  Most Heisenberg-like spin chains flow to a universal free-fermion fixed point
near the magnetic-field induced saturation point. Here we show that an exotic
fixed point, characterized by two species of low-energy excitations with mutual
anyonic statistics, may also emerge in such spin chains if the dispersion
relation has two minima. By using bosonization, two-magnon exact calculations,
and numerical density-matrix-renormalization-group, we demonstrate the
existence of this anyonic-liquid fixed point in an XXZ spin chain with up to
second neighbor interactions. We also identify a range of microscopic
parameters, which support this phase.
