  We use scale invariant scattering theory to exactly determine the lines of
renormalization group fixed points for $O(N)$-symmetric models with quenched
disorder in two dimensions. Random fixed points are characterized by two
disorder parameters: a modulus that vanishes when approaching the pure case,
and a phase angle. The critical lines fall into three classes depending on the
values of the disorder modulus. Besides the class corresponding to the pure
case, a second class has maximal value of the disorder modulus and includes
Nishimori-like multicritical points as well as zero temperature fixed points.
The third class contains critical lines that interpolate, as $N$ varies,
between the first two classes. For positive $N$, it contains a single line of
infrared fixed points spanning the values of $N$ from $\sqrt{2}-1$ to $1$. The
symmetry sector of the energy density operator is superuniversal (i.e.
$N$-independent) along this line. For $N=2$ a line of fixed points exists only
in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless
phase observed in presence of disorder.

Statistical Mechanics;High Energy Physics - Theory