  We study O(n)-symmetric two-dimensional conformal field theories (CFTs) for a
continuous range of n below two. These CFTs describe the fixed point behavior
of self-avoiding loops. There is a pair of known fixed points connected by an
RG flow. When n is equal to two, which corresponds to the Kosterlitz-Thouless
critical theory, the fixed points collide. We find that for n generic these
CFTs are logarithmic and contain negative norm states; in particular, the O(n)
currents belong to a staggered logarithmic multiplet. Using a conformal
bootstrap approach we trace how the negative norm states decouple at n = 2,
restoring unitarity. The IR fixed point possesses a local relevant operator,
singlet under all known global symmetries of the CFT, and, nevertheless, it can
be reached by an RG flow without tuning. Besides, we observe logarithmic
correlators in the closely related Potts model.
