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Quasispin models in quantum physics

Andrii Zymin

Published Sep 12, 2019• 0 citations
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Abstract

Title: Quasispin models in quantum physics Author: Andrii Zymin Department: Institute of Particle and Nuclear Physics Supervisor: Mgr. Pavel Stránský, Ph.D., Institute of Particle and Nuclear Physics Abstract: The use of symmetries in quantum physics helps in a deeper understanding of physical systems and simplifies numerical calculations. This thesis studies models based on the SU(2) algebra, which, in spite of their apparent simplicity, show rather rich behavior and describe a wide spectrum of physical phenomena. We review various realizations of the SU(2) algebra (namely the spin, boson, and fermion realization) and present the most general quantum hamiltonian with one- and two-body interactions, constructed from the SU(2) generators. We perform the classical limit of the hamiltonian and show a numerical study of several particular examples. Keywords: dynamical symmetries, Lipkin model, su(2) algebra, quasispin

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Title: Quasispin models in quantum physics Author: Andrii Zymin Department: Institute of Particle and Nuclear Physics Supervisor: Mgr. Pavel Stránský, Ph.D., Institute of Particle and Nuclear Physics Abstract: The use of symmetries in quantum physics helps in a deeper understanding of physical systems and simplifies numerical calculations. This thesis studies models based on the SU(2) algebra, which, in spite of their apparent simplicity, show rather rich behavior and describe a wide spectrum of physical phenomena. We review various realizations of the SU(2) algebra (namely the spin, boson, and fermion realization) and present the most general quantum hamiltonian with one- and two-body interactions, constructed from the SU(2) generators. We perform the classical limit of the hamiltonian and show a numerical study of several particular examples. Keywords: dynamical symmetries, Lipkin model, su(2) algebra, quasispin
A general algebraic approach incorporating both invariance groups and dynamical symmetry algebras is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum manybody physics models due to symmetries of their Hamiltonians H . Its general ideas are manifested on some recent new examples: (1) G-invariant biphotons and a related SU (2)-invariant treatment of unpolarized light; (2) quasispin clusters in nonlinear models of quantum optics; and (3) construction of composite particles and (para)fields from G -invariant clusters due to internal symmetries.
In order to develop a previously proposed method to investigate thermal effects in quantum many-particle systems, basic ideas are formulated in more general systems obeying su(2)-algebra. It is shown that Hamiltonians of a single orbit shell model with pairing correlation, the Lipkin model, a symmetric rigid rotor model and an asymmetric rigid rotor model have almost the same structure, that is algebraic structure of su(1, 1), by the use of the Holstein-Primakoff boson representation for su(2)-algebra and so-called phase-space doubling introduced to study thermal phenomena. It is also discussed that the interaction term between intrinsic system and external environment approximately represents the interesting physical process in each system at finite temperature.
It is shown that nuclear matter described by quasispin or pseudospin operators may exhibit irreversible behavior, in the sense of modern kinetic theory. With the help of the group-contraction technique, a harmonic oscillator can be associated to the many-body system and it is seen that in the standard Lipkin-Meshkov-Glick model the ground-state phase transition gives rise to a significant change in the spectral properties of the characteristic operators. This fact makes room for the introduction of the microscopic operators of irreversible dynamics. Their meaning and the consequences of their existence are examined and it is seen that the time evolution of the quasispin arrangement closely resembles the approach to equilibrium of a macroscopic system.NUCLEAR STRUCTURE Operators of irreversible Hamiltonian dynamics; quasispin nuclear matter; group contraction, Lipkin-Meshkov-Glick model; ground-state phase transition; entropy; irreversible approach to equilibrium.
The su(2)-algebraic many-fermion model is formulated so as to be able to get the unified understanding of the structures of three simple models: the single-level pairing, the isoscalar proton-neutron pairing and the two-level Lipkin model. The basic idea is to introduce an auxiliary su(2)-algebra, any generator of which commutes with any generator of the starting su(2)-algebra. With the aid of this algebra, the minimum weight states are completely determined in simple form. Further, concerning the two algebras, boson realization is presented. Through this formulation, the behavior of the total fermion in the Lipkin model is notably different from that in the other two models. As the supplementary problem, the boson-fermion realization and the Lipkin model in the isovector pairing model are investigated.
Following an idea, recently, proposed by the present authors for the two-level pairing model, the Lipkin model is reexamined. It is a natural generalization of the method already developed by the present authors with Kuriyama. This model is a schematic model for many-fermion systems and obeys the su(2)-algebra. It is shown that the use of the Schwinger boson representation, the model is expressed in terms of the su(1,1) ⊗ su(1,1)-algebra and with the aid of the MYT mapping method, it is disguised from the orginal form in terms of the Holstein-Primakoff representation. Further, under various coherent states, the classical counterparts are derived. It is concluded that the Lipkin model can be treated in the common ring as that of the two-level pairing model.
Microscopic models, which embody the simplicity and significance of a dynamical symmetry approach to nuclear structure, are reviewed. They can reveal striking features of atomic nuclei when a symmetry dominates and solutions in domains that may otherwise be unreachable. 1. Overview of algebraic fermion models. A theory that invokes group symmetries is driven by an expectation that the wave functions of the quantum mechanical system under consideration can be characterized by their invariance properties under the corresponding symmetry transformations. But even when the symmetries are not exact, if one can find near invariant operators, the associated symmetries can be used to help reduce the dimensionality of a model space to a tractable size. Throughout the years, group-theoretical approaches have identified fundamental symmetries in light to heavy nuclei and achieved a reasonable reproduction of experimental data (for a review of fermion models, see 1). In addition, they provide theoretical predictions for nuclear systems including heavy unstable