Interacting boson model s-bosons (l=0) d-bosons (l=2) Interpretation: “nucleon pairs with l = 0, 2” “quanta of collective excitations” Dynamical algebra:

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Interacting boson model s-bosons (l=0) d-bosons (l=2) Interpretation: “nucleon pairs with l = 0, 2” “quanta of collective excitations” Dynamical algebra: U(6) Subalgebras: U(5), O(6), O(5), O(3), SU(3), [O(6), SU(3)] …generators:…conserves: F Iachello, A Arima (1975) Dynamical symmetries (extension of standard, invariant symmetries) : U(5) O(6) SU(3) [O(6), SU(3)] See eg.: F.Iachello, A. Arima : The Interacting Boson Model, Cambridge University Press, 1987

D Warner, Nature 420, 614 (2002). triangle(s) Parameter space of the model simplest version (IBM-1) can be imaged as the surface of a “symmetry pyramide”. Corresponding points in various triangles are connected by similarity transformations (parameter symmetries), so it is sufficient to investigate dynamics in one of the triangles.

Simplified Hamiltonian d-boson number operator quadrupole operator scaling constant ħω=1 MeV control parameters η, χ symmetry triangle ensures that the thermodynamic limit exists: N→ ∞

1 st order 2 nd order Phase diagram critical exponent 1 st order obtained from variational procedure based on condensate-type ground-state wave functions order parameter: β=0 spherical, β>0 prolate, β<0 oblate. I II III

d-boson number quadrupole operator The Hamiltonian with s- and d- bosons: Classical Limit obtained by Glauber coherent states: If restricted to L=0 states, Hamiltonian is solely a function of quadrupole deformation parameters β – γ (in the intrinsic frame) -> 2D system Classical limit of IBM1 scaling constant a = N/10 MeV

Classical potential in case of prolate deformation section through the plane y = 0 Particularly important values of energy: x y x y γ

Chaos within the Triangle Standard classical measures of chaos: Lyapounov exponents λ D(t)... separation of two neighbouring trajectories at time t Fraction of chaotic phase space volume σ New highly regular arc discovered [Alhassid,Whelan PRL 67 (1991) 816] using both measures : chaotic volume σ and average maximal Lyapounov exponent λ. Fraction of chaotic phase space σ at two values of angular momentum. The arc is clearly visible in both pictures. adapted from Alhassid,Whelan PRL 67 (1991) 816

Chaos within the Triangle Standard quantum measures of chaos: Brody parameter ω (distribution of nearest neighbor level separation S): Distribution of B(E2) strengths (Porter-Thomas distribution) (Alhassid,Whelan) Δ 3 statistics (long range spectral correlations) (Alhassid,Whelan) Wave function entropy (localisation in dynamical-symmetry bases) (Cejnar,Jolie) interpolates between Poisson (ω=0, regular dynamics) and Wigner distribution (ω=1, chaos)