  Bloch's theorem was a major milestone that established the principle of
bandgaps in crystals. Although it was once believed that bandgaps could form
only under conditions of periodicity and long-range correlations for Bloch's
theorem, this restriction was disproven by the discoveries of amorphous media
and quasicrystals. While network and liquid models have been suggested for the
interpretation of Bloch-like waves in disordered media, these approaches based
on searching for random networks with bandgaps have failed in the deterministic
creation of bandgaps. Here, we reveal a deterministic pathway to bandgaps in
random-walk potentials by applying the notion of supersymmetry to the wave
equation. Inspired by isospectrality, we follow a methodology in contrast to
previous methods: we transform order into disorder while preserving bandgaps.
Our approach enables the formation of bandgaps in extremely-disordered
potentials analogous to Brownian motion, and also allows the tuning of
correlations while maintaining identical bandgaps, thereby creating a family of
potentials with 'Bloch-like eigenstates'.
