We analyze shape phase transitions in two-dimensional algebraic models. We apply our analysis to linearto-bent transitions in molecules and point out what observables are particularly sensitive to the transition ͑order parameters͒. We study numerically the scaling behavior of observables and confirm the dependence of the energy gap for phase transitions of U͑n͒-SO͑n +1͒ type. We calculate energies of excited states and show their unusual behavior for some values of the Hamiltonian control parameter. This behavior is due to the doublehumped nature of the potential and can be associated with the concept of monodromy. Finally, we compute numerically thermodynamic quantities, in particular heat capacities, and show their large variation at and around the critical value of the control parameter.

Excited state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions (QPTs) to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-1/2 system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.

The elastic scattering of 6He on 208Pb has been measured at laboratory energies of 14, 16, 18 and 22 MeV. These data were analyzed using phenomenological Woods- Saxon form factors and optical model calculations. A semiclassical polarization po- tential was used to study the e ect of the Coulomb dipole polarizability. Evidence for long range absorption, partially arising from Coulomb dipole polarizability, is reported. The energy variation of the optical potential was found to be consistent with the dispersion relations which connect the real and imaginary parts of the potential

Collisions induced by 9;10;11 Be on a 64 Zn target at the same c.m. energy were studied. For the first time, strong effects of the 11 Be halo structure on elastic-scattering and reaction mechanisms at energies near the Coulomb barrier are evidenced experimentally. The elastic-scattering cross section of the 11 Be halo nucleus shows unusual behavior in the Coulomb-nuclear interference peak angular region. The extracted total-reaction cross section for the 11 Be collision is more than double the ones measured in the collisions induced by 9;10 Be. It is shown that such a strong enhancement of the total-reaction cross section with 11 Be is due to transfer and breakup processes. A hundred years after Rutherford's scattering experiment [1], heavy-ion-elastic-scattering angular distributions (AD) are usually plotted as a ratio of the Rutherford cross section (i.e., pure Coulomb scattering). Such representation usually shows a decrease of the elastic cross section with the angle due to absorption at small impact parameters by nonelastic processes, and an oscillatory behavior. The latter, using the language of optics, is described in terms of refraction by nonabsorbing lenses (Coulomb rainbow model) or diffraction by sharp-edged, nonrefracting apertures (Fraunhofer or Fresnel diffraction model). However, the refraction or diffraction descriptions are oversimplifications of the realistic process; rather, the nucleus behaves as a ''cloudy crystal ball.'' The elasticscattering AD may show a peak resulting from the interference between the Coulomb and nuclear amplitudes (Coulomb-nuclear interference peak) [2], which, in analogy with the Coulomb rainbow model, is sometimes called the rainbow peak. Since elastic scattering is a peripheral process, it does not give information on the interior region of nuclei. It probes the tail of the wave function, and hence one can learn about surface properties, such as size of nuclei and surface diffuseness. Therefore, elastic scattering is an ideal tool to study peculiar nuclear structures as, for example, the nuclear halo. Such structure originates when very weakly bound nucleon(s) can tunnel into the classically forbidden region, giving rise to a diffuse tail surrounding a well-bound core. The behavior of the system in nuclear reactions is mostly determined by the tail of the wave function [3]. The reaction mechanisms may also be affected by the weak binding: at energies around the Coulomb barrier, couplings between the entrance channel and the continuum [4][5][6][7][8], as well as to the various reaction channels [9][10][11][12], are expected to be very important. Direct reactions, such as breakup or transfer, may be favored owing to the low binding energy, the extended tail of halo nuclei, and the large Q values for selected transfer channels.Almost all elastic-scattering and reaction mechanism studies around the barrier with halo nuclei have been performed with the 2n halo nucleus 6 He. All authors agree that, due to the 6 He structure, one has an enhancement of the total-reaction (T...

We study the structure of the eigenstates and the dynamics of a system that undergoes an excited state quantum phase transition (ESQPT). The analysis is performed for two-level pairing models characterized by a U (n + 1) algebraic structure. They exhibit a second order phase transition between two limiting dynamical symmetries represented by the U (n) and SO(n + 1) subalgebras. They are, or can be mapped onto, models of interacting bosons. We show that the eigenstates with energies very close to the ESQPT critical point, EESQPT, are highly localized in the U (n)-basis. Consequently, the dynamics of a system initially prepared in a U (n)-basis vector with energy E ∼ EESQPT may be extremely slow. Signatures of an ESQPT can therefore be found in the structures of the eigenstates and in the speed of the system evolution after a sudden quench. Our findings can be tested experimentally with trapped ions. Introduction.-Quantum phase transitions (QPTs) occur at zero temperature. They correspond to an abrupt change in the character of the ground state of a system when a control parameter passes a critical point [1]. The subject, which permeates condensed matter and nuclear physics, has become one of the highlights of experiments with cold gases, where transitions from a superfluid to a Mott insulator [2] and from a normal to a superradiant phase [3] have been observed. The investigations are not restricted to the properties of the ground state, but extend also to the dynamics of systems undergoing QPTs. In this context, one finds studies about the quantum analogue of the Kibble-Zurek mechanism [4], as well as the relaxation time [5][6][7][8], revivals [9,10], and temporal fluctuations [11,12] at critical points.Recently, the concept of ground state QPT has been generalized to encompass also QPTs occurring at excited states. These so-called ESQPT refer to a singularity in the energy spectrum caused by the clustering of excited levels at a critical energy [13][14][15][16]. This critical point can be reached either for a constant excited energy by varying the control parameter(s) or by fixing the latter and increasing the energy. ESQPTs have been investigated for a broad class of many-body quantum systems, such as the Lipkin-Meshkov-Glick (LMG) [17][18][19], the molecular vibron [15,20] In terms of dynamics, it has been shown that an ESQPT leads to random oscillations of the survival probability in isolated systems [21], to singularities in the evolution of observables [32], and to maximal decoherence in open systems [18,33]. Despite these works, studies of the effects of ESQPTs on systems' evolutions are still scarce.In this Rapid Communication, we provide insights into the