II. Spontaneous symmetry breaking

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

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

The molecular rotor Axial rotor 1 2 3

Born-Oppenheimer Approximation . Electronic motion Vibrations Rotations CO

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

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

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)

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

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

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

Rotating mean field (Cranking model): Reaction of the nucleons to the inertial forces must be taken into account 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) …….

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

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

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

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

Isotropy broken Isotropy conserved

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

Deformed?

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

II.3 Discrete symmetries Combinations of discrete operations

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

Rotational bands in

Chiral bands

Examples for chiral sister bands

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

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

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

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

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

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