Relativistic Quantum Chaos

Quantum chaos is generally referred to as the study of quantum manifestations or fingerprints of nonlinear dynamical and chaotic behaviors in the corresponding classical system, an interdisciplinary field that has been active for about four decades. In closed chaotic Hamiltonian systems, for example, the basic phenomena studied include energy level-spacing statistics and quantum scarring. In open Hamiltonian systems, quantum chaotic scattering has been investigated extensively. Previous works were almost exclusively for nonrelativistic quantum systems described by the Schrodinger equation. Recent years have witnessed a rapid growth of interest in Dirac materials such as graphene, topological insulators, molybdenum disulfide and topological Dirac semimetals. A common feature of these materials is that their physics is described by the Dirac equation in relativistic quantum mechanics, generating phenomena that do not usually emerge in conventional semiconductor materials. This has important consequences. In particular, at the level of basic science, a new field has emerged: Relativistic Quantum Chaos (RQC), which aims to uncover, understand, and exploit relativistic quantum manifestations of classical nonlinear dynamical behaviors including chaos. Practically, Dirac materials have the potential to revolutionize solid-state electronic and spintronic devices, and have led to novel device concepts such as valleytronics. Exploiting manifestations of nonlinear dynamics and chaos in the relativistic quantum regime can have significant applications.

The aim of this article is to give a comprehensive review of the basic results obtained so far in the emergent field of RQC. Phenomena to be discussed in depth include energy level-spacing statistics in graphene or Dirac fermion systems that exhibit various nonlinear dynamical behaviors in the classical limit, relativistic quantum scars (unusually high concentrations of relativistic quantum spinors about classical periodic orbits), peculiar features of relativistic quantum chaotic scattering and quantum transport, manifestations of the Klein paradox and its effects on graphene or 2D Dirac material based devices, chaos based modulation of conductance fluctuations in relativistic quantum dots, regularization of relativistic quantum tunneling by chaos, superpersistent currents in chaotic Dirac rings subject to a magnetic flux, and exploitation of relativistic quantum whispering gallery modes for applications in quantum information science. Computational methods for solving the Dirac equation in various situations will be introduced and physical theories developed so far in RQC will be described. Potential device applications will be discussed.