  In thermal equilibrium the dynamics of phase transitions is largely
controlled by fluctuation-dissipation relations: On the one hand, friction
suppresses fluctuations, while on the other hand the thermal noise is
proportional to friction constants. Out of equilibrium, this balance dissolves
and one can have situations where friction vanishes due to antidamping in the
presence of a finite noise level. We study a wide class of $O(N)$ field
theories where this situation is realized at a phase transition, which we
identify as a critical exceptional point. In the ordered phase, antidamping
induces a continuous limit cycle rotation of the order parameter with an
enhanced number of $2N-3$ Goldstone modes. Close to the critical exceptional
point, however, fluctuations diverge so strongly due to the suppression of
friction that in dimensions $d<4$ they universally either destroy a preexisting
static order, or give rise to a fluctuation-induced first order transition.
This is demonstrated within a non-perturbative approach based on
Dyson-Schwinger equations for $N=2$, and a generalization for arbitrary $N$,
which can be solved exactly in the long wavelength limit. We show that in order
to realize this physics it is not necessary to drive a system far out of
equilibrium: Using the peculiar protection of Goldstone modes, the transition
from an $xy$ magnet to a ferrimagnet is governed by an exceptional critical
point once weakly perturbed away from thermal equilibrium.
