  The recent experimental advancement to realise ultracold gases scattering off
an eight-fold optical potential [Phys. Rev. Lett. 122, 110404 (2019)] heralds
the beginning of a new technique to study the properties of quasicrystalline
structures. Quasicrystals possess long-range order but are not periodic, and
are still little studied in comparison to their periodic counterparts. Here, we
consider an ultracold bosonic gas in an eight-fold symmetric lattice and assume
a toy model where the atoms occupy the ground states of the local minima of the
potential. The ground state phases of the system are studied, with particular
interest in the local nature of the phases. The usual Mott-insulator, density
wave, and supersolid phases of the standard and extended Bose-Hubbard model are
observed. For non-zero long-range interactions, we find that density wave
states can spontaneously break the eight-fold symmetry, and may even possess no
rotational symmetry. We find the local variation in the number of nearest
neighbours to play a vital role in the phase transitions, local structure, and
global symmetries of the ground states. This variation in the number of nearest
neighbours is not a unique property of the considered eight-fold lattice, and
we expect our results to be generalisable to any quasicrystalline potential
where there are only small position dependent variations in the site energy,
tunnelling and interactions.
