Exact results for the O(N) model with quenched disorder

We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for 0(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 root 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.