  Magnetic field emerging from the geometric curvature of quantum structure of
electrons in a crystal bends electronic trajectory causing anomalous linear and
nonlinear electrical Hall effects that have been observed in low symmetry
crystals with narrow electronic band gap. We present first-principles
theoretical analysis to show that dynamical lowering of crystal symmetry by
lattice vibrations results in oscillations in the quantum geometry of electrons
which have observable nonlinear Hall signatures. Using these, we introduce a
vibrational spectroscopy based on Geometry of Quantum Electronic Structure
(GQuES) making specific predictions for the transport and radiative GQuES
spectra of 2D materials. As the crystal symmetry is commonly lowered by
dynamical fields, GQuES is applicable to a wide range of materials and
excitations spanning sub-GHz, THz and infrared frequencies.
