When an intense laser is emitted on a solid, the material can emit light at many multiples of the laser’s color, so‑called high harmonics. These harmonics act like fingerprints of how electrons move and interact on an extremely short timescale. We review recent theoretical advances for describing the all-optical spectroscopic technique of high-order harmonic spectroscopy in condensed matter systems. Theoretical methods for characterizing the underlying ultrafast electron processes, their interactions beyond the electric dipole, and the presence of excitons will be discussed. With it, predictions are made concerning the spectroscopic signatures of beyond-electric-dipole interactions, exciton processes, phase transitions, or other aspects of the ultrafast generation process.
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