II. Spontaneous symmetry breaking

II.1 Weinberg’s chairHamiltonian rotational invariant

)( weight the

withnsorientatio allover averagean

ison that distributidensity a have

IM| :momentumangular

good of seigenstate

IMKD

Why do we see the chair shape?

States of different IM are so dense that the tiniest interactionWith the surroundings generates a wave packet that is well oriented.

IM

IM IMaca ||Spontaneously broken symmetry

momentumangular

]s m kg[eV10~eV10~ levels rotational of distanceenergy

eV10~ levels rotational of scaleenergy

1-215-49-2

49-2

J

JJJ

Tiniest external fields generate a superposition of the |JM>that is oriented in space, which is stable.

Spontaneous symmetry breakingMacroscopic (“infinite”) system

The molecular rotor

3NH

1

2

3 21 Axial rotor

3

23

1

23

2

2

1 JJJH

3

2

1

2)1(

2

1 KKIIE

0],[0],[0],[ 23 JHJHJH z

3

23

1

22

21

2

1 JJJH

aKMI |,,| :seigenstate

function Wigner D iKIMK

iMIMK edeD )(),,(

),,(8

12,,|,, :rotor ofn orientatiofor

amplitudey probabilit2/1

2

I

MKDI

KMI

symmetry. rotational breaksly spontanousthat

structure intrinsic"" thedescibes | a

.

.

Born-Oppenheimer Approximation

Electronic motion

Vibrations

Rotations eVrot410~

eVel 1~

CO

eVvib110~

Microscopic (“finite system”)

Rotational levels become observable.

eV 10 :scale intrinsiceV10~ :molecules 1-6-2

Spontaneous symmetry breaking=

Appearance of rotational bands.

Energy scale of rotational levels in

HCl

)1()()1(

)1()(

IBIEIE

JIIBIIE

Microwave absorptionspectrum

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 independentlyinside the deformed potential (intrinsic degrees of freedom)

The nucleonic motion is much fasterthan the rotation (adiabatic approximation)

Nucleons are indistinguishable

),,(),,()( rotKrotin

rotin

x

EEE

2

)1( 2KIIEE in

The nucleus does not have an orientation degree of freedomwith respect to the symmetry axis.

03

2

K

Axial symmetryin

iKin e )(3R

K

2/1

2),,(

8

12

IMKD

I

Single particle and collective degrees of freedom become entangled at high spin and low deformation.

Limitations:scale intrinsic~MeV10~ :scaleenergy rotational 1-

2

Rotationalbands in

Er163

Adiabatic regimeCollective model

II.3 Microscopic approach:

Retains the simple picture of an anisotropic object going round.

Mean field theory + concept of spontaneous symmetry breaking for interpretation.

Rotating mean field (Cranking model):

Start from the Hamiltonian in a rotating frame

zjvtH 12'momentumangular

ninteractiobody - twoeffective

energy kinetic

12

zj

v

t

Mean field approximation:find state |> of (quasi) nucleons moving independently inmean field generated by all nucleons.mfV

(routhian) frame rotating thein nhamiltonia field mean '

},| { :tencyselfconsis , -' , |'|' 12

h

VvJVtheh mfzmf

Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) …….

Reaction of thenucleons to the inertial forces must be taken into account

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 TACS. Frauendorf Nuclear Physics A557, 259c (1993)

Quantization of single particlemotion determines relation J().

Spontaneous symmetry breaking

Symmetry operation S and

.|'|'|'

energy same the withsolutions field mean are states All

1||| and ,''

HHE

hh

|SS

|S

|SSS

Full two-body Hamiltonian H’ Mean field approximation

Mean field Hamiltonian h’ and m.f. state h’|>=e’|>.

Symmetry restoration |Siic

'' HH SS

Spontaneous symmetry breaking

Which symmetries can be broken?

Combinations of discrete operations

rotation withreversal time- )(

inversion space-

angleby axis-zabout rotation - )(

y

z

TR

P

R

zJHH ' is invariant under

axis-zabout rotation - )(zRBroken by m.f. rotational

bands

Obeyed by m.f.spinparitysequence

broken by m.f.doublingofstates

zmf jVth '

zJiz e )( axis-z about the Rotation R

peaked.sharply is 1|||

.''but ''

|R

RRRR

z

zzzz hhHH

Rotational degree of freedom and rotational bands.

Deformed charge distribution

nucleons on high-j orbitsspecify orientation

.|2

1II|momentumangular good of State

.energy same thehave )(| nsorientatio All

deiI

z |R

deformed

Er163

spherical

Pb200

Isotropybroken

Isotropyconserved

Current in rotating Yb162

Lab frame Body fixed frame

J. Fleckner et al. Nucl. Phys. A339, 227 (1980)

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.

.energy same thehave )(| nsorientatio All

peaked.sharply is 1|||

.''but ''

|R

|R

RRRR

z

z

zzzz hhHH

II.3 Discrete symmetries

Combinations of discrete operations

rotation withreversal time- )(

inversion space-

angleby axis-zabout rotation - )(

y

z

TR

P

R

Common bands

by axis-zabout rotation - )(

rotation withreversal time- 1 )(

inversion space - 1

z

y

R

TR

P

PAC solutions(Principal Axis Cranking)

nI

e iz

2

signature ||)(

R

TAC solutions (planar)(Tilted Axis Cranking) Many cases of strongly brokensymmetry, i.e. no signature splitting

Rotationalbands in

Er163

Chiral bands

Examples for chiral sister bands

7513459 Pr 1

2/112/11hh

5910445 Rh 2/11

12/9 hg

7513560 Nd 1

2/112

2/11hh

Chirality

mirror

It is impossible to transform one configurationinto the other by rotation.

mirror

mass-less particles

Only left-handed neutrinos:Parity violation in weak interaction

Reflection asymmetric shapes,

two reflection planes

Simplex quantum number

I

i

z

parity

e

)(

||

)(

S

PRS

Parity doubling

Th226

01

10

TS

TS

II.4 Spontaneous breaking of isospin symmetry

Form a condensate“isovector pair field”

02

ˆ

0

ˆ

np

ppnn

np

ppnn

y

z

The relative strengths of pp, nn, and pn

pairing are determined by the isospin symmetry

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

2

)1( :energy nalisorotatio

|)0,,( :state nalisorotatio

| :state intrinsic

intrinsic

0

TTH)E(T,T

D

z

TTz

A

MeVTTTTE z

75

2

1,

2

)1()(

:alExperiment

exp