  We unveil a novel mechanism for quasicondensation of hard-core bosons in the
presence of quasiperiodicity-induced multifractal single-particle states. The
new critical state, here dubbed fractal quasicondensate, is characterized by
natural orbitals with multifractal properties and by an occupancy of the lowest
natural orbital, {\lambda}0 ~ L{\gamma}, which grows with system size but with
a nonuniversal scaling exponent, {\gamma} < 1/2. In contrast to fractal
quasicondensates obtained when the chemical potential lies in a region of
multifractal single-particle states, placing the chemical potential in regions
of localized or delocalized states yields, respectively, no condensation or the
usual 1D quasicondensation with {\gamma} = 1/2. Our findings are established by
studying one-dimensional hardcore bosons subjected to various quasiperiodic
potentials, including the well-known Aubry-Andre model, employing a mapping to
non-interacting fermionics that allows for numerically exact results. We
discuss how to test our findings in state-of-the-art ultracold atom
experiments.
