Unveiling the universal class categorization of Altland-Zirnbauer in mesoscopic systems: probing entanglement and conductance fluctuations

Abstract

In this thesis, we present the ten classes of Altland-Zirnbauer symmetry and how they are used to describe universal quantum transport in mesoscopic systems, particularly chaotic quantum dots. We perform numerical simulations using Random Matrix Theory (RMT) to explore transport properties such as conductance and shot noise power, as well as entanglement quantifiers including concurrence, entanglement formation, and Bell inequality. We begin by introducing the description of our system, the chaotic quantum billiard. Then, we derive the MahauxWeidenmüller formula for the scattering matrix, and consequently, the conductance and shot noise power. We then present the possible fundamental symmetries present in mesoscopic systems, such as time-reversal symmetry, spin rotation, chirality, and particle-hole symmetry. In this way, we explore how these symmetries impose constraints on the Hamiltonian and, consequently, the scattering matrix. Finally, we present numerical results showing the influence of these fundamental symmetries on transport and entanglement properties. We provide a statistical analysis of the conductance and shot noise power for Schrödinger, Dirac, and Andreev billiards. Specifically, we observe that chiral and particle-hole symmetries play a role in the universal conductance fluctuations. In addition to transport properties, we perform numerical simulations of quantum entanglement in chaotic quantum billiards. We observe violations of the Bell inequality for all billiards and analyze their dependencies on the number of resonances and barrier transparency. We demonstrate that it is easier to violate the Bell inequality in chiral/sublattice and particle-hole degrees of freedom than in orbital degrees of freedom. Moreover, our results exhibit a unique and peculiar behavior: the realization of entanglement in the Andreev billiard always results in the production of Bell pairs in the single-channel limit, regardless of the presence of time-reversal symmetry and tunneling rates. The entangled pairs are formed by Majorana modes in an interface of different topological phases and may serve as a useful tool for entanglement generation and control, as well as quantum computing.