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Recent Progress in Materials

Elementary Collective Effects in Systems Containing Small Fermion-Numbers

Angelo Plastino, Diana Monteoliva, Angel Ricardo Plastino

Published Feb 6, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.21926/rpm.2401004

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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.
We investigate finite systems of N paired fermions, common in atomic nuclei, for example. These systems exhibit quantum mechanical features akin to those of superconductors. We discover, however, some specific N dependent effects that can not be attained in the thermodynamics limit of ordinary superconductivity. In particular, an important fact is uncovered: there is a strong correlation between the temperature T and the number of fermions N. A certain temperature increase ΔT produces, in thermal quantifiers (such as the entropy), quite different effects if N=4 or N=25. In fact, whether a given temperature value should be regarded as high or low can not be ascertained independent of the N value.
Finite quantum many fermion systems are essential for our current understanding of Nature. They are at the core of molecular, atomic, and nuclear physics. In recent years, the application of information and complexity measures to the study of diverse types of many-fermion systems has opened a line of research that elucidates new aspects of the structure and behavior of this class of physical systems. In this work we explore the main features of information and information-based complexity indicators in exactly soluble many-fermion models of the Lipkin kind. Models of this kind have been extremely useful in shedding light on the intricacies of quantum many body physics. Models of the Lipkin kind play, for finite systems, a role similar to the one played by the celebrated Hubbard model of solid state physics. We consider two many fermion systems and show how their differences can be best appreciated by recourse to information theoretic tools. We appeal to information measures as tools to compare the structural details of different fermion systems. We will discover that few fermion systems are endowed by a much larger complexity-degree than many fermion ones. The same happens with the coupling-constants strengths. Complexity augments as they decrease, without reaching zero. Also, the behavior of the two lowest lying energy states are crucial in evaluating the system's complexity.
interacting fermions. This model serves as a simplified proxy for an atomic nucleus, capturing its essential features in a more manageable form compared to a realistic nuclear model, which would require the diagonalization of matrices with millions of elements, making the extraction of qualitative features a significant challenge. Specifically, we aim to correlate Cf with particle number fluctuations and temperature, using the paradigmatic Lipkin model. Our analysis reveals intriguing dependencies of Cf on the total fermion number, showcasing distinct behaviors at different temperatures. Notably, we find that the degree of quantum mixedness exhibits a strong dependence on the total fermion number, with varying trends across different temperature regimes. Remarkably, this dependence remains unaffected by the strength of the fermion-fermion interaction (as long as it is non-zero), underscoring the robustness of the observed phenomena. Through comprehensive numerical simulations, we provide illustrative graphs depicting these dependencies, offering valuable insights into the fundamental characteristics of quantum many-body fermion systems. Our findings illuminate the intricate dynamics of the degree of mixedness, a crucial quantum property, with potential implications for diverse fields ranging from condensed matter physics to quantum information science.
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
Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.
The interaction among fundamental particles in nature leads to many interesting effects in quantum statistical mechanics; examples include superconductivity for charged systems and superfluidity in cold gases. It is a huge challenge for mathematical physics to understand the collective behavior of systems containing a large number of particles, emerging from known microscopic interactions. In this workshop we brought together researchers working on different aspects of many-body quantum mechanics to discuss recent developments, exchange ideas and propose new challenges and research directions.