1. Symmetries Galore “Not all is lost inside the triangle” (A. Leviatan, Seville, March, 2014) R. F. Casten Yale CERN, August, 2014

2. Themes and challenges of Modern Science Complexity out of simplicity -- Microscopic How the world, with all its apparent complexity and diversity can be constructed out of a few elementary building blocks and their interactions Simplicity out of complexity – Macroscopic How the world of complex systems can display such remarkable regularity and simplicity What is the force that binds nuclei? Why do nuclei do what they do? What are the simple patterns that nuclei display and what is their origin ? We will look at a model that lives in both camps

3. The macroscopic perspective (many-body quantal system) often exploits the idea of symmetries These describe the basic structure of the object. Geometrical symmetries describe the shape. Symmetry descriptions are usually analytic and parameter-free ( except for scale). (Scary group theory comment.) One model, the Interacting Boson Approximation (IBA) model, is expressed directly in terms of symmetries. Unfortunately, very few physical systems, especially in atomic nuclei, manifest a symmetry very well. We will see that this statement is now being greatly modified and symmetries may play a much larger role than heretofore.

4. 330 4+4+ 2+2+ 0 100 0+0+ E (keV) JπJπ Simple Observables - Even-Even Nuclei 1000 2+2+ Relative B(E2) values

5. E(J) Rot.  ( ħ 2 /2 I )J(J+1) R 4/2 = 3.33 Nuclei that are non-spherical can rotate (1952)! Rotational energies follow the quantum symmetric top: E(0) Rot.  0, E(2) Rot.  6, E(4) Rot.  20 Example of a geometrical symmetry – ellipsoidal, axially symmetric nuclei 0+0+ 2+2+ 4+4+ 6+6+ 8+8+

6. 6 + 690 4 + 330 0 + 0 2 + 100 J E (keV) ? Without rotor paradigm Paradigm Benchmark 700 333 100 0 Rotor ( ħ 2 /2 I ) J(J + 1) Amplifies structural differences Centrifugal stretching Deviations Identify additional degrees of freedom Value of paradigms

7. 0+0+ 2+2+ 4+4+ 6+6+ 8+8+ Rotational states Vibrational excitations Rotational states built on (superposed on) vibrational modes Ground or equilibrium state Rotor E(I)  ( ħ 2 /2 I )I(I+1) R 4/2 = 3.33 So, identification of the characteristic predictions of a symmetry both tells us the basic nature of the system and can highlight specific deviations from the perfect symmetry which can point to additional degrees of freedom. Real life example

8. The IBA – A collective model built on a highly truncated shell model foundation (only configurations with pairs of nucleons coupled to states with angular momentum 0 (s bosons) or 2 (d bosons) Embodies the finite number of valence nucleons. Like the shell model but opposed to traditional collective models, the predictions depend on N and Z

9. Shell Model Configurations Fermion configurations Boson configurations (by considering only configurations of pairs of fermions with J = 0 or 2.) s bosons and d bosons as the basic building blocks of the collective states The IBA Roughly, gazillions !! Need to simplify Huge truncation of the shell model

10. Symmetries of the IBA U(5) vibrator SU(3) rotor O(6) γ-soft U(6) Magical group theory stuff happens here Sph. Def. R 4/2 = 2.0 R 4/2 = 3.33 R 4/2 = 2.5 s and d bosons: 6-Dim. problem Three Dynamic symmetries, nuclear shapes IBA Symmetry Triangle What are these symmetries? Idealized structures whose predictions follow analytic formulas, with states labeled by good quantum numbers.

11. Proliferation of Symmetries ENTER PDS,QDS: Recent work (Leviatan, Van Isacker, Alhassid, Bonatsos, Cejnar, Pietralla, Cakirli, rfc,..): symmetries not limited to vertices: triangle permeated by symmetry elements: AoR, QDS, PDS AoR PDS QDS O(6) PDS – SU(3) QDS However, most nuclei do not exhibit the idealized symmetries: So, is their role just as benchmarks? Most nuclei lie inside the triangle where chaos and disorder are thought to reign.

12. PDS, QDS: what are these things? They are various situations in which some of the features of a Dyn.Sym. persist even though there is considerable symmetry-breaking. PDS: Some of the levels have the pure symmetry [such as SU(3)] and others are severely mixed QDS: Some of the degeneracies characteristic of a symmetry persist and some of the wave function correlations persist.

13. SU(3) O(3) Characteristic signatures: Degenerate bands within a group Vanishing B(E2) values between groups....  Typical SU(3) Scheme (for N valence nucleons)  vibrations What do real nuclei look like – what are the data??

