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 unusual states of this system: a 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.
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