  The dominance of interactions over kinetic energy lies at the heart of
strongly correlated quantum matter, from fractional quantum Hall liquids, to
atoms in optical lattices and twisted bilayer graphene. Crystalline phases
often compete with correlated quantum liquids, and transitions between them
occur when the energy cost of forming a density wave approaches zero. A prime
example occurs for electrons in high magnetic fields, where the instability of
quantum Hall liquids towards a Wigner crystal is heralded by a roton-like
softening of density modulations at the magnetic length. Remarkably,
interacting bosons in a gauge field are also expected to form analogous liquid
and crystalline states. However, combining interactions with strong synthetic
magnetic fields has been a challenge for experiments on bosonic quantum gases.
Here, we study the purely interaction-driven dynamics of a Landau gauge
Bose-Einstein condensate in and near the lowest Landau level (LLL). We observe
a spontaneous crystallization driven by condensation of magneto-rotons,
excitations visible as density modulations at the magnetic length. Increasing
the cloud density smoothly connects this behaviour to a quantum version of the
Kelvin-Helmholtz hydrodynamic instability, driven by the sheared internal flow
profile of the rapidly rotating condensate. At long times the condensate
self-organizes into a persistent array of droplets, separated by vortex
streets, which are stabilized by a balance of interactions and effective
magnetic forces.
