We employ a realistic nuclear mean-field theory using the phenomenological, Woods-Saxon Hamiltonian with newly adjusted parameters containing no parametric correlations; originally present correlations are removed employing the Monte Carlo approach. We find very large neutron shell gaps at $N=136$ for all the four octupole deformations ${\alpha }_{3\mu =0,1,2,3}$. These shell gaps generate well-pronounced double potential-energy minima in the standard multipole $({\alpha }_{20},{\alpha }_{22},{\alpha }_{3\mu },{\alpha }_{40})$ representation, often at ${\alpha }_{20}=0$, which in turn generate exotic symmetries $C_{2v}$, $D_{2d}$, $T_d$, and $D_{3h}$, discussed in detail. The main goal of the article is to formulate spectroscopic criteria for experimental identification. Calculations employing macroscopic-microscopic method are performed for nuclei with $Z\ge 82$ and $N\ge 126$ in multidimensional deformation spaces to analyze the expected exotic symmetries and octupole shape instabilities in the mass table “northeast” of the doubly magic ${}^{208}\mathrm{Pb}$ nucleus. Whereas the proton-unperturbed properties of neutron-generated octupole shell effects are illustrated in detail for exotic ${}_{Z=82}{\mathrm{Pb}}_{N>126}$ nuclei, our discussion is extended into even-even $Z>82$ nuclei approaching the less exotic $Z/N$ ratios, to encourage experiments which could identify the predicted exotic symmetries. In addition to the tetrahedral point group symmetry, $T_d$, of which experimental evidence has recently been published, we present $D_{2d}$ symmetry resulting from a superposition of axially symmetric quadrupole and tetrahedral symmetries and two new point group symmetries, $D_{3h}$ and $C_{2v}$, associated with the octupole ${\alpha }_{33}$ and ${\alpha }_{31}$ energy minima, respectively. The multidimensional $n>2$ deformation spaces are treated as usual by projecting the total potential energies onto the $n=2$ subspace. Using the representation theory of point groups we formulate quantum mechanical criteria for experimental identification of exotic symmetries through analysis of the specific properties of the collective rotational bands generated by the symmetries. The resulting band structures happen to be markedly distinct from the structure of the bands generated by ellipsoidal symmetry quantum rotors; those various rotational properties are discussed in detail.

We present the quantum model of the asymptotic dynamics underlying the Belinski–Khalatnikov–Lifshitz (BKL) scenario. The symmetry of the physical phase space enables making use of the affine coherent states quantization. Our results show that quantum dynamics is regular in the sense that during quantum evolution the expectation values of all considered observables are finite. The classical singularity of the BKL scenario is replaced by the quantum bounce that presents a unitary evolution of considered gravitational system. Our results suggest that quantum general relativity has a good chance to be free from singularities.

We introduce the concept of the nuclear octupole fourfold (i.e., applying simultaneously to all the four octupole deformations ${\alpha }_{30}$, ${\alpha }_{31}$, ${\alpha }_{32}$, and ${\alpha }_{33}$) neutron “magic number” $N=196$ and discuss the physical consequences of its presence. Our theoretical predictions are obtained using the realistic phenomenological mean-field approach with the deformed Woods-Saxon potential, the latter employing the new parametrization optimized in our preceding articles. Correlations among 4 parameters in the set of 12 parameters of the Woods-Saxon potential are detected and removed employing Monte Carlo approach leading to stabilization of the predictive power of the modeling. Our main focus is examining the impact of the four-fold octupole magic number $N=196$ on the stability properties of superheavy nuclei with $114\le Z\le 130$ and $166\le N\le 206$. Calculations suggest that majority of the examined nuclei are either spherical or octupole deformed, octupole-tetrahedral geometry playing the dominating role lowering the ground-state energy by up to 8 MeV. The origin and manifestations of this domination are illustrated and discussed. It turns out that, in several cases, alternative point-group symmetries may lead to noticeable lowering of the nuclear energy; this concerns the $C_{2v}$ geometry associated with ${\alpha }_{31}$, the $D_{3h}$ geometry related to ${\alpha }_{33}$, and $D_{2d}$ corresponding to the combination of ${\alpha }_{32}$ and ${\alpha }_{20}$ quadrupole component.

The quantum Belinski–Khalatnikov–Lifshitz scenario presents an unitary evolution of the system. However, the affine coherent states quantization applied to the quantization of the underlying classical scenario depends on the choice of the group parametrization. Using the two simplest parameterizations of the affine group, we show that qualitative features of our quantum system do not depend on the choice. It means that the quantum bounce replacing a singular classical scenario is expected to be a generic feature of the considered system. This paper complements our recent article (Góźdź et al. in Eur Phys J C 79:45, 2019).

Within this work we present a consistent approach to quadrupole-octupole collective vibrations coupled with the rotational motion. A realistic collective Hamiltonian with variable mass-parameter tensor and potential obtained through the macroscopic-microscopic Strutinsky-like method with particle-number-projected BCS (Bardeen–Cooper–Schrieffer) approach in full vibrational and rotational, nine-dimensional collective space is diagonalized in the basis of projected harmonic oscillator eigensolutions. This orthogonal basis of zero-, one-, two-, and three-phonon oscillator-like functions in vibrational part, coupled with the corresponding Wigner function is, in addition, symmetrized with respect to the so-called symmetrization group, appropriate to the collective space of the model. In the present model it is $D_4$ group acting in the body-fixed frame. This symmetrization procedure is applied in order to provide the uniqueness of the Hamiltonian eigensolutions with respect to the laboratory coordinate system. The symmetrization is obtained using the projection onto the irreducible representation technique. The model generates the quadrupole ground-state spectrum as well as the lowest negative-parity spectrum in ${}^{156}\mathrm{Gd}$ nucleus. The interband and intraband $B\left(E1\right)$ and $B\left(E2\right)$ reduced transition probabilities are also calculated within those bands and compared with the recent experimental results for this nucleus. Such a collective approach is helpful in searching for the fingerprints of the possible high-rank symmetries (e.g., octahedral and tetrahedral) in nuclear collective bands.