Geometrized dynamics and search of structural instabilities in SU(2) models

The structural instabilities or general nonthermodynamic phase transitions of the time-dependent Hartree-Fock flow on the Bloch sphere are investigated for different SU(2) model Hamiltonians by means of a simple geometrical construction. In particular, the generalized Lipkin-Meshkov-Glick model is found to exhibit a variety of instabilities, opposite to the standard Lipkin-Meshkov-Glick model which possesses only one ground-state (thermodynamic) phase transition. The relationship between the fixed points of the time-dependent Hartree-Fock flow and the tangency points between energy surfaces and the Bloch sphere is established. It is found that different Hamiltonians of the class under study give rise to flow patterns whose invariant sets may contain either rotations or librations, which in turn may be degenerate as well as nondegenerate, in contrast to previously investigated models.