Phase transitions and quasi-dynamical symmetry in nuclear collective models, III: The U(5) to SU(3) phase transition in the IBM

This paper studies the dependence of the U(5) to SU(3) phase transition in the interacting boson model (IBM) on the finite number N of bosons. Such investigations illuminate the relationship between a first order transition in a macroscopic system, which exhibits a discontinuous change of some of its properties at a highly singular critical point, and a transition in a corresponding finite particle system, which shows a critical point smeared by fluctuations. The IBM model Hamiltonian H(α)=(1−α)H1+αH2 of this paper is an interpolation between a U(5)-invariant Hamiltonian H1, given by the d-boson number operator, and an SU(3) Hamiltonian H2, given by a quadrupole-quadrupole operator. For N sufficiently large, the low energy eigenstates of H(α) fall into either a U(5) phase or an SU(3) phase depending on the value of α. An exception is a narrow zone around a critical value of α. The width of this critical zone decreases as N increases. In this zone the spectra for the N values considered show an approximate X(5) dynamical symmetry to within an N-dependent scale factor. The states in the SU(3) phase show an SU(3) quasi-dynamical symmetry that becomes more well-defined as N increases. The states in the U(5) phase show a U(5) quasi-dynamical symmetry albeit one that closely approaches a pure U(5) dynamical symmetry for smaller values of α and which also becomes better defined the larger the particle number. The concluding section reviews studies of phase transitions in nuclear physics.