14. Totally typical example Similar in many ways to SU(3). But note that the excited  vibrational excitations are not degenerate as they should be in SU(3). Also there are collective B(E2) values from the  band to the ground band. Most deformed rotors are not SU(3).  vibrations

15. Clearly, SU(3) is severely broken Or so we thought

16. B(E2) values in deformed rotor nuclei (typical values) 300 Wu 12 Wu 2 Wu  So, “clearly”, this violates the predictions of SU(3). So, realistic calculations have broken the symmetries (mixed their basis states). The degeneracies, quantum numbers, selection rules of the symmetry no longer apply. …. Or so we thought

17. What is an SU(3)-PDS? A bit weird!! It is based on the IBA SU(3) symmetry BUT breaks it while retaining pure SU(3) symmetry ONLY the ground and  bands which preserve SU(3) exactly. All other states are severely mixed! Why would we need such a thing? We have excellent fits to the data with numerical IBA calculations that break SU(3). Why would we think  band preserves SU(3)?

18. (  ) Partial Dynamical Symmetry (PDS) SU(3) So, expect PDS to predict vanishing B(E2) values between these bands as in SU(3). How can that possibly work since empirically these B(E2) values are collective !? BUT,  to ground B(E2)s CAN be finite in the PDS PDS: ONLY  and ground bands are pure SU(3).

19. SU(3) PDS :  B(E2) values , ground states pure SU(3), others mixed. Generalized T(E2) T(E2) PDS = e [ A (s + d + d + s) + B (d + d)] = e [T(E2) SU(3) + C (s + d + d + s) ] M(E2: J  – J gr ) eC -------------------- = ------------------------------------ eC cancels M(E2: J  – J ’ gr ) eC Relative INTERband  to gr B(E2)’s are parameter-free! First term gives zero from SU(3) selection rules. But not second. Hence, relative  to ground B(E2) values: So, Leviatan: PDS works well for 168 Er (Leviatan, PRL, 1996). Accidental or a new paradigm? Tried extensive test for 47 Rare earth nuclei. Key data will be relative  to ground B(E2) values.

20. Illustrative results

21. Transitional nuclei

22. 5:100:70 Alaga Lets look into these predictions and comparisons a little deeper. Compare to “Alaga Rules” – what you would get for a pure rotor for RATIOS of squares of transition matrix elements [B(E2) values] from one rotational band to another.

23. 168-Er: The Alaga rules, valence space, and collectivity PDS has pure , gr band K, so why differ from Alaga? Ans: PDS (from IBA) is valence space model: predictions are N val – dep. PDS differs from Alaga solely due to N-Dep effects that arise from the fermion underpinning (Pauli Principle). Valence collective models that agree with the PDS have minimal mixing. Predictions deviating from the PDS signal configuration mixing.

24. Using the PDS to better understand collective model calculations (and collectivity in nuclei) PDS B(E2:  – gr): the sole reason they differ from the Alaga rules is that they take into account the finite number of valence nucleons. Therefore, IBA calculations (like WCD) that agree with the PDS differ from the Alaga rules purely because of finite valence nucleon number effects. Not heretofore recognized. IBA – CQF deviates further from the Alaga rules, agrees better with the data (and has one fewer parameter). IBA – CQF: The differences from the PDS are due to mixing Can use the PDS to disentangle valence space from mixing!

25. Again, as one enters the triangle expect vertex symmetries to be highly broken. Consider structures descending from O(6) Now a highly selective O(6) PDS

26. 

27. Calculate wave functions throughout triangle, expand in O(6) basis   good along a line descending from O(6) Remnants of pure O(6) (for the ground state band only) persist along a line in the triangle extending into the region of well-deformed rotational nuclei

28. Order, Chaos, and the Arc of regularity

29. Whelan, Alhassid, ca 1989 Order and chaos? What happens generally inside the triangle Arc of Regularity is a narrow zone in the triangle with high degree of order amidst regions of chaotic behavior. What is going on along the AoR? 6 +, 4 +, 3 +, 2 +, 0 + 4 +, 2 +, 0 + 2 + 0 + 3 2 1 0 n d

30. Role of the arc as a “boundary” between two classes of structures  AoR – E(0 + 2 ) ~ E(2 + 2 ) Not just a theoretical curiosity: 8 nuclei in the rare earth region have been found to lie along the arc

31. The symmetry underlying the Arc of Regularity is an SU(3)-based Quasi Dynamical Symmetry QDS All SU(3) degeneracies and all analytic ratios of 0 + bandhead energies persist along the Arc.

