Physics of Nuclei: Key Role of an Emergent Symmetry.

We show through first-principles nuclear structure calculations that the special nature of the strong nuclear force determines highly regular patterns heretofore unrecognized in nuclei that can be tied to an emergent approximate symmetry. This symmetry is ubiquitous and mathematically tracks with a symplectic symmetry group. This, in turn, has important implications for understanding the physics of nuclei: we find that nuclei are made of only a few equilibrium shapes, deformed or not, with associated vibrations and rotations. It also opens the path for ab initio large-scale modeling of open-shell intermediate-mass nuclei without the need for renormalized interactions and effective charges.

Equations-of-motion approach to quantum mechanics: application to a model phase transition.

We present a generalized equations-of-motion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a five-dimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10 x 10 matrices give better results than obtained by diagonalizing 1000 x 1000 matrices.

Scaling properties and asymptotic spectra of finite models of phase transitions as they approach macroscopic limits.

The asymptotic spectra and scaling properties of a mixed-symmetry Hamiltonian, which exhibits a second-order phase transition in its macroscopic limit, are examined for a system of N interacting bosons. A second interacting boson-model Hamiltonian, which exhibits a first-order phase transition, is also considered. The latter shows many parallel characteristics and some notable differences, leaving it open to question as to the nature of its asymptotic critical-point properties.

Partially solvable pair-coupling models with seniority-conserving interactions.

Given a single j-shell Hamiltonian, the algebraic conditions for conservation of seniority are derived from a quasispin tensor decomposition of the two-nucleon interaction. This makes it possible to construct useful solvable and partially solvable shell-model Hamiltonians with eigenstates classified by a spectrum generating algebra. Applications are made to the low-lying energy levels of the N = 50 nuclear isotones.