixed frame J. Fleckner et al. Nucl. Phys. A339, 227 (1980) Moments of inertia reflect the complex flow. No simple formula.

20. Deformed?

21. Rotor composed of current loops, which specify the orientation. Orientation specified by the magnetic dipole moment. Magnetic rotation.

22. II.3 Discrete symmetries Combinations of discrete operations

23. Common bands PAC solutions (Principal Axis Cranking) TAC solutions (planar) (Tilted Axis Cranking) Many cases of strongly broken symmetry, i.e. no signature splitting

24. Rotational bands in

25. Chiral bands

26. Examples for chiral sister bands

27. Chirality mirror It is impossible to transform one configuration into the other by rotation.

28. mirror mass-less particles Only left-handed neutrinos: Parity violation in weak interaction

29. Reflection asymmetric shapes, two reflection planes Simplex quantum number Parity doubling

31. II.4 Spontaneous breaking of isospin symmetry Form a condensate “isovector pair field”

32. The relative strengths of pp, nn, and pn pairing are determined by the isospin symmetry

33. Symmetry restoration –Isorotations (strong symmetry breaking – collective model)