  The emerging possibilities to steer and control electronic motion on subcycle
time scales with strong electric fields enable studying the nonperturbative
optical response and Bloch bands' topological properties, originated from
Berry's trilogy: connection, curvature, and phase. This letter introduces a
theoretical framework for the nonperturbative electron dynamics in
two-dimensional (2D) crystalline solids induced by the few-cycle and
strong-field optical lasers. In the presented model, the expression associated
with the Bloch band topology and broken crystal symmetry merges
self-consistently in the system observables such as High Harmonic Generation
(HHG). This singles out our work from recent HHG calculations from the
strongly-driven systems. Concisely, in our theoretical experiment on 2D
materials in the strong-field optical regime, we show that Bloch band topology
and broken symmetry manifest themselves in several ways: the momentum-resolved
attosecond interferometry of electron wave packets, anomalous and chiral
velocity in both intraband and interband dynamics, anomalous Hall current and
respective HHG highly sensitive to the laser waveform, multiple plateau-cutoff
structures in both longitudinal and transverse HHG, the formation of even
harmonics in the perpendicular polarization with respect to the driving laser,
singular jumps across the phase diagram of the HHG, attosecond chirp, and
ultrafast valley polarization induced by the chiral gauge field that is robust
to lattice imperfections and scattering. The link between HHG and solid-state
band geometry offers an all-optical reconstruction of electron band structure
by optical means, and accelerates studies on the non-equilibrium Floquet
engineering, topologically-protected nonlinear spin and edge currents,
valleytronics, quantum computing and high-temperature superconductivity on
sub-femtosecond time scales.
