Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.

We uncover an alternative mechanism for anomalous Hall effect. In particular, we investigate the magnetisation dynamics of an insulating ferromagnet (FM) deposited on the surface of a three-dimensional topological insulator (TI), subject to an external voltage. The spin-polarised current on the TI surface induces a spin-transfer torque on the magnetisation of the top FM while its dynamics can change the transmission probability of the surface electrons through the exchange coupling and hence the current. We find a host of nonlinear dynamical behaviors including multistability, chaos, and phase synchronisation. Strikingly, a dynamics mediated Hall-like current can arise, which exhibits a nontrivial dependence on the channel conductance. We develop a physical understanding of the mechanism that leads to the anomalous Hall effect. The nonlinear dynamical origin of the effect stipulates that a rich variety of final states exist, implying that the associated Hall current can be controlled to yield desirable behaviors. The phenomenon can find applications in Dirac-material based spintronics.

An outstanding and fundamental problem in contemporary physics is to include and probe the many-body effect in the study of relativistic quantum manifestations of classical chaos. We address this problem using graphene systems described by the Hubbard Hamiltonian in the setting of resonant tunneling. Such a system consists of two symmetric potential wells separated by a potential barrier, and the geometric shape of the whole domain can be chosen to generate integrable or chaotic dynamics in the classical limit. Employing a standard mean-field approach to calculating a large number of eigenenergies and eigenstates, we uncover a class of localized states with near-zero tunneling in the integrable systems. These states are not the edge states typically seen in graphene systems, and as such they are the consequence of many-body interactions. The physical origin of the non-edge-state type of localized states can be understood by the one-dimensional relativistic quantum tunneling dynamics through the solutions of the Dirac equation with appropriate boundary conditions. We demonstrate that, when the geometry of the system is modified to one with chaos, the localized states are effectively removed, implying that in realistic situations where many-body interactions are present, classical chaos is capable of facilitating greatly quantum tunneling. This result, besides its fundamental importance, can be useful for the development of nanoscale devices such as graphene-based resonant-tunneling diodes.