Quantum Many-Body Theory

<p>This thesis is a collection of theoretical investigations into different aspects of the broad subject of quantum many-body theory. The results are grouped into three main parts, which in turn are divided into separate self-contained sections. Some of the work is presented in the form of published papers and papers that have been submitted for publication. The first section of Part A introduces some of the concepts involved in many-body problems, by developing methods to evaluate expectation values of the form . In the rest of Part A I consider collective excitations of finite quantum systems. The calculations are confined to nuclei because the results can then be compared with the extensive investigations that have been made into collective nuclear modes. In Section AII, wavefunctions are proposed for rotational excitations of even-even nuclei. Both isoscalar and isovector nuclear modes are discussed. In particular, the l2,m> isoscalar states are investigated for both spherical and deformed even-even nuclei, and the simplest isovector wavefunction is shown to give a good description of the giant dipole resonance. In section AIII wavefunctions are proposed for compressional vibrational states of spherical nuclei. Section AIV discusses sum rules for nuclear transitions of a given electric multipolarity. It is found that the 2+ and 1- states investigated in section AII and all but one of the vibrational states discussed in AIII each exhaust a large part of the appropriate sum rule. In Part B I consider the problem of how to describe flow in quantum fluids. In particular, we want to be able to identify the physical motion represented by any given many-body wavefunction. Section BI derives a guantum mechanical velocity field for a many-body system, paying special attention to the need for a quantum continuity equation. It is found that when the wavefunction has the usual time dependence e-iwt , that the quantum velocity formula averages over all oscillatory motion, so that much of the physical nature of the flow field is lost. In section BII a particular wavefunction is proposed to represent the quantum excitation corresponding to any given potential flow field. The results obtained by considering specific examples are very encouraging. In Part C I investigate the properties of surfaces. Section CI presents a theoretical description of the tension, energy and thickness of a classical liquid-vapour interface. In section CII the classical results are extended to describe the surface of a quantum system, namely superfluid helium four. Problems occur for the quantum system if the correlations arising from the zero-point-motion of the phonon modes are included in the ground state wavefunction. Finally, in section CIII discuss generalized virial theorems that give the change in the free energy of a system undergoing an infinitesimal deformation. For example, a particular deformation gives the expression used in CII, for the surface tension of a plane quantum surface.</p>

Quantum Many-Body Theory

<p>This thesis is a collection of theoretical investigations into different aspects of the broad subject of quantum many-body theory. The results are grouped into three main parts, which in turn are divided into separate self-contained sections. Some of the work is presented in the form of published papers and papers that have been submitted for publication. The first section of Part A introduces some of the concepts involved in many-body problems, by developing methods to evaluate expectation values of the form . In the rest of Part A I consider collective excitations of finite quantum systems. The calculations are confined to nuclei because the results can then be compared with the extensive investigations that have been made into collective nuclear modes. In Section AII, wavefunctions are proposed for rotational excitations of even-even nuclei. Both isoscalar and isovector nuclear modes are discussed. In particular, the l2,m> isoscalar states are investigated for both spherical and deformed even-even nuclei, and the simplest isovector wavefunction is shown to give a good description of the giant dipole resonance. In section AIII wavefunctions are proposed for compressional vibrational states of spherical nuclei. Section AIV discusses sum rules for nuclear transitions of a given electric multipolarity. It is found that the 2+ and 1- states investigated in section AII and all but one of the vibrational states discussed in AIII each exhaust a large part of the appropriate sum rule. In Part B I consider the problem of how to describe flow in quantum fluids. In particular, we want to be able to identify the physical motion represented by any given many-body wavefunction. Section BI derives a guantum mechanical velocity field for a many-body system, paying special attention to the need for a quantum continuity equation. It is found that when the wavefunction has the usual time dependence e-iwt , that the quantum velocity formula averages over all oscillatory motion, so that much of the physical nature of the flow field is lost. In section BII a particular wavefunction is proposed to represent the quantum excitation corresponding to any given potential flow field. The results obtained by considering specific examples are very encouraging. In Part C I investigate the properties of surfaces. Section CI presents a theoretical description of the tension, energy and thickness of a classical liquid-vapour interface. In section CII the classical results are extended to describe the surface of a quantum system, namely superfluid helium four. Problems occur for the quantum system if the correlations arising from the zero-point-motion of the phonon modes are included in the ground state wavefunction. Finally, in section CIII discuss generalized virial theorems that give the change in the free energy of a system undergoing an infinitesimal deformation. For example, a particular deformation gives the expression used in CII, for the surface tension of a plane quantum surface.</p>

