On the Theory of Ergodicity, Localization, and Memory in Quantum Chaotic Systems

Abstract

This dissertation discusses a range of phenomena in which quantum mechanics reshapes the nature of chaos. In contrast to classical dynamics, where chaos is associated with exponential sensitivity and rapid loss of memory, quantum mechanics can suppress these tendencies through interference, uncertainty, spectral discreteness, and wave effects leading to novel forms of ergodicity, localization, transport, and memory. The systems studied here span from quantum-chaotic geometries to condensed-matter systems with dynamical disorder. The dissertation begins by examining the tension between ergodicity and localization in single-particle chaotic systems considering the phenomenon of quantum scarring, where probability density is enhanced near unstable classical periodic orbits. Building on this setting, it establishes a fresh ergodicity theorem for stacks of neighboring eigenstates and introduces the complementary notion to the quantum scar, namely antiscarring, in which probability density is reduced along the scar-generating periodic orbit. From there, the focus shifts from stationary eigenstates to generic nonstationary quantum states. A central finding is that quantum dynamics need not forget their beginnings, even at arbitrarily long times. This idea is formalized through the concept of the quantum birthmark: a persistent imprint of the initial state and its evolutes that survives in the infinite-time limit and produces a fundamental form of nonergodic behavior. The birthmark framework identifies both a universal contribution, rooted in global symmetries, and a revival-enhanced contribution, arising from short-time recurrences before the Heisenberg time. In this way, quantum birthmarks provide a unified framework for quantum memory as well as for various localization effects including scarring. We bring these ideas into direct contact with experiments through electrostatically confined graphene quantum dots, where relativistic wave dynamics, openness, and Klein tunneling create a rich setting for quantum-chaotic behavior in real materials. In close collaboration with experimentalists, the dissertation develops theoretical descriptions of scanning tunneling microscopy signatures in confined graphene structures, including the first visualization and interpretation of a genuine Heller-type quantum scar. The dissertation further studies elliptical graphene dots to reveal anisotropic Klein tunneling of quasi-resonant modes and introduces time-domain methods for simulating local density of states and mode filtering in open Dirac systems, both massless and massive. A different aspect of quantum chaos appears in periodic systems with weak superlattice modulation, where transport can become organized into extended channels termed superwires, in addition to branched flow dynamics and diffusion. Rather than arising from static confinement, these channels are generated dynamically and are linked by dynamical tunneling between disconnected regions of classical phase space. The dissertation develops the theoretical picture underlying these structures, showing how chaos-assisted tunneling, flat-band physics, and the paraxial approximation together explain the formation of multiple parallel transport pathways and their controllable suppression. Finally, the dissertation turns to condensed-matter systems in which the source of chaos is dynamical disorder arising from the lattice itself. This motivates the development of Quantum Acoustics, a nonperturbative time-domain framework for electron-lattice dynamics that enables the examination of disorder, transport, and backaction directly in coordinate space. Within this framework, static disorder gives rise to Anderson localization, while dynamical disorder instead results in transient localization, Planckian diffusion, and anomalous optical response. In regimes of strong mutual electron-lattice interaction, the same formalism further suggests routes toward transient polaron formation, charge ordering, and pseudogap-like phenomena.