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FERMION SYSTEMS WITH FUZZY SYMMETRIES (LEVERAGING THE KNOWN TO UNDERSTAND THE UNKNOWN)

Jerry Paul Draayer, K. D. Sviratcheva, T. Dytrych, C. Bahri, K. P. Drumev, James P. Vary

Published Jan 1, 2007• 0 citations
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Abstract

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

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FERMION SYSTEMS WITH FUZZY SYMMETRIES: (LEVERAGING THE KNOWN TO UNDERSTAND THE UNKNOWN)

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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.

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Symmetries

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This chapter provides the quantum mechanical definition of the multipole moments, and study those of their properties that follow from symmetry considerations. It offers both the power and limitations of symmetry arguments. The interaction between the electromagnetic moments of atoms and nuclei determines the hyperfine structure of the spectrum, and because of this hyperfine spectroscopy is a powerful technique for studying the electromagnetic properties of nuclei. The chapter describes the quantum theory of systems that have a Correspondence Principle limit in classical point mechanics. It explores the detailed connection between the abstract formalism and the wave functions of Schrodinger's theory by computing the x-p transformation function. The chapter offers the symmetry properties of two-nucleon systems. The statistical mechanics of systems obeying the exclusion principle was worked out by Fermi and Dirac and is called Fermi-Dirac statistics; particles that obey this type of statistics are called fermions.

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Elementary Collective Effects in Systems Containing Small Fermion-Numbers

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Collective effects are a typical nuclear feature that refers to the behavior of a group of N fermions (protons and neutrons) within the atomic nucleus. Our interest lies in light nuclei only. Thus, we consider here, using the Lipkin model, a small-N fermion system at low-temperature T and discover -collective phenomena. Our fermion-model simplicity allows one to gain insight into collective fermion behavior. We focus attention, for N<20, on several quantifiers. These include standard ones related to thermal behavior, such as entropy, and quantifiers of another kind, like quantum purity.

Recent Progress in Materials•Feb 6, 2024•0 citations

Group-theoretical approach to interaction renormalization forab initiosolutions of the nuclear problem

K. D. Sviratcheva, Jerry Paul Draayer, T. Dytrych

For many problems in nuclear, high-energy and solid-state physics a crucial step toward a solution involves a diagonalization or simplification of the Hamiltonian of a many-body quantum-mechanical system. Problems of interest include renormalization of the Hamiltonian to accommodate excluded basis spaces, decoupling of a model space region of special interest from irrelevant configurations, or near-diagonalization to invoke perturbative treatments. In such applications, the Similarity Renormalization Group (SRG) approach (or flow equations ) can provide solutions where more conventional methods fail. We show that when SRG employs a near symmetry of a system, utilizing the Casimir invariant operators, it evolves the Hamiltonian of the system toward a (block-)diagonal form that can be realized in terms of a finite number of integrity basis operators. In particular, for atomic nuclei, an SRG flow equation that utilizes the SU(3) symmetry, which often dominates the nuclear dynamics, is found to naturally evolve a nucleon-nucleon (NN) interaction toward a block-diagonal form. A remarkable consequence is that a renormalized SU(3)-preserving many-body Hamiltonian expressed in terms of only the SU(3) integrity basis realizes energy spectra identical to the ones of the original realistic NN interaction, which reflects the low-energy symmetries as well as symmetry-breaking patterns of quantum chromodynamics. Such a scheme is especially suitable for ab initio SU(3) symmetry-adapted shell model calculations, which can further allow for perturbative treatments of the nuclear interaction. Moreover, the use of symmetries dramatically reduces the size of the problem, as the flow equation can be rewritten in terms of SU(3) reduced matrix elements of the NN interaction. This is crucial for maintaining the unitarity of the SRG transformations that generate many-body interactions.

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PARTIAL DYNAMICAL SYMMETRY IN A FERMIONIC MANY-BODY SYSTEM

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The concept of partial symmetry is introduced for an interacting fermion system. The associated Hamiltonians are shown to be closely related to a realistic nuclear quadrupole-quadrupole interaction. An application to $^{12}$C is presented.

arXiv:nucl-th/0008069•Nov 1, 2001•0 citations

Duality symmetry conjugates of the quantum Rabi model : effective bosonic, fermionic and coupling-only dynamical properties

Joseph Akeyo Omolo

Symmetry transformations have proved useful in determining the algebraic structure and internal dynamical properties of physical systems. In the quantum Rabi model, invariance under parity symmetry transformation has been used to obtain exact solutions of the eigenvalue equation and very good approximations of the internal dynamics of the interacting atom-light system. In this article, two symmetry operators, characterized as "duality" symmetry operators, have been introduced which transform the quantum Rabi Hamiltonian into duality conjugates. The parity and duality symmetry operators constitute an algebraically closed set of symmetry transformation operators of the quantum Rabi model. The closed $SU(2)$ Lie algebra provides the standard eigenvalues and eigenstates of the parity symmetry operator. It is established that Jaynes-Cummings and anti-Jaynes-Cummings operators are duality symmetry conjugates. Symmetric or antisymmetric linear combinations of the Rabi Hamiltonian and a corresponding duality conjugate yield the familiar spin-dependent force driven bosonic , coupling-only or quantized light mode quadrature-driven fermionic Hamiltonian. It is established that the effective bosonic, fermionic and coupling-only Hamiltonians are exact, not approximate forms of the quantum Rabi Hamiltonian as they have generally been interpreted. The effective bosonic form generates the dynamics of the light mode driven by the atomic spin-dependent force, while the fermionic form generates the dynamics of the atomic spin driven by the quantized light mode quadrature-dependent force, thus providing a complete picture of the quantum Rabi dynamics.

