  Ferrofluids show unusual hydrodynamic effects due to the magnetic nature of
their constituents. For increasing magnetization a classical ferrofluid
undergoes a Rosensweig instability and creates self-organized ordered surface
structures or droplet crystals. A Bose-Einstein condensate with strong dipolar
interactions is a quantum ferrofluid that also shows superfluidity. The field
of dipolar quantum gases is motivated by the search for new phases that break
continuous symmetries. The simultaneous breaking of continuous symmetries like
the phase invariance for the superfluid state and the translational symmetry
for a crystal provides the basis of novel states of matter. However,
interaction-induced crystallization in a superfluid has not been observed. Here
we use in situ imaging to directly observe the spontaneous transition from an
unstructured superfluid to an ordered arrangement of droplets in an atomic
dysprosium Bose-Einstein condensate. By utilizing a Feshbach resonance to
control the interparticle interactions, we induce a finite-wavelength
instability and observe discrete droplets in a triangular structure, growing
with increasing atom number. We find that these states are surprisingly
long-lived and measure a hysteretic behaviour, which is typical for a
crystallization process and in close analogy to the Rosensweig instability. Our
system can show both superfluidity and, as shown here, spontaneous
translational symmetry breaking. The presented observations do not probe
superfluidity in the structured states, but if the droplets establish a common
phase via weak links, this system is a very good candidate for a supersolid
ground state.
