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A missing atom or a disordered bond can trap an electronic state, connect conducting paths, or change how a crystal responds to an electric field. We study which of these effects are enforced by topology, and how local wavefunctions reveal them when translation symmetry is lost.

Rotating one atomic layer relative to another creates a long-period pattern that can slow electrons until their interactions dominate. We study how the resulting wavefunctions select fractional phases, magnetism, and collective order. Local-orbital models and continuum theory connect that microscopic structure to optical and transport measurements.

An electron in a crystal spreads over atoms and bonds, and its wavefunction changes as it moves through the band. We connect this spatial structure to band topology and quantum geometry, then identify its signatures in tunnelling images, optical spectra, and electrical response. Our material models keep the orbital structure that makes those signatures possible.

An electric field moves charge between atoms, across a sample, and through collective modes. We connect these motions to the geometry of electronic wavefunctions and identify measurements that distinguish them. Optical spectral weight, capacitance, kinetic inductance, and local Hall fields let us test microscopic theory in working devices.