arXiv:2112.12514•Dec 23, 2021•0 citations
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
This chapter provides the quantum mechanical definition of the multipole moments, and study those of their properties that follow from symmetry considerations. It offers both the power and limitations of symmetry arguments. The interaction between the electromagnetic moments of atoms and nuclei determines the hyperfine structure of the spectrum, and because of this hyperfine spectroscopy is a powerful technique for studying the electromagnetic properties of nuclei. The chapter describes the quantum theory of systems that have a Correspondence Principle limit in classical point mechanics. It explores the detailed connection between the abstract formalism and the wave functions of Schrodinger's theory by computing the x-p transformation function. The chapter offers the symmetry properties of two-nucleon systems. The statistical mechanics of systems obeying the exclusion principle was worked out by Fermi and Dirac and is called Fermi-Dirac statistics; particles that obey this type of statistics are called fermions.
Collective effects are a typical nuclear feature that refers to the behavior of a group of N fermions (protons and neutrons) within the atomic nucleus. Our interest lies in light nuclei only. Thus, we consider here, using the Lipkin model, a small-N fermion system at low-temperature T and discover -collective phenomena. Our fermion-model simplicity allows one to gain insight into collective fermion behavior. We focus attention, for N<20, on several quantifiers. These include standard ones related to thermal behavior, such as entropy, and quantifiers of another kind, like quantum purity.
For many problems in nuclear, high-energy and solid-state physics a crucial step toward a solution involves a diagonalization or simplification of the Hamiltonian of a many-body quantum-mechanical system. Problems of interest include renormalization of the Hamiltonian to accommodate excluded basis spaces, decoupling of a model space region of special interest from irrelevant configurations, or near-diagonalization to invoke perturbative treatments. In such applications, the Similarity Renormalization Group (SRG) approach (or flow equations ) can provide solutions where more conventional methods fail. We show that when SRG employs a near symmetry of a system, utilizing the Casimir invariant operators, it evolves the Hamiltonian of the system toward a (block-)diagonal form that can be realized in terms of a finite number of integrity basis operators. In particular, for atomic nuclei, an SRG flow equation that utilizes the SU(3) symmetry, which often dominates the nuclear dynamics, is found to naturally evolve a nucleon-nucleon (NN) interaction toward a block-diagonal form. A remarkable consequence is that a renormalized SU(3)-preserving many-body Hamiltonian expressed in terms of only the SU(3) integrity basis realizes energy spectra identical to the ones of the original realistic NN interaction, which reflects the low-energy symmetries as well as symmetry-breaking patterns of quantum chromodynamics. Such a scheme is especially suitable for ab initio SU(3) symmetry-adapted shell model calculations, which can further allow for perturbative treatments of the nuclear interaction. Moreover, the use of symmetries dramatically reduces the size of the problem, as the flow equation can be rewritten in terms of SU(3) reduced matrix elements of the NN interaction. This is crucial for maintaining the unitarity of the SRG transformations that generate many-body interactions.
The concept of partial symmetry is introduced for an interacting fermion system. The associated Hamiltonians are shown to be closely related to a realistic nuclear quadrupole-quadrupole interaction. An application to 12^{12}12C is presented.
Symmetry transformations have proved useful in determining the algebraic structure and internal dynamical properties of physical systems. In the quantum Rabi model, invariance under parity symmetry transformation has been used to obtain exact solutions of the eigenvalue equation and very good approximations of the internal dynamics of the interacting atom-light system. In this article, two symmetry operators, characterized as "duality" symmetry operators, have been introduced which transform the quantum Rabi Hamiltonian into duality conjugates. The parity and duality symmetry operators constitute an algebraically closed set of symmetry transformation operators of the quantum Rabi model. The closed SU(2)SU(2)SU(2) Lie algebra provides the standard eigenvalues and eigenstates of the parity symmetry operator. It is established that Jaynes-Cummings and anti-Jaynes-Cummings operators are duality symmetry conjugates. Symmetric or antisymmetric linear combinations of the Rabi Hamiltonian and a corresponding duality conjugate yield the familiar spin-dependent force driven bosonic , coupling-only or quantized light mode quadrature-driven fermionic Hamiltonian. It is established that the effective bosonic, fermionic and coupling-only Hamiltonians are exact, not approximate forms of the quantum Rabi Hamiltonian as they have generally been interpreted. The effective bosonic form generates the dynamics of the light mode driven by the atomic spin-dependent force, while the fermionic form generates the dynamics of the atomic spin driven by the quantized light mode quadrature-dependent force, thus providing a complete picture of the quantum Rabi dynamics.