  Systems of soft-core particles interacting via a two-scale potential are
studied. The potential is responsible for peaks in the structure factor of the
liquid state at two different but comparable length scales, and a similar
bimodal structure is evident in the dispersion relation. Dynamical density
functional theory in two dimensions is used to identify two novel states of
this system, the crystal-liquid state, in which the majority of the particles
are located on lattice sites but a minority remains free and so behaves like a
liquid, and a 12-fold quasicrystalline state. Both are present even for deeply
quenched liquids and are found in a regime in which the liquid is unstable with
respect to modulations on the smaller scale only. As a result the system
initially evolves towards a small scale crystal state; this state is not a
minimum of the free energy, however, and so the system subsequently attempts to
reorganize to generate the lower energy larger scale crystals. This dynamical
process generates a disordered state with quasicrystalline domains, and takes
place even when this large scale is linearly stable, i.e., it is a nonlinear
process. With controlled initial conditions a perfect quasicrystal can form.
The results are corroborated using Brownian dynamics simulations.
