First-order quantum phase transitions: Test ground for emergent chaoticity,regularity and persisting symmetries

We present a comprehensive analysis of the emerging order and chaos and enduring symmetries, accompanying a generic (high-barrier) first-order quantum phase transition (QPT). The interacting boson model Hamiltonian employed, describes a QPT between spherical and deformed shapes, associated with its U(5) and SU(3) dynamical symmetry limits. A classical analysis of the intrinsic dynamics reveals a rich but simply-divided phase space structure with a Hénon–Heiles type of chaotic dynamics ascribed to the spherical minimum and a robustly regular dynamics ascribed to the deformed minimum. The simple pattern of mixed but well-separated dynamics persists in the coexistence region and traces the crossing of the two minima in the Landau potential. A quantum analysis discloses a number of regular low-energy U(5)-like multiplets in the spherical region, and regular SU(3)-like rotational bands extending to high energies and angular momenta, in the deformed region. These two kinds of regular subsets of states retain their identity amidst a complicated environment of other states and both occur in the coexistence region. A symmetry analysis of their wave functions shows that they are associated with partial U(5) dynamical symmetry (PDS) and SU(3) quasi-dynamical symmetry (QDS), respectively. The pattern of mixed but well-separated dynamics and the PDS or QDS characterization of the remaining regularity, appear to be robust throughout the QPT. Effects of kinetic collective rotational terms, which may disrupt this simple pattern, are considered.     

Quantum phase transitions in Bose–Fermi systems

Quantum phase transitions in a system of N bosons with angular momentum L = 0, 2 (s, d) and a single fermion with angular momentum j are investigated both classically and quantum mechanically. It is shown that the presence of the odd fermion strongly influences the location and nature of the phase transition, especially the critical value of the control parameter at which the phase transition occurs. Experimental evidence for the U(5)–SU(3) (spherical to axially-deformed) transition in odd–even nuclei is presented.     

Rapid cooling of the neutron star in Cassiopeia A triggered by neutron superfluidity in dense matter

We propose that the observed cooling of the neutron star in Cassiopeia A is due to enhanced neutrino emission from the recent onset of the breaking and formation of neutron Cooper pairs in the (3)P(2) channel. We find that the critical temperature for this superfluid transition is ?0.5×10(9) K. The observed rapidity of the cooling implies that protons were already in a superconducting state with a larger critical temperature. This is the first direct evidence that superfluidity and superconductivity occur at supranuclear densities within neutron stars. Our prediction that this cooling will continue for several decades at the present rate can be tested by continuous monitoring of this neutron star.     

Ion trap MS using high trapping gas pressure enables unequivocal structural analysis of three isobaric compounds in a mixture by using energy‐resolved mass spectrometry and the survival yield technique

Recently, it has been shown that energy‐resolved mass spectrometry (MS) can provide quantitative information from two isomeric or isobaric compounds in mixtures by using the survival yield (SY) technique together with “gas‐phase collisional purification” (GPCP) strategy (Anal. Chem., 2016, 88, p.10821). Herein, we present an improvement and an extension of this concept to the structural analysis of a model mixture of three isobaric compounds (two peptides and a polyether). By using default collision‐induced dissociation (CID) tandem MS parameters on an ion trap instrument, the previous approach did not show any signs of isobaric contamination. However, by modifying CID conditions and using a threefold increase of the He trapping gas pressure (to reach 3.00·10^?5 mbar), the SY curve was unexpectedly and strongly shifted to higher excitation voltages with two plateaus appearing. Those plateaus, indicating clearly the presence of three isobaric compounds, were taken as reliable indicators to perform GPCP at carefully selected excitation voltages in order to selectively fragment one compound after the other. In this way, CID mass spectra of each compound were correctly recovered, both in terms of fragment ion peaks and in terms of relative intensities, from energy‐resolved MSⁿ spectra of the three compounds mixture. This feature enables their unequivocal structural analysis as if samples were free from isobaric interferences. In this paper, we also discuss the possibility for recovering SY curves for pure compounds directly from the mixture. Clearly, in this case, the higher He trapping gas pressure made it possible to use the SY technique, for the first time, for the structural analysis in the case of mixtures of three isobaric compounds. This observation, quite unexpected, demonstrates that the trapping gas pressure is of paramount importance although it is usually not considered in energy‐resolved MS for structural and/or quantitative analysis.     

A tale of two algorithms

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of ‘numbers’, extending the rational number system, adequate for measuring continuous quantities. Moreover, that such ‘numbers’ are in one-to-one correspondence with points on a ‘number line’. But typically real ‘numbers’ are not systematically presented via any constructive method. While taken for granted, they are one of the most commonly used mathematical objects. This paper proposes a geometric algorithm, extending the long division algorithm, which leads to a constructive definition of real numbers. It proceeds to describe a direct algorithm for adding ‘real numbers’. Combined use of the two algorithms enables a smooth and meaningful presentation, offering a double image (geometric and numerical) of real numbers in decimal notation. An early such presentation is of both conceptual and practical importance.