Rigorous treatment of pairwise and many-body electrostatic interactions among dielectric spheres at the Debye–Hückel level

Electrostatic interactions among colloidal particles are often described using the venerable (two-particle) Derjaguin–Landau–Verwey–Overbeek (DLVO) approximation and its various modifications. However, until the recent development of a many-body theory exact at the Debye–Hückel level (Yu in Phys Rev E 102:052404, 2020), it was difficult to assess the errors of such approximations and impossible to assess the role of many-body effects. By applying the exact Debye–Hückel level theory, we quantify the errors inherent to DLVO and the additional errors associated with replacing many-particle interactions by the sum of pairwise interactions (even when the latter are calculated exactly). In particular, we show that: (1) the DLVO approximation does not provide sufficient accuracy at shorter distances, especially when there is an asymmetry in charges and/or sizes of interacting dielectric spheres; (2) the pairwise approximation leads to significant errors at shorter distances and at large and moderate Debye lengths and also gets worse with increasing asymmetry in the size of the spheres or magnitude or placement of the charges. We also demonstrate that asymmetric dielectric screening, i.e., the enhanced repulsion between charged dielectric bodies immersed in media with high dielectric constant, is preserved in the presence of free ions in the medium. Graphic abstract

Microscopic Model to Take into Account Complex Configurations for Pygmy and Giant Resonances

A microscopic model for taking into account quasiparticle–phonon interaction in magic nuclei is considered within nuclear quantum many-body theory. This model is of interest for constructing a microscopic theory of pygmy and giant multipole resonances—first of all, a description of their fine structure. This article reports on a continuation and development of our earlier study [1]. Basic physics results of that study are confirmed here, and new results are obtained: (i) exact (not approximate, as in [1]) expressions for the first and second variations of the vertex in the phonon field are found and employed; (ii) a new equation involving, in addition to the known effective interaction, the total amplitude for particle–hole interaction is derived for the vertex, which is the main ingredient in the theory of finite Fermi systems; (iii) the required two-phonon configurations are obtained owing to the last result. The new equation for the vertex now contains complex configurations such as $$1p1h\otimes\textrm{phonon}$$ and two-phonon ones, along with numerous ground-state correlations.

Superconducting State

For the past almost fifty years, scientists have been trying to explain the phenomenon of superconductivity. The mechanism is the key ingredient of microscopic theory, which was developed by Bardeen, Cooper, and Schrieffer in 1957. The theory also introduced the basic concepts of pairing, coherence length, energy gap, and so on. Since then, microscopic theory has undergone an intensive development. This book provides a very detailed theoretical treatment of the key mechanisms of superconductivity, including the current state of the art (phonons, magnons, plasmons). In addition, the book contains descriptions of the properties of the key superconducting compounds that are of the most interest for science and applications. For many years, there has been a search for new materials with higher values of the main parameters, such as the critical temperature and critical current. At present, the possibility of observing superconductivity at room temperature has become perfectly realistic. That is why the book is especially concerned with high-Tc systems such as high-Tc oxides, hydrides with record values for critical temperature under high pressure, nanoclusters, and so on. A number of interesting novel superconducting systems have been discovered recently, including topological materials, interface systems, and intercalated graphene. The book contains rigorous derivations based on statistical mechanics and many-body theory. The book also provides qualitative explanations of the main concepts and results. This makes the book accessible and interesting for a broad audience.