  Artificial quasicrystals are nowadays routinely manufactured, yet only two
naturally occurring examples are known. We present a class of systems with the
potential to be realized both artificially and in nature, in which the lowest
energy state is a one-dimensional quasicrystal. These systems are based on
incommensurately charge-ordered materials, in which the quasicrystalline phase
competes with the formation of a regular array of discommensurations as a way
of interpolating between incommensurate charge order at high temperatures and
commensurate order at low temperatures. The nonlocal correlations
characteristic of the quasicrystalline state emerge from a free-energy
contribution localized in reciprocal space. We present a theoretical phase
diagram showing that the required material properties for the appearance of
such a ground state allow for one-dimensional quasicrystals to form in real
materials. The result is a potentially wide class of one-dimensional
quasicrystals.
