Mesoscopic Fluctuations: Nuclei, Quantum Dots, and Beyond
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Mesoscopic Fluctuations: Nuclei, Quantum Dots, and Beyond J. G. G. S. Ramos1 · A. L. R. Barbosa2 · D. Bazeia1 · C. Lewenkopf3 Accepted: 28 October 2020 © Sociedade Brasileira de F´ısica 2020
Abstract This paper is dedicated to review the contributions of Prof. Mahir Hussein to the theory of random matrices and its applications to mesoscopic systems, with focus on his more recent works on conductance fluctuations of chaotic quantum dots. Keywords Mesoscopic fluctuations · Electronic transport · Random matrices
1 Introduction Random matrices were originally proposed by Wigner to model the statistical properties of compound nucleus reactions [1]. In such processes, the involved many-body states can be pictured as a combination of a very large number of particle-hole excitations states (of the order of 106 components). Thus, a microscopic calculation of matrix elements, transition rates, and cross sections is neither feasible nor insightful. By assuming that the matrix elements of the underlying Hamiltonian are Gaussian distributed, the Random Matrix Theory (RMT) provides a very powerful statistical description of complex many-body systems based on very little system specific information, namely, their fundamental symmetries (originally translated into orthogonal, unitary, and symplectic universal symmetry classes), mean level spacing, and number of open channels to set the energy scales. The Bohigas-Gianonni-Schmit conjecture [2] that relates the spectral properties of quantum systems with a classical chaotic dynamics of systems with few degrees of freedom
C. Lewenkopf
to those predicted by RMT expanded the applications of the theory to a large variety of problems in different fields [1]. For instance, RMT became very popular to describe the electronic transport properties in low-dimensional microand nanostructures in condensed matter [3, 4], particularly, exploring its connection to the stochastic scattering matrix [5–7]. Hussein gave significant contributions to the field, more specifically, in applications of RMT to nuclear physics, the problem of enhanced symmetry breaking processes in compound nuclear reactions, and the description of the crossover between symmetry classes using the maximum entropy approach and deformed Gaussian ensembles [8, 9]. In this paper, we review the works of Hussein on ballistic mesoscopic fluctuations and present recent developments fostered by his original ideas, as well as an outlook on possible future extensions. The presentation is organized as follows. In Section 2, we review the main elements of mesoscopic physics of ballistic systems. In Section 3, we present the density of fluctuations maxima, a statistical measure put forward by Hussein and collaborators for chaotic quantum dots. Section 4 summarizes some of the developments based on the latter related to condensed matter physics. Finally, we present our conclusions and outlook in Section 5
[email protected] 1
Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970, Jo˜ao Pessoa, PB, Brazil
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