  We present an analytic microscopic theory showing that in a large class of
spin-$\frac{1}{2}$ quasiperiodic quantum kicked rotors, a dynamical analog of
the integer quantum Hall effect (IQHE) emerges from an intrinsic chaotic
structure. Specifically, the inverse of the Planck's quantum ($h_e$) and the
rotor's energy growth rate mimic the `filling fraction' and the `longitudinal
conductivity' in conventional IQHE, respectively, and a hidden quantum number
is found to mimic the `quantized Hall conductivity'. We show that for an
infinite discrete set of critical values of $h_e$, the long-time energy growth
rate is universal and of order of unity (`metallic' phase), but otherwise
vanishes (`insulating' phase). Moreover, the rotor insulating phases are
topological, each of which is characterized by a hidden quantum number. This
number exhibits universal behavior for small $h_e$, i.e., it jumps by unity
whenever $h_e$ decreases, passing through each critical value. This intriguing
phenomenon is not triggered by the like of Landau band filling, well-known to
be the mechanism for conventional IQHE, and far beyond the canonical
Thouless-Kohmoto-Nightingale-Nijs paradigm for quantum Hall transitions.
Instead, this dynamical phenomenon is of strong chaos origin; it does not occur
when the dynamics is (partially) regular. More precisely, we find that, for the
first time, a topological object, similar to the topological theta angle in
quantum chromodynamics, emerges from strongly chaotic motion at microscopic
scales, and its renormalization gives the hidden quantum number.Our analytic
results are confirmed by numerical simulations.Our findings indicate that rich
topological quantum phenomena can emerge from chaos and might point to a new
direction of study in the interdisciplinary area straddling chaotic dynamics
and condensed matter physics.