32. Summary AoR PDS QDS O(6) PDS – SU(3) QDS Elements of structural symmetries abound throughout the triangle

33. Principal Collaborators: R. Burcu Cakirli D. Bonatsos Klaus Blaum Aaron Couture Thanks to Ami Leviatan, Piet Van Isacker, Michal Macek, Norbert Pietralla, C. Kremer for discussions.

34. BACKUPS

35. States labelled by quantum numbers Degenerate bands within irrep (  ) T(E2) = eQ = e[(s † d + d † s) - (d † d ) (2) ] (  ) SU(3)

37. Creation and destruction operators as Example: Consider the case we have just discussed – the spherical vibrator. 0 2 4,2,0 0 E 2E 1 2 Why is the B(E2: 4 – 2) = 2 x B(E2: 2– 0) ?? Difficult to see with Shell Model wave functions with 1000’s of components However, as we have seen, it is trivial using destruction operators WITHOUT EVER KNOWING ANYTHING ABOUT THE DETAILED STRUCTURE OF THESE VIBRATIONS !!!! These operators give the relationships between states. “Ignorance operators”

39. Finite Valence Space effects on Collectivity Note: 5 N’s up, 5 N’s down: 1 with N Alaga Most dramatic, direct evidence for explicit valence space size in emerging collectivity PDS B(E2: g to Gr) R.F. Casten, D.D. Warner and A. Aprahamian, Phys. Rev. C 28, 894 (1983).

41. ~ 0.03 ~ 0.1 x ~0.03 ~ 0.003

42. Concepts of group theory Generators, Casimirs, Representations, conserved quantum numbers, degeneracy splitting Generators of a group: Set of operators, O i that close on commutation. [ O i, O j ] = O i O j - O j O i = O k i.e., their commutator gives back 0 or a member of the set For IBA, the 36 operators s † s, d † s, s † d, d † d are generators of the group U(6). Generators : define and conserve some quantum number. Ex.: 36 Ops of IBA all conserve total boson number = n s + n d N = s † s + d † Casimir: Operator that commutes with all the generators of a group. Therefore, its eigenstates have a specific value of the q.# of that group. The energies are defined solely in terms of that q. #. N is Casimir of U(6). Representations of a group: The set of degenerate states with that value of the q. #. A Hamiltonian written solely in terms of Casimirs can be solved analytically

43. Let’s illustrate group chains and degeneracy- breaking. Consider a Hamiltonian that is a function ONLY of: s † s + d † d That is: H = a(s † s + d † d) = a (n s + n d ) = aN In H, the energies depend ONLY on the total number of bosons, that is, on the total number of valence nucleons. ALL the states with a given N are degenerate. That is, since a given nucleus has a given number of bosons, if H were the total Hamiltonian, then all the levels of the nucleus would be degenerate. This is not very realistic (!!!) and suggests that we should add more terms to the Hamiltonian. I use this example though to illustrate the idea of successive steps of degeneracy breaking being related to different groups and the quantum numbers they conserve. The states with given N are a “representation” of the group U(6) with the quantum number N. U(6) has OTHER representations, corresponding to OTHER values of N, but THOSE states are in DIFFERENT NUCLEI (numbers of valence nucleons).

44. H’ = H + b d † d = aN + b n d Now, add a term to this Hamiltonian: Now the energies depend not only on N but also on n d States of a given n d are now degenerate. They are “representations” of the group U(5). States with different n d are not degenerate

45. N N + 1 N + 2 ndnd 1 2 0 a 2a E 00 b 2b H’ = aN + b d † d = a N + b n d U(6) U(5) H’ = aN + b d † d Etc. with further terms

46. Concept of a Dynamical Symmetry N Spectrum generating algebra !! Each successive term: Introduces a new sub-group A new quantum number to label the states described by that group Adds an eigenvalue term that is a function of the new quantum number, hence Breaks a previous degeneracy

47. Now, WHY are spin increasing transitions so small in Alaga rules and so much larger in the data than in the PDS? 0 2 4 22 J int, R J tot J int, ~R 0, 0 0, 2 0, 4 2, 0  R = 4 !! ~ Forbidden Deviations from Alaga:  – gr. band mixing. E2 matrix element : ME(E2) = [  unpert +  mix ] Can be larger or smaller than the unperturbed. BUT, if unperturbed is forbidden, then ME(E2) = [  mix ] > unpert (which is zero). So mixing always increases forbidden/weak transitions.

48. Review of phonon creation and destruction operators is a b-phonon number operator. For the IBA a boson is the same as a phonon – think of it as a collective excitation with ang. mom. 0 (s) or 2 (d). What is a creation operator? Why useful? A)Bookkeeping – makes calculations very simple. B) “Ignorance operator”: We don’t know the structure of a phonon but, for many predictions, we don’t need to know its microscopic basis.