Symmetry vs. Chaos * in nuclear collective dynamics signatures & consequences Pavel Cejnar Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic ipnp.troja.mff.cuni.cz Kazimierz /15 * According to Empedocles (cca BC), the real world, Cosmos, is an interference of Sphairos, an exquisite world of perfect order originating in symmetry, and Chaos, a world of complete disorder which results from a lack of symmetry.

Invariant symmetry  commutation of Hamiltonian with generators of a certain group (group of invariant symmetry)  conservation laws 2/15 All kinds of dynamical symmetry relevant in physics of nuclear collective motions. These motions are therefore mostly thought to be regular... Symmetry Dynamical symmetry  symmetry of a particular system with respect to a dynamical group – a higher group than the one following from the invariance requirements  invariant symmetry with respect to the dynamical group can be broken, but a number of motion integrals remains preserved  dynamical symmetry integrability perfect order

Quantum chaos  no genuinely quantum definition of chaos (linearity & quasi periodicity of quantum mechanics)  chaos studied in connection with classical limit  Bohigas conjecture (1984): Chaos on quantum level affects statistical properties of discrete energy spectra. Chaotic systems yield spectral correlations consistent with Gaussian random matrix model. Classical chaos  exponential sensitivity to initial conditions (“butterfly wing effect”)  practical loss of predictability  quasi ergodic trajectories in the phase space Chaos ω=0.62 Nuclei show neat signatures of quantum chaos! data from neutron and proton resonances: Bohigas, Haq, Pandey (1983) ensemble of low-energy levels: Von Egidy et al. (1987) (Wigner) Nearest-neighbor spacing Brody distribution interpolates between Poisson (ω=0) … order Wigner (ω=1) … chaos 3/15

1) Single-particle dynamics Nucleonic motions in deformed nuclear potentials ? Origin of chaos in atomic nuclei 2) Collective dynamics Nuclear vibrations and rotations a) Interacting Boson Model (IBM) Iachello, Arima 1975 Alhassid, Whelan, Novoselsky : PRL 65, 2971 (1990); PRC 43, 2637 (1991); PRC 45, 1677 (1992); PRL 67, 816 (1991); NPA 556, 42 (1993) Paar, Vorkapic, Dieperink : PLB 205, 7 (1988); PRC 41, 2397 (1990), PRL 69, 2184 (1992) Mizusaki et al.: PLB 269, 6 (1991) Canetta, Maino : PLB 483, 55 (2000) Cejnar, Jolie, Macek, Casten, Dobeš, Stránský : PLB 420, 241 (1998); PRE 58, 387 (1998); PRL 93, (2004); PRC 75, (2007), PRC 80, (2009), PRC 82, (2010), PRL 105, (2010) b) Geometric Collective Model (GCM) Bohr 1952 Cejnar, Stránský, Kurian, Hruška : PRL 93, (2004); PRC 74, (2006); PRE 79, (2009), PRE 79, (2009), JP Conf.Ser. 239, (2010) a) Interacting Boson Model (IBM) Iachello, Arima 1975 Alhassid, Whelan, Novoselsky : PRL 65, 2971 (1990); PRC 43, 2637 (1991); PRC 45, 1677 (1992); PRL 67, 816 (1991); NPA 556, 42 (1993) Paar, Vorkapic, Dieperink : PLB 205, 7 (1988); PRC 41, 2397 (1990), PRL 69, 2184 (1992) Mizusaki et al.: PLB 269, 6 (1991) Canetta, Maino : PLB 483, 55 (2000) Cejnar, Jolie, Macek, Casten, Dobeš, Stránský : PLB 420, 241 (1998); PRE 58, 387 (1998); PRL 93, (2004); PRC 75, (2007), PRC 80, (2009), PRC 82, (2010), PRL 105, (2010) b) Geometric Collective Model (GCM) Bohr 1952 Cejnar, Stránský, Kurian, Hruška : PRL 93, (2004); PRC 74, (2006); PRE 79, (2009), PRE 79, (2009), JP Conf.Ser. 239, (2010) Arvieu, Brut, Carbonell, Touchard : PRA 35, 2389 (1987) Rozmej, Arvieu : NPA 545, C497 (1992) Heiss, Nazmitdinov, Radu : PRL 72, 2351 (1994); PRL 73, 1235 (1994); PRC 52, 3032 (1995) Arvieu, Brut, Carbonell, Touchard : PRA 35, 2389 (1987) Rozmej, Arvieu : NPA 545, C497 (1992) Heiss, Nazmitdinov, Radu : PRL 72, 2351 (1994); PRL 73, 1235 (1994); PRC 52, 3032 (1995) Many-body dynamics Complex interactions of all particles in the nucleus too difficult => two complementary simplifications: 4/15

Angular momentum → 0 Geometric Collective Model  effectively 2D system Principal Axes System Shape variables …corresponding tensor of momenta quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D ) Hamiltonian A. Bohr 1952 Gneuss et al neglect … 5/15

