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Physical Review C

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

Dora M. Jezek, E. Susana Hernández

Published Apr 1, 1987• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.35.1555

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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.
We analyze the possible bifurcations of the stationary points of the self-consistent flow on the Bloch sphere. We propose a generalized SU(2) model Hamiltonian for which, by means of a geometrically simple classification, we find the topologically invariant regions of the mean field flow in the space of interaction parameters. The borders of these regions are the bifurcation sets corresponding to general nonthermodynamic phase transitions. When these separatrices are crossed, the mean field flow undergoes a qualitative change. We also examine the consequences of the catastrophical configurations on the exact dynamics in quasispin space.
The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For twobody interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the "librations" and "rotations" in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.NUCLEAR STRUCTURE Lipkin-Meshkov-Glick model, TDHF, transformation of TDHF equations to Hamilton's canonical form, coherent states, ground state phase transition.
We investigate the time-dependent Hartree-Fock flow pattern of a two-level many fermion system interacting via a two-body interaction which does not preserve the parity symmetry of standard SU(2) models. The geometrical features of the time-dependent Hartree-Fock energy surface are analyzed and a phase instability is clearly recognized. The time evolution of one-body observables along self-consistent and exact trajectories are examined together with the overlaps between both orbits. Typical time scales for the determinantal motion can be set and the validity of the time-dependent Hartree-Fock approach in the various regions of quasispin phase space is discussed.
An analysis of the classical and quantum phase transitions of the Lipkin-Meshkov-Glick model is presented. It is shown that the classical dynamics is ruled by the energy surface of the system. Applying the catastrophe formalism to this energy surface the separatrix is obtained. It determines the regions in the control parameter space where there are phase transitions. Special attention is given to the compositions of ground and first-excited energy states, which are well described by the even and odd SU(2) coherent states. Phase transitions are shown to be associated with a change in the wave functions from collective to single-particle behavior. Evaluating the distribution of nearest-neighbor spacings it is shown that the separatrix of the system emerges as a useful tool to describe the global behavior of the quantum-level structure and their corresponding wave functions.
We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground-state quantum phase transition where a bifurcation occurs, showing a change in stability associated with an excited-state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest-energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find excellent agreement between them for large sizes of the system in a wide region of the parameter space except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid.
We report on quantum revivals and classical periodicities of wave packets centered around the ground state within the Lipkin-Meshkov-Glick model. Special attention is paid to the behavior at first- and second-order phase transitions. In line with previous studies, we find that away from criticality, characteristic times exhibit smooth, nonsingular behavior, but upon approaching the transition points they diverge as power laws with associated critical exponents. Finite-size effects are studied and the observed phenomenology is discussed in the framework of the time-energy uncertainty relation.