Over the past decades, high harmonic spectroscopy (HHS) has emerged as a powerful tool for all-optical probing of topological properties of solids. There are outstanding questions regarding the universal nature of the spectral features of harmonics and their relationship to the nontrivial topological properties. Here, we present a systematic theoretical study of HHS in topological materials, including lower-order and higher-order topological insulators (LOTIs and HOTIs), focusing on observables such as helicity, circular dichroism, ellipticity dependence, and channel-resolved intensity yields. Using the Haldane, Kane-Mele, and breathing kagome lattice models, we theoretically extend all-optical approaches from the LOTI to the HOTI regime by explicitly incorporating contributions from bulk, edge, and corner states. Depending on the crystalline system, our calculations suggest that these observables can encode topological information through distinct modifications of the HHG spectra in topological phases. In particular, we identify significant enhancements of the harmonic intensity yields, reaching up to two orders of magnitude relative to trivial phases, together with distinct spectral signatures associated with edge and corner contributions revealed through channel-resolved intensity yields. These results show that channel-resolved HHS provides a promising route for probing topological states in both LOTIs and HOTIs.

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• Charge dynamics
• Edge states
• Electronic structure
• Flat bands
• Phase transitions
• Quantum phase transitions
• Spin-orbit coupling
• Symmetry protected topological states
• Topological insulators
• Topological phase transition
• Topological phases of matter
• Ultrafast magnetic effects