Geometric Collective Model neglect higher-order terms …corresponding tensor of momenta quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D ) Principal Axes System A. Bohr 1952 Gneuss et al Hamiltonian sphericalprolate oblate B A C >0 Shape-phase diagram Shape variables neglect … 5/15

independent scales energy timecoordinates External parameters … but a fixed value of Planck constant 1) “shape” parameter 2) “classicality” parameter path crossing all parabolas  all equivalent classes of Hamiltonians integrability Geometric Collective Model Two essential parameters B A prolate oblate spherical C >0 Hamiltonian Stránský, Cejnar… 2004 …… neglect … 6/15

Geometric Collective Model chaosorder Stránský, Cejnar… 2004 …… Classical chaos Regular phase space fraction map of the degree of chaos 7/15

Geometric Collective Model chaosorder Stránský, Cejnar… 2004 …… Classical chaos Regular phase space fraction map of the degree of chaos 7/15 x y x y x y x y convex concave convex concave change of the shape of the border of the accessible domain in the xy plane

Classical chaos J = 0, E = 0, A = −5.05, B = C = K = 1 50,000 passages of 52 randomly chosen trajectories through the section y=0 Geometric Collective Model x y x Stránský, Cejnar… 2004 …… Poincaré section examples of trajectories and a Poincaré section 8/15

J = 0J = 0 Two quantization options (a) 2D system (b) 5D system restricted to 2D (true geometric model of nuclei) The 2 options differ also in the metric (measure) for calculating matrix elements. ► Possibility to test Bohigas conjecture in different quantization schemes. with additional constraints (to avoid quasi-degeneracies due to the symmetry of V ) even / odd classicality parameter Geometric Collective Model Stránský, Cejnar… 2004 …… Quantum chaos 9/15

Geometric Collective Model Stránský, Cejnar… 2004 …… Quantum chaos comparison of classical and quantal measures f reg … classical regular fraction 1−ω … adjunct of Brody parameter 10/15

1) Quasi-2D angular momentum 2) Hamiltonian perturbation Choice of P Geometric Collective Model Stránský, Cejnar… 2004 …… Quantum chaos departure from integrable regime 2) 1) 11/15 Visual method by A. Peres (1984)

B=0.62 Geometric Collective Model Stránský, Cejnar… 2004 …… Quantum chaos 1) complex mixture of regular and chaotic patterns: ordered vs. chaotic states 2) 12/15 Visual method by A. Peres (1984)

■ ● ♦ B=0.62 Geometric Collective Model Stránský, Cejnar… 2004 …… Quantum chaos complex mixture of regular and chaotic patterns: ordered vs. chaotic states 1) 2) 12/15 Visual method by A. Peres (1984)

Macek, Cejnar, Jolie… /15 Interacting Boson Model More than 1 classical control parameter But there exist regions of almost full compatibility  multi-dimensional chaotic map. J = 0, E = 0 with the GCM.

Interacting Boson Model Macek, Dobeš, Cejnar… /15 Consequences of regularity for the adiabatic separation of intrinsic and collective motions: rotational bands exist even at very high excitation energies if the corresponding region of the J =0 spectrum is regular. Product of and correlation coefficients for intrinsic wave functions in the given “band” Energy ratio for 4 + and 2 + states in a given “band” Selection of hypothetical bands of rotational states based on the maximal correlation of the intrinsic SU(3) structures. Classical regular fraction in the respective energy region J N = 30 0 intrinsic wave functions for various band members in the SU(3) basis

GCM J = 0, E = 0 Conclusions Conclusions Nuclear collective motions exhibit an intricate interplay or regular and chaotic features (despite the presumption that collective = regular). Models of collective motions may serve as a laboratory for general studies of chaos (profit from the coexistence of simplicity and complexity). Order/Chaos have relevant nuclear-structure consequences (e.g. for the adiabatic separation of collective and intrinsic motions etc.). Thanks to Thanks to Pavel Stránský (now UNAM Mexico) Michal Macek (soon Uni Jerusalem) Conclusions Conclusions Nuclear collective motions exhibit an intricate interplay or regular and chaotic features (despite the presumption that collective = regular). Models of collective motions may serve as a laboratory for general studies of chaos (profit from the coexistence of simplicity and complexity). Order/Chaos have relevant nuclear-structure consequences (e.g. for the adiabatic separation of collective and intrinsic motions etc.). Thanks to Thanks to Pavel Stránský (now UNAM Mexico) Michal Macek (soon Uni Jerusalem) 15/15

Appendices

J=0, A=−0.84, B=C=K=1 Dependence on energy

GCM

GCM

Peres lattices A visual method to study quantum chaos in 2D systems due to Asher Peres, PRL 53, 1711 (1984) ► In any system there exists infinite number of integrals of motions: e.g. time averages of an arbitrary quantity along individual orbits (note: in fully or partly chaotic systems, these integrals do not allow one to build action-angle variables since they are strongly nonanalytic) ( ) ► One can construct a lattice: energy versus value of ► Quantum counterparts of such observables can be found: ► In a fully regular (integrable) system, the lattice is always ordered (the new integral of motion is constant on tori => it is a function of actions) ► In a chaotic system, the lattice is disordered ► In a mixed regular & chaotic system, the lattice is partly ordered & disordered regularmixedchaotic

Relation to the regular fractionGCM classicality parameter B=1.09 2D even