Hall Effect in Floquet Systems: Generic Superweak Chaos, Quantum and Topological Properties

Basic Hamiltonians of condensed-matter physics are, in general, classically nonintegrable and exhibit chaos. However, chaos is usually ignored, due to several reasons, e.g., when the chaos is weak. For example, Bloch electrons in magnetic fields are described by integrable effective Hamiltonians derived by Peierls substitution, e.g., the Harper Hamiltonian. The latter was used by Thouless et al. (TKNN) to show that, within linear-response theory, a magnetic band carries an integer Hall conductance satisfying a Diophantine equation (DE). Later, it was shown that the DE is a general result of magnetic translational invariance of the exact, nonintegrable Hamiltonian. Weak/strong chaos was studied to a much larger extent for Floquet (time-periodic) 1D Hamiltonians, which were shown to have a topologically nontrivial quasienergy (QE) spectrum, whose Chern numbers satisfy the DE.

To exhibit the delicate and nontrivial nature of weak chaos, we introduce a novel Floquet system, the "kicked Hall system", and study its classical, quantum, and topological properties. This system are charged particles periodically kicked by a 1D periodic potential V(x) of strength K in the presence of perpendicular magnetic (B) and electric (E) fields.

For E=0, this is a generalized version of the Zaslavsky system, exhibiting weak chaos and slow global diffusion on "stochastic webs" for arbitrarily small K << 1. We show that under generic conditions the Hall effect from B and E transforms the weak chaos into"superweak chaos (SWC), i.e., the system behaves as if the effective kick strength for K << 1 were K^2 rather than K. The global diffusion of SWC is then much slower than the already slow one in the E=0 case, where SWC is a rare phenomenon. Thus, the Hall effect relatively stabilizes the system for K << 1. Quantally, the SWC is shown to be a (quasi)classical fingerprint of "quantum antiresonance" (QAR), with a QE spectrum consisting of just one flat (infinitely degenerate) band. Thus, for nonzero E, QAR occurs much more generically than for E=0. We show that under quantum-resonance conditions the QE spectrum is topologically nontrivial, satisfying the DE.